Abstract

We examine localized surface modes in the core of a photonic crystal fiber composed of a finite nonlinear (Kerr) hexagonal waveguide array containing a topological defect in the form of a central void. Using the coupled-modes approach, we find the fundamental surface mode and the staggered and unstaggered ring-shaped modes, and their linear stability windows, for two void diameters. We find that, for a small void diameter, the unstable unstaggered ring mode of the system always requires less power and its instability gain at low powers is smaller than in the case without the void. Also, for the small void case, the unstaggered ring mode does not require a minimum power threshold, in sharp contrast with the case without the void. For a larger void, most of these observations hold, as well. We follow numerically the dynamical evolution of these ring modes to reveal their decay channels at long propagation distances.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  16. M. I. Molina, R. A. Vicencio, and Y. S. Kivshar, “Discrete solitons and nonlinear surface modes in semi-infinite waveguide arrays,” Opt. Lett. 31, 1693–1695 (2006).
    [CrossRef]
  17. A. Szameit, M. I. Molina, M. Heinrich, F. Dreisow, R. Keil, S. Nolte, and Y. S. Kivshar, “Observation of localized modes at phase slips in two-dimensional photonic lattices,” Opt. Lett. 35, 2738–2740 (2010).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  24. M. I. Molina, and Y. S. Kivshar, “Discrete and surface solitons in photonic graphene nanoribbons,” Opt. Lett. 35, 2895–2897(2010).
    [CrossRef]
  25. P. D. Rasmussen, F. H. Bennet, D. N. Neshev, A. A. Sukhorukov, C. R. Rosberg, W. Krolikowski, O. Bang, and Y. S. Kivshar, “Observation of two-dimensional nonlocal gap solitons,” Opt. Lett. 34, 295–297 (2009).
    [CrossRef]
  26. F. H. Bennet, I. A. Amuli, A. A. Sukhorukov, W. Krolikowski, D. N. Neshev, and Y. S. Kivshar, “Focusing-to-defocusing crossover in nonlinear periodic structures,” Opt. Lett. 35, 3213–3215(2010).
    [CrossRef]
  27. F. H. Bennet and J. Farnell, “Waveguide arrays in selectively infiltrated photonic crystal fibres,” Opt. Commun. 283, 4069–4073 (2010).
    [CrossRef]
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    [CrossRef]
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  30. M. I. Molina, R. A. Vicencio, and Y. S. Kivshar, “Discrete solitons and nonlinear surface modes in semi-infinite waveguide arrays,” Opt. Lett. 31, 1693–1695 (2006).
    [CrossRef]
  31. M. I. Molina and Y. S. Kivshar, “Nonlinear localized modes at phase slips in two-dimensional photonic lattices,” Phys. Rev. A 80, 063812 (2009).
    [CrossRef]

2011 (1)

M. I. Molina, “Boundary-induced Anderson localization in photonic lattices,” Phys. Lett. A 375, 2056–2058 (2011).
[CrossRef]

2010 (5)

2009 (3)

2008 (2)

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

M. I. Molina and Y. S. Kivshar, “Nonlinear localized modes at phase-slip defects in waveguide arrays,” Opt. Lett. 33, 917–919 (2008).
[CrossRef]

2007 (1)

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef]

2006 (4)

2005 (1)

2004 (3)

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[CrossRef]

D. K. Campbell, S. Flach, and Y. S. Kivshar, “Localizing energy through nonlinearity and discreteness,” Phys. Today 57(1), 43–49 (2004).
[CrossRef]

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, Phys. Rev. Lett., “Observation of vortex-ring ‘discrete’ solitons in 2D photonic lattices,” 92, 123904 (2004).
[CrossRef]

2003 (1)

2002 (1)

S. D. Hart, G. R. Maskaly, B. Temelkuran, P. H. Prideaux, J. D. Joannopoulos, and Y. Fink, “External reflection from omnidirectional dielectric mirror fibers,” Science 296, 510–513 (2002).
[CrossRef]

2001 (1)

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

1997 (1)

D. S. Wiersma, P. Bartolini, A. Legendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

1992 (1)

V. D. Freilikher and S. A. Gredeskul, “III Localization of waves in media with one-dimensional disorder,” Prog. Opt. 30, 137–203 (1992).
[CrossRef]

1987 (2)

E. Yablonovich, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef]

S. John, Phys. Rev. Lett. “Strong localization of photons in certain disordered dielectric superlattices,” 58, 2486–2489 (1987).
[CrossRef]

1984 (1)

S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53, 2169–2172 (1984);
[CrossRef]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

Aitchison, J. S.

