Abstract

The nonlinear (NL) diffraction of wave packets in honeycomb lattices near Dirac points is studied. Strong nonlinearity can significantly deform the diffraction patterns from conical to triangular structure. This is described by a mean field discrete NL Dirac system and in the continuous limit by a higher-order NL Dirac system, which, in turn, is consistent with the trigonal warping of the dispersion relation. The anticontinuous limit is also examined and similar properties are obtained.

© 2011 Optical Society of America

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  1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005).
    [CrossRef] [PubMed]
  2. M. I. Molina and Y. S. Kivshar, Opt. Lett. 35, 2895 (2010).
    [CrossRef] [PubMed]
  3. O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
    [CrossRef] [PubMed]
  4. O. Bahat-Treidel, O. Peleg, and M. Segev, Opt. Lett. 33, 2251(2008).
    [CrossRef] [PubMed]
  5. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
    [CrossRef]
  6. M. Ablowitz, S. Nixon, and Y. Zhu, Phys. Rev. A 79, 053830(2009).
    [CrossRef]
  7. O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, Phys. Rev. Lett. 104, 063901(2010).
    [CrossRef] [PubMed]
  8. L. H. Haddad and L. D. Carr, Physica D 238, 1413 (2009).
    [CrossRef]
  9. O. Bahat-Treidel, O. Peleg, M. Segev, and H. Buljan, Phys. Rev. A 82, 013830 (2010).
    [CrossRef]
  10. P. G. Kevrekidis, B. A. Malomed, and Y. B. Gaididei, Phys. Rev. E 66, 016609 (2002).
    [CrossRef]
  11. C. R. Rosberg, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, and Y. S. Kivshar, Opt. Lett. 32, 397 (2007).
    [CrossRef] [PubMed]
  12. P. R. Wallace, Phys. Rev. 71, 622 (1947).
    [CrossRef]
  13. M. J. Ablowitz and Y. Zhu, Phys. Rev. A 82, 013840 (2010).
    [CrossRef]
  14. T. Ando, T. Nakanishi, and R. Saito, J. Phys. Soc. Jpn. 67, 2857.
    [CrossRef]

2010

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, Phys. Rev. Lett. 104, 063901(2010).
[CrossRef] [PubMed]

O. Bahat-Treidel, O. Peleg, M. Segev, and H. Buljan, Phys. Rev. A 82, 013830 (2010).
[CrossRef]

M. J. Ablowitz and Y. Zhu, Phys. Rev. A 82, 013840 (2010).
[CrossRef]

M. I. Molina and Y. S. Kivshar, Opt. Lett. 35, 2895 (2010).
[CrossRef] [PubMed]

2009

L. H. Haddad and L. D. Carr, Physica D 238, 1413 (2009).
[CrossRef]

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
[CrossRef]

M. Ablowitz, S. Nixon, and Y. Zhu, Phys. Rev. A 79, 053830(2009).
[CrossRef]

2008

2007

C. R. Rosberg, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, and Y. S. Kivshar, Opt. Lett. 32, 397 (2007).
[CrossRef] [PubMed]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef] [PubMed]

2005

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005).
[CrossRef] [PubMed]

2002

P. G. Kevrekidis, B. A. Malomed, and Y. B. Gaididei, Phys. Rev. E 66, 016609 (2002).
[CrossRef]

1947

P. R. Wallace, Phys. Rev. 71, 622 (1947).
[CrossRef]

Ablowitz, M.

M. Ablowitz, S. Nixon, and Y. Zhu, Phys. Rev. A 79, 053830(2009).
[CrossRef]

Ablowitz, M. J.

M. J. Ablowitz and Y. Zhu, Phys. Rev. A 82, 013840 (2010).
[CrossRef]

Ando, T.

