Abstract

We present a theoretical study of vortical plasmonic lattice solitons, which form in two-dimensional arrays of metallic nanowires embedded into nonlinear media with both focusing and defocusing Kerr nonlinearities. Their existence, stability, and subwavelength spatial confinement are investigated in detail.

© 2011 Optical Society of America

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  1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
    [CrossRef] [PubMed]
  2. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradulin, Phys. Rep. 408, 131 (2005).
    [CrossRef]
  3. E. Feigenbaum and M. Orenstein, Opt. Lett. 32, 674 (2007).
    [CrossRef] [PubMed]
  4. Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, Phys. Rev. Lett. 99, 153901 (2007).
    [CrossRef] [PubMed]
  5. A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, Opt. Express 16, 21209 (2008).
    [CrossRef] [PubMed]
  6. F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, Phys. Rev. Lett. 104, 106802 (2010).
    [CrossRef] [PubMed]
  7. A. Marini, A. V. Gorbach, and D. V. Skryabin, Opt. Lett. 35, 3532 (2010).
    [CrossRef] [PubMed]
  8. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1(2008).
    [CrossRef]
  9. A. Szameit, T. Pertsch, S. Nolte, A. Tunnermann, and F. Lederer, Phys. Rev. A 77, 043804 (2008).
    [CrossRef]
  10. N. K. Efremidis and K. Hizanidis, Opt. Express 13, 10571 (2005).
    [CrossRef] [PubMed]
  11. N. K. Efremidis, P. Zhang, Z. Chen, D. N. Christodoulides, C. E. Ruter, and D. Kip, Phys. Rev. A 81, 053817 (2010).
    [CrossRef]
  12. R. Iliew, C. Etrich, M. Augustin, E. B. Kley, S. Nolte, A. Tunnermann, and F. Lederer, Phys. Status Solidi A 204, 3689 (2007).
    [CrossRef]
  13. X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, Phys. Rev. Lett. 97, 073901 (2006).
    [CrossRef] [PubMed]
  14. S. M. Xiao, V. P. Drachev, A. V. Kildishev, X. J. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, Nature 466, 735 (2010).
    [CrossRef] [PubMed]
  15. B. A. Malomed and P. G. Kevrekidis, Phys. Rev. E 64, 026601 (2001).
    [CrossRef]
  16. E. A. Ostrovskaya and Y. S. Kivshar, Phys. Rev. Lett. 93, 160405 (2004).
    [CrossRef] [PubMed]
  17. L. Brzozowski, E. H. Sargent, A. S. Thorpe, and M. Extavour, Appl. Phys. Lett. 82, 4429 (2003).
    [CrossRef]

2010

F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, Phys. Rev. Lett. 104, 106802 (2010).
[CrossRef] [PubMed]

A. Marini, A. V. Gorbach, and D. V. Skryabin, Opt. Lett. 35, 3532 (2010).
[CrossRef] [PubMed]

N. K. Efremidis, P. Zhang, Z. Chen, D. N. Christodoulides, C. E. Ruter, and D. Kip, Phys. Rev. A 81, 053817 (2010).
[CrossRef]

S. M. Xiao, V. P. Drachev, A. V. Kildishev, X. J. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, Nature 466, 735 (2010).
[CrossRef] [PubMed]

2008

A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, Opt. Express 16, 21209 (2008).
[CrossRef] [PubMed]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1(2008).
[CrossRef]

A. Szameit, T. Pertsch, S. Nolte, A. Tunnermann, and F. Lederer, Phys. Rev. A 77, 043804 (2008).
[CrossRef]

2007

E. Feigenbaum and M. Orenstein, Opt. Lett. 32, 674 (2007).
[CrossRef] [PubMed]

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

R. Iliew, C. Etrich, M. Augustin, E. B. Kley, S. Nolte, A. Tunnermann, and F. Lederer, Phys. Status Solidi A 204, 3689 (2007).
[CrossRef]

2006

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, Phys. Rev. Lett. 97, 073901 (2006).
[CrossRef] [PubMed]

2005

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradulin, Phys. Rep. 408, 131 (2005).
[CrossRef]

N. K. Efremidis and K. Hizanidis, Opt. Express 13, 10571 (2005).
[CrossRef] [PubMed]

2004

E. A. Ostrovskaya and Y. S. Kivshar, Phys. Rev. Lett. 93, 160405 (2004).
[CrossRef] [PubMed]

2003

L. Brzozowski, E. H. Sargent, A. S. Thorpe, and M. Extavour, Appl. Phys. Lett. 82, 4429 (2003).
[CrossRef]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
[CrossRef] [PubMed]

2001

B. A. Malomed and P. G. Kevrekidis, Phys. Rev. E 64, 026601 (2001).
[CrossRef]

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1(2008).
[CrossRef]

Augustin, M.