Amuli, I. A.

Avidan, A.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Bang, O.

Bartal, G.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef]

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, Phys. Rev. Lett., “Observation of vortex-ring ‘discrete’ solitons in 2D photonic lattices,” 92, 123904 (2004).
[CrossRef]

Bartolini, P.

D. S. Wiersma, P. Bartolini, A. Legendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

Bennet, F. H.

Birks, T. A.

Campbell, D. K.

D. K. Campbell, S. Flach, and Y. S. Kivshar, “Localizing energy through nonlinearity and discreteness,” Phys. Today 57(1), 43–49 (2004).
[CrossRef]

Christodoulides, D. N.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, Phys. Rev. Lett., “Observation of vortex-ring ‘discrete’ solitons in 2D photonic lattices,” 92, 123904 (2004).
[CrossRef]

Cohen, O.

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, Phys. Rev. Lett., “Observation of vortex-ring ‘discrete’ solitons in 2D photonic lattices,” 92, 123904 (2004).
[CrossRef]

Couny, F.

Dreisow, F.

Eggleton, B.

Eisenberg, H. S.

Farnell, J.

F. H. Bennet and J. Farnell, “Waveguide arrays in selectively infiltrated photonic crystal fibres,” Opt. Commun. 283, 4069–4073 (2010).
[CrossRef]

Farr, L.

Fink, Y.

S. D. Hart, G. R. Maskaly, B. Temelkuran, P. H. Prideaux, J. D. Joannopoulos, and Y. Fink, “External reflection from omnidirectional dielectric mirror fibers,” Science 296, 510–513 (2002).
[CrossRef]

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef]

Flach, S.

D. K. Campbell, S. Flach, and Y. S. Kivshar, “Localizing energy through nonlinearity and discreteness,” Phys. Today 57(1), 43–49 (2004).
[CrossRef]

Fleischer, J. W.

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, Phys. Rev. Lett., “Observation of vortex-ring ‘discrete’ solitons in 2D photonic lattices,” 92, 123904 (2004).
[CrossRef]

Freilikher, V. D.

V. D. Freilikher and S. A. Gredeskul, “III Localization of waves in media with one-dimensional disorder,” Prog. Opt. 30, 137–203 (1992).
[CrossRef]

Giessen, H.

Gissibl, T.

Gredeskul, S. A.

V. D. Freilikher and S. A. Gredeskul, “III Localization of waves in media with one-dimensional disorder,” Prog. Opt. 30, 137–203 (1992).
[CrossRef]

Hart, S. D.

S. D. Hart, G. R. Maskaly, B. Temelkuran, P. H. Prideaux, J. D. Joannopoulos, and Y. Fink, “External reflection from omnidirectional dielectric mirror fibers,” Science 296, 510–513 (2002).
[CrossRef]

Heinrich, M.

Hudock, J.

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, Phys. Rev. Lett., “Observation of vortex-ring ‘discrete’ solitons in 2D photonic lattices,” 92, 123904 (2004).
[CrossRef]

Irman, A.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[CrossRef]

Joannopoulos, J. D.

S. D. Hart, G. R. Maskaly, B. Temelkuran, P. H. Prideaux, J. D. Joannopoulos, and Y. Fink, “External reflection from omnidirectional dielectric mirror fibers,” Science 296, 510–513 (2002).
[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. “Strong localization of photons in certain disordered dielectric superlattices,” 58, 2486–2489 (1987).
[CrossRef]

S. John, “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Phys. Rev. Lett. 53, 2169–2172 (1984);
[CrossRef]

S. John, Localization of Light and the Photonic Band Gap Concept (Springer Verlag, 2005).

Johnson, S. G.

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

Keil, R.

Kevrekidis, P. G.

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

Kivshar, Y. S.