T. Ando, T. Nakanishi, and R. Saito, J. Phys. Soc. Jpn. 67, 2857.
[CrossRef]

Bahat-Treidel, O.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, Phys. Rev. Lett. 104, 063901(2010).
[CrossRef] [PubMed]

O. Bahat-Treidel, O. Peleg, M. Segev, and H. Buljan, Phys. Rev. A 82, 013830 (2010).
[CrossRef]

O. Bahat-Treidel, O. Peleg, and M. Segev, Opt. Lett. 33, 2251(2008).
[CrossRef] [PubMed]

Bartal, G.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef] [PubMed]

Buljan, H.

O. Bahat-Treidel, O. Peleg, M. Segev, and H. Buljan, Phys. Rev. A 82, 013830 (2010).
[CrossRef]

Carr, L. D.

L. H. Haddad and L. D. Carr, Physica D 238, 1413 (2009).
[CrossRef]

Castro Neto, A. H.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
[CrossRef]

Christodoulides, D. N.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef] [PubMed]

Dubonos, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005).
[CrossRef] [PubMed]

Firsov, A. A.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005).
[CrossRef] [PubMed]

Freedman, B.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef] [PubMed]

Gaididei, Y. B.

P. G. Kevrekidis, B. A. Malomed, and Y. B. Gaididei, Phys. Rev. E 66, 016609 (2002).
[CrossRef]

Geim, A. K.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
[CrossRef]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005).
[CrossRef] [PubMed]

Grigorieva, I. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005).
[CrossRef] [PubMed]

Grobman, M.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, Phys. Rev. Lett. 104, 063901(2010).
[CrossRef] [PubMed]

Guinea, F.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
[CrossRef]

Haddad, L. H.

L. H. Haddad and L. D. Carr, Physica D 238, 1413 (2009).
[CrossRef]

Jiang, D.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005).
[CrossRef] [PubMed]

Katsnelson, M. I.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005).
[CrossRef] [PubMed]

Kevrekidis, P. G.

P. G. Kevrekidis, B. A. Malomed, and Y. B. Gaididei, Phys. Rev. E 66, 016609 (2002).
[CrossRef]

Kivshar, Y. S.

Krolikowski, W.

Malomed, B. A.

P. G. Kevrekidis, B. A. Malomed, and Y. B. Gaididei, Phys. Rev. E 66, 016609 (2002).
[CrossRef]

Manela, O.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef] [PubMed]

Molina, M. I.

Morozov, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005).
[CrossRef] [PubMed]

Nakanishi, T.

T. Ando, T. Nakanishi, and R. Saito, J. Phys. Soc. Jpn. 67, 2857.
[CrossRef]

Neshev, D. N.

Nixon, S.

M. Ablowitz, S. Nixon, and Y. Zhu, Phys. Rev. A 79, 053830(2009).
[CrossRef]

Novoselov, K. S.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
[CrossRef]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005).
[CrossRef] [PubMed]

Peleg, O.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, Phys. Rev. Lett. 104, 063901(2010).
[CrossRef] [PubMed]

O. Bahat-Treidel, O. Peleg, M. Segev, and H. Buljan, Phys. Rev. A 82, 013830 (2010).
[CrossRef]

O. Bahat-Treidel, O. Peleg, and M. Segev, Opt. Lett. 33, 2251(2008).
[CrossRef] [PubMed]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef] [PubMed]

Pereg-Barnea, T.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, Phys. Rev. Lett. 104, 063901(2010).
[CrossRef] [PubMed]

Peres, N. M. R.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
[CrossRef]

Rosberg, C. R.

Saito, R.

T. Ando, T. Nakanishi, and R. Saito, J. Phys. Soc. Jpn. 67, 2857.
[CrossRef]

Segev, M.