R. Iliew, C. Etrich, M. Augustin, E. B. Kley, S. Nolte, A. Tunnermann, and F. Lederer, Phys. Status Solidi A 204, 3689 (2007).
[CrossRef]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
[CrossRef] [PubMed]

Bartal, G.

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

Brzozowski, L.

L. Brzozowski, E. H. Sargent, A. S. Thorpe, and M. Extavour, Appl. Phys. Lett. 82, 4429 (2003).
[CrossRef]

Chan, C. T.

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, Phys. Rev. Lett. 97, 073901 (2006).
[CrossRef] [PubMed]

Chen, Z.

N. K. Efremidis, P. Zhang, Z. Chen, D. N. Christodoulides, C. E. Ruter, and D. Kip, Phys. Rev. A 81, 053817 (2010).
[CrossRef]

Chettiar, U. K.

S. M. Xiao, V. P. Drachev, A. V. Kildishev, X. J. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, Nature 466, 735 (2010).
[CrossRef] [PubMed]

Christodoulides, D. N.

N. K. Efremidis, P. Zhang, Z. Chen, D. N. Christodoulides, C. E. Ruter, and D. Kip, Phys. Rev. A 81, 053817 (2010).
[CrossRef]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1(2008).
[CrossRef]

Davoyan, A. R.

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
[CrossRef] [PubMed]

Drachev, V. P.

S. M. Xiao, V. P. Drachev, A. V. Kildishev, X. J. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, Nature 466, 735 (2010).
[CrossRef] [PubMed]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
[CrossRef] [PubMed]

Efremidis, N. K.

N. K. Efremidis, P. Zhang, Z. Chen, D. N. Christodoulides, C. E. Ruter, and D. Kip, Phys. Rev. A 81, 053817 (2010).
[CrossRef]

N. K. Efremidis and K. Hizanidis, Opt. Express 13, 10571 (2005).
[CrossRef] [PubMed]

Etrich, C.

R. Iliew, C. Etrich, M. Augustin, E. B. Kley, S. Nolte, A. Tunnermann, and F. Lederer, Phys. Status Solidi A 204, 3689 (2007).
[CrossRef]

Extavour, M.

L. Brzozowski, E. H. Sargent, A. S. Thorpe, and M. Extavour, Appl. Phys. Lett. 82, 4429 (2003).
[CrossRef]

Fan, X.

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, Phys. Rev. Lett. 97, 073901 (2006).
[CrossRef] [PubMed]

Feigenbaum, E.

Genov, D. A.

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

Gorbach, A. V.

Hizanidis, K.

Hu, B.

F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, Phys. Rev. Lett. 104, 106802 (2010).
[CrossRef] [PubMed]

Iliew, R.

R. Iliew, C. Etrich, M. Augustin, E. B. Kley, S. Nolte, A. Tunnermann, and F. Lederer, Phys. Status Solidi A 204, 3689 (2007).
[CrossRef]

Kevrekidis, P. G.

B. A. Malomed and P. G. Kevrekidis, Phys. Rev. E 64, 026601 (2001).
[CrossRef]

Kildishev, A. V.

S. M. Xiao, V. P. Drachev, A. V. Kildishev, X. J. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, Nature 466, 735 (2010).
[CrossRef] [PubMed]

Kip, D.

N. K. Efremidis, P. Zhang, Z. Chen, D. N. Christodoulides, C. E. Ruter, and D. Kip, Phys. Rev. A 81, 053817 (2010).
[CrossRef]

Kivshar, Y. S.

Kley, E. B.