M. I. Molina, and Y. S. Kivshar, “Discrete and surface solitons in photonic graphene nanoribbons,” Opt. Lett. 35, 2895–2897(2010).
[CrossRef]

F. H. Bennet, I. A. Amuli, A. A. Sukhorukov, W. Krolikowski, D. N. Neshev, and Y. S. Kivshar, “Focusing-to-defocusing crossover in nonlinear periodic structures,” Opt. Lett. 35, 3213–3215(2010).
[CrossRef]

A. Szameit, M. I. Molina, M. Heinrich, F. Dreisow, R. Keil, S. Nolte, and Y. S. Kivshar, “Observation of localized modes at phase slips in two-dimensional photonic lattices,” Opt. Lett. 35, 2738–2740 (2010).
[CrossRef]

M. I. Molina and Y. S. Kivshar, “Nonlinear localized modes at phase slips in two-dimensional photonic lattices,” Phys. Rev. A 80, 063812 (2009).
[CrossRef]

P. D. Rasmussen, F. H. Bennet, D. N. Neshev, A. A. Sukhorukov, C. R. Rosberg, W. Krolikowski, O. Bang, and Y. S. Kivshar, “Observation of two-dimensional nonlocal gap solitons,” Opt. Lett. 34, 295–297 (2009).
[CrossRef]

M. I. Molina and Y. S. Kivshar, “Nonlinear localized modes at phase-slip defects in waveguide arrays,” Opt. Lett. 33, 917–919 (2008).
[CrossRef]

M. I. Molina, R. A. Vicencio, and Y. S. Kivshar, “Discrete solitons and nonlinear surface modes in semi-infinite waveguide arrays,” Opt. Lett. 31, 1693–1695 (2006).
[CrossRef]

M. I. Molina, R. A. Vicencio, and Y. S. Kivshar, “Discrete solitons and nonlinear surface modes in semi-infinite waveguide arrays,” Opt. Lett. 31, 1693–1695 (2006).
[CrossRef]

D. K. Campbell, S. Flach, and Y. S. Kivshar, “Localizing energy through nonlinearity and discreteness,” Phys. Today 57(1), 43–49 (2004).
[CrossRef]

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

Knight, J. C.

Krolikowski, W.

Kuhlmey, B.

Lahini, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Legendijk, A.

D. S. Wiersma, P. Bartolini, A. Legendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

Lodahl, P.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[CrossRef]

Malomed, B. A.

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

Mandelik, D.

Manela, O.

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, Phys. Rev. Lett., “Observation of vortex-ring ‘discrete’ solitons in 2D photonic lattices,” 92, 123904 (2004).
[CrossRef]

Mangan, B. J.

Maskaly, G. R.

S. D. Hart, G. R. Maskaly, B. Temelkuran, P. H. Prideaux, J. D. Joannopoulos, and Y. Fink, “External reflection from omnidirectional dielectric mirror fibers,” Science 296, 510–513 (2002).
[CrossRef]

Mason, M. W.

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).

Modotto, D.

Molina, M. I.

Morales-Molina, L.

Morandotti, R.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

R. Morandotti, H. S. Eisenberg, D. Mandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, and J. S. Aitchison, “Interactions of discrete solitons with structural defects,” Opt. Lett. 28, 834–836 (2003).
[CrossRef]

Neshev, D. N.

Nikolaev, I. S.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[CrossRef]

Nolte, S.

Overgaag, K.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[CrossRef]

Pozzi, F.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Pricking, S.

Prideaux, P. H.

S. D. Hart, G. R. Maskaly, B. Temelkuran, P. H. Prideaux, J. D. Joannopoulos, and Y. Fink, “External reflection from omnidirectional dielectric mirror fibers,” Science 296, 510–513 (2002).
[CrossRef]

Rasmussen, P. D.

Righini, R.

D. S. Wiersma, P. Bartolini, A. Legendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

Roberts, P. J.

Rosberg, C. R.

Russell, P. St. J.

Sabert, H.

