Abstract

We point out that 2D photonic crystals (PhCs) can support surface bands that are pinned to Dirac points. These bands can be made very flat by optimizing the parameters of the system. Surface modes are found at the interface of two different cladding materials: one is a PhC with Dirac linear dispersion for the TE mode, and the other is a PhC that has a broad TE gap at the Dirac frequency.

© 2012 Optical Society of America

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  1. S. John, Phys. Rev. Lett. 58, 2486 (1987).
    [CrossRef]
  2. E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
    [CrossRef]
  3. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light(Princeton University, 2008).
  4. M. Plihal and A. A. Maradudin, Phys. Rev. B 44, 8565(1991).
    [CrossRef]
  5. R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, Phys. Rev. A 75, 063813 (2007).
    [CrossRef]
  6. F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
    [CrossRef]
  7. S. Raghu and F. D. M. Haldane, Phys. Rev. A 78, 033834 (2008).
    [CrossRef]
  8. T. Ochiai and M. Onoda, Phys. Rev. B 80, 155103 (2009).
    [CrossRef]
  9. S. R. Zandbergen and M. J. A. de Dood, Phys. Rev. Lett. 104, 043903 (2010).
    [CrossRef]
  10. O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. Christodoulides, Phys. Rev. Lett. 98, 103901 (2007).
    [CrossRef]
  11. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
    [CrossRef]
  12. A. K. Geim and K. S. Novoselov, Nat. Mater. 6, 183(2007).
    [CrossRef]
  13. A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
    [CrossRef]
  14. Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, A. Szameit, N. Malkova, Z. Chen, and M. Segev, in CLEO: QELS-Fundamental Science, OSA Technical Digest (Optical Society of America, 2012), paper QF2H.6.
  15. S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 173 (2001).
    [CrossRef]
  16. S. G. Johnson, The NLopt nonlinear-optimization package, http://ab-initio.mit.edu/nlopt .
  17. M. J. D. Powell, in Proceedings of the 40th Workshop on Large Scale Nonlinear Optimization (Springer, 2006), pp. 255–297.
  18. K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 54, 17954 (1996).
    [CrossRef]
  19. S. Ryu and Y. Hatsugai, Phys. Rev. Lett. 89, 077002(2002).
    [CrossRef]
  20. R. S. K. Mong and V. Shivamoggi, Phys. Rev. B 83, 125109 (2011).
    [CrossRef]
  21. J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, Phys. Rev. B 86, 035141 (2012).
    [CrossRef]

2012 (1)

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, Phys. Rev. B 86, 035141 (2012).
[CrossRef]

2011 (1)

R. S. K. Mong and V. Shivamoggi, Phys. Rev. B 83, 125109 (2011).
[CrossRef]

2010 (1)

S. R. Zandbergen and M. J. A. de Dood, Phys. Rev. Lett. 104, 043903 (2010).
[CrossRef]

2009 (2)

T. Ochiai and M. Onoda, Phys. Rev. B 80, 155103 (2009).
[CrossRef]

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

2008 (2)

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef]

S. Raghu and F. D. M. Haldane, Phys. Rev. A 78, 033834 (2008).
[CrossRef]

2007 (3)

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

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, Phys. Rev. A 75, 063813 (2007).
[CrossRef]

A. K. Geim and K. S. Novoselov, Nat. Mater. 6, 183(2007).
[CrossRef]

2004 (1)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

2002 (1)

S. Ryu and Y. Hatsugai, Phys. Rev. Lett. 89, 077002(2002).
[CrossRef]

2001 (1)

1996 (1)

K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 54, 17954 (1996).
[CrossRef]

1991 (1)

M. Plihal and A. A. Maradudin, Phys. Rev. B 44, 8565(1991).
[CrossRef]

1987 (2)

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef]

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef]

Bartal, G.

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

Bazaliy, Ya. B.

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, Phys. Rev. A 75, 063813 (2007).
[CrossRef]

Beenakker, C. W. J.

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, Phys. Rev. A 75, 063813 (2007).
[CrossRef]

Chan, C. T.

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, Phys. Rev. B 86, 035141 (2012).
[CrossRef]

Chen, Z.

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, A. Szameit, N. Malkova, Z. Chen, and M. Segev, in CLEO: QELS-Fundamental Science, OSA Technical Digest (Optical Society of America, 2012), paper QF2H.6.