R. Iliew, C. Etrich, M. Augustin, E. B. Kley, S. Nolte, A. Tunnermann, and F. Lederer, Phys. Status Solidi A 204, 3689 (2007).
[CrossRef]

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1(2008).
[CrossRef]

A. Szameit, T. Pertsch, S. Nolte, A. Tunnermann, and F. Lederer, Phys. Rev. A 77, 043804 (2008).
[CrossRef]

R. Iliew, C. Etrich, M. Augustin, E. B. Kley, S. Nolte, A. Tunnermann, and F. Lederer, Phys. Status Solidi A 204, 3689 (2007).
[CrossRef]

Lee, J. C. W.

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, Phys. Rev. Lett. 97, 073901 (2006).
[CrossRef] [PubMed]

Liu, Y.

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

Malomed, B. A.

B. A. Malomed and P. G. Kevrekidis, Phys. Rev. E 64, 026601 (2001).
[CrossRef]

Maradulin, A. A.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradulin, Phys. Rep. 408, 131 (2005).
[CrossRef]

Marini, A.

Mihalache, D.

F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, Phys. Rev. Lett. 104, 106802 (2010).
[CrossRef] [PubMed]

Ni, X. J.

S. M. Xiao, V. P. Drachev, A. V. Kildishev, X. J. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, Nature 466, 735 (2010).
[CrossRef] [PubMed]

Nolte, S.

A. Szameit, T. Pertsch, S. Nolte, A. Tunnermann, and F. Lederer, Phys. Rev. A 77, 043804 (2008).
[CrossRef]

R. Iliew, C. Etrich, M. Augustin, E. B. Kley, S. Nolte, A. Tunnermann, and F. Lederer, Phys. Status Solidi A 204, 3689 (2007).
[CrossRef]

Orenstein, M.

Ostrovskaya, E. A.

E. A. Ostrovskaya and Y. S. Kivshar, Phys. Rev. Lett. 93, 160405 (2004).
[CrossRef] [PubMed]

Panoiu, N. C.

F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, Phys. Rev. Lett. 104, 106802 (2010).
[CrossRef] [PubMed]

Pertsch, T.

A. Szameit, T. Pertsch, S. Nolte, A. Tunnermann, and F. Lederer, Phys. Rev. A 77, 043804 (2008).
[CrossRef]

Ruter, C. E.

N. K. Efremidis, P. Zhang, Z. Chen, D. N. Christodoulides, C. E. Ruter, and D. Kip, Phys. Rev. A 81, 053817 (2010).
[CrossRef]

Sargent, E. H.

L. Brzozowski, E. H. Sargent, A. S. Thorpe, and M. Extavour, Appl. Phys. Lett. 82, 4429 (2003).
[CrossRef]

Segev, M.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1(2008).
[CrossRef]

Shadrivov, I. V.

Shalaev, V. M.

S. M. Xiao, V. P. Drachev, A. V. Kildishev, X. J. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, Nature 466, 735 (2010).
[CrossRef] [PubMed]

Silberberg, Y.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1(2008).
[CrossRef]

Skryabin, D. V.

Smolyaninov, I. I.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradulin, Phys. Rep. 408, 131 (2005).
[CrossRef]

Stegeman, G. I.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1(2008).
[CrossRef]

Szameit, A.

A. Szameit, T. Pertsch, S. Nolte, A. Tunnermann, and F. Lederer, Phys. Rev. A 77, 043804 (2008).
[CrossRef]

Thorpe, A. S.

L. Brzozowski, E. H. Sargent, A. S. Thorpe, and M. Extavour, Appl. Phys. Lett. 82, 4429 (2003).
[CrossRef]

Tunnermann, A.

A. Szameit, T. Pertsch, S. Nolte, A. Tunnermann, and F. Lederer, Phys. Rev. A 77, 043804 (2008).
[CrossRef]

R. Iliew, C. Etrich, M. Augustin, E. B. Kley, S. Nolte, A. Tunnermann, and F. Lederer, Phys. Status Solidi A 204, 3689 (2007).
[CrossRef]

Wang, G. P.

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, Phys. Rev. Lett. 97, 073901 (2006).
[CrossRef] [PubMed]

Xiao, S. M.