O. Bahat-Treidel, O. Peleg, M. Segev, and H. Buljan, Phys. Rev. A 82, 013830 (2010).
[CrossRef]

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, Phys. Rev. Lett. 104, 063901(2010).
[CrossRef] [PubMed]

O. Bahat-Treidel, O. Peleg, and M. Segev, Opt. Lett. 33, 2251(2008).
[CrossRef] [PubMed]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef] [PubMed]

Shapira, N.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, Phys. Rev. Lett. 104, 063901(2010).
[CrossRef] [PubMed]

Sukhorukov, A. A.

Wallace, P. R.

P. R. Wallace, Phys. Rev. 71, 622 (1947).
[CrossRef]

Zhu, Y.

M. J. Ablowitz and Y. Zhu, Phys. Rev. A 82, 013840 (2010).
[CrossRef]

M. Ablowitz, S. Nixon, and Y. Zhu, Phys. Rev. A 79, 053830(2009).
[CrossRef]

J. Phys. Soc. Jpn.

T. Ando, T. Nakanishi, and R. Saito, J. Phys. Soc. Jpn. 67, 2857.
[CrossRef]

Nature

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rev.

P. R. Wallace, Phys. Rev. 71, 622 (1947).
[CrossRef]

Phys. Rev. A

M. J. Ablowitz and Y. Zhu, Phys. Rev. A 82, 013840 (2010).
[CrossRef]

O. Bahat-Treidel, O. Peleg, M. Segev, and H. Buljan, Phys. Rev. A 82, 013830 (2010).
[CrossRef]

M. Ablowitz, S. Nixon, and Y. Zhu, Phys. Rev. A 79, 053830(2009).
[CrossRef]

Phys. Rev. E

P. G. Kevrekidis, B. A. Malomed, and Y. B. Gaididei, Phys. Rev. E 66, 016609 (2002).
[CrossRef]

Phys. Rev. Lett.

O. Bahat-Treidel, O. Peleg, M. Grobman, N. Shapira, M. Segev, and T. Pereg-Barnea, Phys. Rev. Lett. 104, 063901(2010).
[CrossRef] [PubMed]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
[CrossRef] [PubMed]

Physica D

L. H. Haddad and L. D. Carr, Physica D 238, 1413 (2009).
[CrossRef]

Rev. Mod. Phys.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

(a) Honeycomb lattice, (b) its Brillouin zone, (c) the lowest two dispersion surfaces near K, (d) dispersion structure in a larger neighborhood.

Fig. 2
Fig. 2

Evolution patterns of a broad envelope in the NLS Eq. (1) with NL coefficient σ equal to (a) 0, (b) 1, (c) 5, (d)  5 .

Fig. 3
Fig. 3

Evolution patterns of a broaden envelope in Eq. (2) with effective NL coefficient g σ τ equal to (a) 0, (b) 0.1, (c) 1, (d)  1 .

Fig. 4
Fig. 4

Evolution patterns (a) in the NLD equation ( g σ τ = 20 ) and (b) in the HONLD Eq. (4) when ν = 0.2 , σ ˜ = 0 . Contours of (c) the dispersion relation ω + ( q ) and (d) the magnitude of its gradient | ω + | .

Fig. 5
Fig. 5

Evolution patterns in the HONLD Eq. (4) with the NL coefficient σ ˜ equal to (a) 0, (b) 1, (c) 5, (d)  5 . Here, ν = 0.1 .

Fig. 6
Fig. 6

The evolution patterns of the discrete equation when g σ τ equals (a) 0, (b) 1, (c) 2, (d)  2 .

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

i z ψ + 2 ψ V ( r ) ψ + σ | ψ | 2 ψ = 0 .
i d a p d z + τ L b p + g σ | a p | 2 a p = 0 i d b p d z + τ L + a p + g σ | b p | 2 b p = 0 ,
i Z a + + b + σ ˜ | a | 2 a = 0 i Z b + a + σ ˜ | b | 2 b = 0 ,
i Z a + + b 1 2 ν Δ + b + σ ˜ | a | 2 a = 0 i Z b + a 1 2 ν Δ a + σ ˜ | b | 2 b = 0 .

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