Christodoulides, D.

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

de Dood, M. J. A.

S. R. Zandbergen and M. J. A. de Dood, Phys. Rev. Lett. 104, 043903 (2010).
[CrossRef]

Dresselhaus, G.

K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 54, 17954 (1996).
[CrossRef]

Dresselhaus, M. S.

K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 54, 17954 (1996).
[CrossRef]

Dubonos, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Firsov, A. A.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Freedman, B.

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

Fujita, M.

K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 54, 17954 (1996).
[CrossRef]

Geim, A. K.

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

A. K. Geim and K. S. Novoselov, Nat. Mater. 6, 183(2007).
[CrossRef]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Grigorieva, I. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Guinea, F.

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

Haldane, F. D. M.

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef]

S. Raghu and F. D. M. Haldane, Phys. Rev. A 78, 033834 (2008).
[CrossRef]

Hatsugai, Y.

S. Ryu and Y. Hatsugai, Phys. Rev. Lett. 89, 077002(2002).
[CrossRef]

Heinrich, M.

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, A. Szameit, N. Malkova, Z. Chen, and M. Segev, in CLEO: QELS-Fundamental Science, OSA Technical Digest (Optical Society of America, 2012), paper QF2H.6.

Jiang, D.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Joannopoulos, J. D.

S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 173 (2001).
[CrossRef]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light(Princeton University, 2008).

John, S.

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef]

Johnson, S. G.

S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 173 (2001).
[CrossRef]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light(Princeton University, 2008).

Malkova, N.

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, A. Szameit, N. Malkova, Z. Chen, and M. Segev, in CLEO: QELS-Fundamental Science, OSA Technical Digest (Optical Society of America, 2012), paper QF2H.6.

Manela, O.

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

Maradudin, A. A.

M. Plihal and A. A. Maradudin, Phys. Rev. B 44, 8565(1991).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light(Princeton University, 2008).

Mei, J.

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, Phys. Rev. B 86, 035141 (2012).
[CrossRef]

Mong, R. S. K.

R. S. K. Mong and V. Shivamoggi, Phys. Rev. B 83, 125109 (2011).
[CrossRef]

Morozov, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Nakada, K.

K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 54, 17954 (1996).
[CrossRef]

Neto, A. H. C.

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

Novoselov, K. S.

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

A. K. Geim and K. S. Novoselov, Nat. Mater. 6, 183(2007).
[CrossRef]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Ochiai, T.

T. Ochiai and M. Onoda, Phys. Rev. B 80, 155103 (2009).
[CrossRef]

Onoda, M.

T. Ochiai and M. Onoda, Phys. Rev. B 80, 155103 (2009).
[CrossRef]

Peleg, O.

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

Peres, N. M. R.

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

Plihal, M.

M. Plihal and A. A. Maradudin, Phys. Rev. B 44, 8565(1991).
[CrossRef]

Plotnik, Y.

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, A. Szameit, N. Malkova, Z. Chen, and M. Segev, in CLEO: QELS-Fundamental Science, OSA Technical Digest (Optical Society of America, 2012), paper QF2H.6.

Powell, M. J. D.

M. J. D. Powell, in Proceedings of the 40th Workshop on Large Scale Nonlinear Optimization (Springer, 2006), pp. 255–297.

Raghu, S.

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef]

S. Raghu and F. D. M. Haldane, Phys. Rev. A 78, 033834 (2008).
[CrossRef]

Rechtsman, M. C.

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, A. Szameit, N. Malkova, Z. Chen, and M. Segev, in CLEO: QELS-Fundamental Science, OSA Technical Digest (Optical Society of America, 2012), paper QF2H.6.

Ryu, S.

S. Ryu and Y. Hatsugai, Phys. Rev. Lett. 89, 077002(2002).
[CrossRef]

Segev, M.

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

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, A. Szameit, N. Malkova, Z. Chen, and M. Segev, in CLEO: QELS-Fundamental Science, OSA Technical Digest (Optical Society of America, 2012), paper QF2H.6.

Sepkhanov, R. A.

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, Phys. Rev. A 75, 063813 (2007).
[CrossRef]

Shivamoggi, V.

R. S. K. Mong and V. Shivamoggi, Phys. Rev. B 83, 125109 (2011).
[CrossRef]

Song, D.