S. M. Xiao, V. P. Drachev, A. V. Kildishev, X. J. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, Nature 466, 735 (2010).
[CrossRef] [PubMed]

Ye, F.

F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, Phys. Rev. Lett. 104, 106802 (2010).
[CrossRef] [PubMed]

Yuan, H. K.

S. M. Xiao, V. P. Drachev, A. V. Kildishev, X. J. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, Nature 466, 735 (2010).
[CrossRef] [PubMed]

Zayats, A. V.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradulin, Phys. Rep. 408, 131 (2005).
[CrossRef]

Zhang, P.

N. K. Efremidis, P. Zhang, Z. Chen, D. N. Christodoulides, C. E. Ruter, and D. Kip, Phys. Rev. A 81, 053817 (2010).
[CrossRef]

Zhang, X.

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

Appl. Phys. Lett.

L. Brzozowski, E. H. Sargent, A. S. Thorpe, and M. Extavour, Appl. Phys. Lett. 82, 4429 (2003).
[CrossRef]

Nature

S. M. Xiao, V. P. Drachev, A. V. Kildishev, X. J. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, Nature 466, 735 (2010).
[CrossRef] [PubMed]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rep.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradulin, Phys. Rep. 408, 131 (2005).
[CrossRef]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1(2008).
[CrossRef]

Phys. Rev. A

A. Szameit, T. Pertsch, S. Nolte, A. Tunnermann, and F. Lederer, Phys. Rev. A 77, 043804 (2008).
[CrossRef]

N. K. Efremidis, P. Zhang, Z. Chen, D. N. Christodoulides, C. E. Ruter, and D. Kip, Phys. Rev. A 81, 053817 (2010).
[CrossRef]

Phys. Rev. E

B. A. Malomed and P. G. Kevrekidis, Phys. Rev. E 64, 026601 (2001).
[CrossRef]

Phys. Rev. Lett.

E. A. Ostrovskaya and Y. S. Kivshar, Phys. Rev. Lett. 93, 160405 (2004).
[CrossRef] [PubMed]

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, Phys. Rev. Lett. 97, 073901 (2006).
[CrossRef] [PubMed]

F. Ye, D. Mihalache, B. Hu, and N. C. Panoiu, Phys. Rev. Lett. 104, 106802 (2010).
[CrossRef] [PubMed]

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, Phys. Rev. Lett. 99, 153901 (2007).
[CrossRef] [PubMed]

Phys. Status Solidi A

R. Iliew, C. Etrich, M. Augustin, E. B. Kley, S. Nolte, A. Tunnermann, and F. Lederer, Phys. Status Solidi A 204, 3689 (2007).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Top panels, schematic of a 2D square array of nanowires and the band predicted by the coupled mode theory. Bottom panel, the normalized coupling constant versus separation distance, determined by using the coupled mode theory and exact numerical simulations. The radius of the nanowires is a = 40 nm .

Fig. 2
Fig. 2

The normalized electric field amplitude (a),(c) and phase (b),(d) of vortical PLSs in focusing (top panels) and defocusing (middle panels) nonlinear media. (a),(b)  β = 3.82 , δ n nl = 0.11 . (c),(d)  β = 5.2 , δ n nl = 0.18 . Radius R (e) and power P (f) versus β. The solid and dashed curves stand for d = 7 a and d = 8 a , respectively. In the inset, δ n nl versus β for d = 8 a , in focusing medium. In all plots, λ = 1550 nm and a = 40 nm .

Fig. 3
Fig. 3

Unstable propagation of vortical PLS at β = 3.82 (top panels), showing the intensity profile at z = 0 and at z = 90 μm . Evolution of maximum electric field amplitude with propagation distance for the unstable vortical PLS (bottom panel). In all plots, λ = 1550 nm and a = 40 nm .

Equations (1)

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i d ϕ m , n d z + κ ( ϕ m , n + 1 + ϕ m , n 1 + ϕ m + 1 , n + ϕ m 1 , n ) + μ ( ϕ m + 1 , n + 1 + ϕ m + 1 , n 1 + ϕ m 1 , n + 1 + ϕ m 1 , n 1 ) + γ | ϕ m , n | 2 ϕ m , n = 0 ,

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