Schwartz, T.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef]

Segev, M.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef]

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, Phys. Rev. Lett., “Observation of vortex-ring ‘discrete’ solitons in 2D photonic lattices,” 92, 123904 (2004).
[CrossRef]

Silberberg, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

R. Morandotti, H. S. Eisenberg, D. Mandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, and J. S. Aitchison, “Interactions of discrete solitons with structural defects,” Opt. Lett. 28, 834–836 (2003).
[CrossRef]

Sorel, M.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

R. Morandotti, H. S. Eisenberg, D. Mandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, and J. S. Aitchison, “Interactions of discrete solitons with structural defects,” Opt. Lett. 28, 834–836 (2003).
[CrossRef]

Stanley, C. R.

Sukhorukov, A. A.

Szameit, A.

Temelkuran, B.

S. D. Hart, G. R. Maskaly, B. Temelkuran, P. H. Prideaux, J. D. Joannopoulos, and Y. Fink, “External reflection from omnidirectional dielectric mirror fibers,” Science 296, 510–513 (2002).
[CrossRef]

Tomlinson, A.

van Driel, A. F.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[CrossRef]

Vanmaekelbergh, D.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[CrossRef]

Vicencio, R. A.

Vieweg, M.

Vos, W. L.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[CrossRef]

Wiersma, D. S.

D. S. Wiersma, P. Bartolini, A. Legendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

Williams, D. P.

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, D.

Yablonovich, E.

E. Yablonovich, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef]

J. Lightwave Technol. (1)

Nature (3)

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature 430, 654–657 (2004).
[CrossRef]

D. S. Wiersma, P. Bartolini, A. Legendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
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Figures (7)

Fig. 1.
Fig. 1.

Hexagonal waveguide array with small central void. (a) Example of a fundamental inner surface mode (β=7.5, P=8.33). Amount of shading denotes the distribution of optical intensity. (b) Power versus propagation constant curve. The lower (upper) curve corresponds to the presence (absence) of the central void. Solid (dashed) curve denotes stable (unstable) portions.

Fig. 2.
Fig. 2.

Hexagonal waveguide array with small central void. (a) Example of an unstaggered “ring” surface mode (β=8.0, P=37.4). Amount of shading denotes the distribution of optical intensity. (b) and (c) show the power versus (b) propagation constant curve and (c) the instability gain for this kind of mode, which is always linearly unstable. Curves labeled A (B) denote the presence (absence) of a central void.

Fig. 3.
Fig. 3.

Hexagonal waveguide array with small central void. (a) Example of a staggered “ring” surface mode (β=5.75, P=47.95). Amount of shading denotes the distribution of the mode amplitude, going from white (amplitude=3) to dark gray (amplitude=+3). (b) Power versus propagation constant curve. Solid (dashed) curve refers to stable (unstable) branches.

Fig. 4.
Fig. 4.

Hexagonal waveguide array with larger central void. (a) Example of an unstaggered ring surface mode (β=8.0, P=75.3). Amount of shading denotes the distribution of optical amplitude. (b) Power versus propagation constant curve for this kind of mode, which is always linearly unstable. (c) Example of a staggered ring surface mode (β=6.5, P=105). Amount of shading denotes the distribution of optical amplitude, going from white (amplitude=3) to dark gray (amplitude=+3). (d) Power versus propagation constant curve for this kind of mode. In (b) and (d), the higher (lower) curve corresponds to the presence (absence) of the large central void, while the solid (dashed) portions denote stable (unstable) branches.

Fig. 5.
Fig. 5.

Dynamical evolution of all six sites in the initially unstaggered ring. (Un(0)=2.5 at each of the six ring sites, Un(0)=0 otherwise).

Fig. 6.
Fig. 6.

Evolution of unstable ring-mode configuration over large propagation distance: (a) z=0, (b) z=50, (c) z=100, (d) z=136, (e) z=137, (f) z=138, (g) z=150, and (h) z=200. The initial amplitude in all six sites around the void is 2, with no phase differences.

Fig. 7.
Fig. 7.

Dynamical evolution of all six sites in the initially unstaggered ring, without the center void. Lowest box shows the evolution at the ring center (Un(0)=3 at each of the six ring sites; Un(0)=0 otherwise).

Equations (2)

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idUn⃗dz+Vm⃗n⃗Um⃗+γ|Un⃗|2Un⃗=0,
βUn⃗+Vm⃗n⃗Um⃗+γ|Un⃗|2Un⃗=0,

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