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, A. Szameit, N. Malkova, Z. Chen, and M. Segev, in CLEO: QELS-Fundamental Science, OSA Technical Digest (Optical Society of America, 2012), paper QF2H.6.

Szameit, A.

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, A. Szameit, N. Malkova, Z. Chen, and M. Segev, in CLEO: QELS-Fundamental Science, OSA Technical Digest (Optical Society of America, 2012), paper QF2H.6.

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light(Princeton University, 2008).

Wu, Y.

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, Phys. Rev. B 86, 035141 (2012).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef]

Zandbergen, S. R.

S. R. Zandbergen and M. J. A. de Dood, Phys. Rev. Lett. 104, 043903 (2010).
[CrossRef]

Zhang, Y.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Zhang, Z.-Q.

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, Phys. Rev. B 86, 035141 (2012).
[CrossRef]

Nat. Mater. (1)

A. K. Geim and K. S. Novoselov, Nat. Mater. 6, 183(2007).
[CrossRef]

Opt. Express (1)

Phys. Rev. A (2)

R. A. Sepkhanov, Ya. B. Bazaliy, and C. W. J. Beenakker, Phys. Rev. A 75, 063813 (2007).
[CrossRef]

S. Raghu and F. D. M. Haldane, Phys. Rev. A 78, 033834 (2008).
[CrossRef]

Phys. Rev. B (5)

T. Ochiai and M. Onoda, Phys. Rev. B 80, 155103 (2009).
[CrossRef]

K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 54, 17954 (1996).
[CrossRef]

R. S. K. Mong and V. Shivamoggi, Phys. Rev. B 83, 125109 (2011).
[CrossRef]

J. Mei, Y. Wu, C. T. Chan, and Z.-Q. Zhang, Phys. Rev. B 86, 035141 (2012).
[CrossRef]

M. Plihal and A. A. Maradudin, Phys. Rev. B 44, 8565(1991).
[CrossRef]

Phys. Rev. Lett. (6)

S. Ryu and Y. Hatsugai, Phys. Rev. Lett. 89, 077002(2002).
[CrossRef]

S. R. Zandbergen and M. J. A. de Dood, Phys. Rev. Lett. 104, 043903 (2010).
[CrossRef]

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

F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008).
[CrossRef]

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef]

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef]

Rev. Mod. Phys. (1)

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

Science (1)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
[CrossRef]

Other (4)

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light(Princeton University, 2008).

Y. Plotnik, M. C. Rechtsman, D. Song, M. Heinrich, A. Szameit, N. Malkova, Z. Chen, and M. Segev, in CLEO: QELS-Fundamental Science, OSA Technical Digest (Optical Society of America, 2012), paper QF2H.6.

S. G. Johnson, The NLopt nonlinear-optimization package, http://ab-initio.mit.edu/nlopt .

M. J. D. Powell, in Proceedings of the 40th Workshop on Large Scale Nonlinear Optimization (Springer, 2006), pp. 255–297.

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Figures (3)

Fig. 1.
Fig. 1.

(a) TE PBS of 2D photonic graphene-like system. Dirac points are found between the second and the third TE photonic band. The inset depicts triangular structure of high-ϵ (ϵh=12) rods in low-ϵ (ϵl=1) media, with r1/a=0.31. (b) Surface plot of the 2D TE band structure (second and third band) with the characteristic hexagonal pattern: Dirac points with conical dispersion.

Fig. 2.
Fig. 2.

Surface bands within the Dirac pseudogap. We plot results with termination parameter t (see text for details) for three particular junctures of the two cladding materials: t=0.55 (red solid), t=0.6 (black dot-dashed), and t=0.8 (magenta dashed). PBSs of bulk of two cladding PhCs are shown for both lower cladding (blue) and upper cladding (green) material. In the insets on the right we plot the electric-field energy density of three states (at k=0) from the top surface bands (up to bottom: t=0.55, t=0.6, and t=0.8). Energy density is in red color, superimposed on the high dielectric pattern (translucent green).

Fig. 3.
Fig. 3.

Superflat dispersion of the surface band between two Dirac points (top red curve). Here t=0.66, r1/a0.306, and r2/a0.480. An additional (bottom) surface band is also present. Blue and green areas are PBSs of two independent claddings, as in Fig. 2. In the inset on the right we plot the electric-field energy density of the flat surface mode (at k=0).

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