Abstract

We study theoretically the dispersion properties of Bloch modes and nonlinearly-induced defect states in two-dimensional waveguide arrays. We define the conditions for achieving anomalous group-velocity dispersion and discuss possibilities for generation of spatiotemporal solitons.

© 2008 Optical Society of America

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References

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  1. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003).
  2. G. I. Stegeman and M. Segev, "Optical spatial solitons and their interactions: Universality and diversity," Science 286, 1518-1523 (1999).
    [CrossRef] [PubMed]
  3. G. P. Agrawal, Nonlinear Fiber Optics, fourth ed. (Academic Press, New York, 2007).
  4. Y. Silberberg, "Collapse of optical pulses," Opt. Lett. 15, 1282-1284 (1990).
    [CrossRef] [PubMed]
  5. A. B. Blagoeva, S. G. Dinev, A. A. Dreischuh, and A. Naidenov, "Light bullets formation in a bulk media," IEEE J. Quantum Electron. QE-27, 2060 (1991).
    [CrossRef]
  6. F. Wise and P. Di Trapani, "The hunt for light bullets - spatiotemporal solitons," Opt. Photon. News 13, 28-32 (2002).
    [CrossRef]
  7. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, "Spatiotemporal optical solitons," J. Opt. B: Quantum Semicl. Opt. 7, R53-R72 (2005).
    [CrossRef]
  8. H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar Ad, D. Ross, and J. S. Aitchison, "Kerr spatiotemporal self-focusing in a planar glass waveguide," Phys. Rev. Lett. 87, 043902-4 (2001).
    [CrossRef] [PubMed]
  9. D. Cheskis, S. Bar Ad, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, and D. Ross, "Strong spatiotemporal localization in a silica nonlinear waveguide array," Phys. Rev. Lett. 91, 223901-4 (2003).
    [CrossRef] [PubMed]
  10. N. I. Nikolov, D. Neshev, O. Bang, and W. Z. Krolikowski, "Quadratic solitons as nonlocal solitons," Phys. Rev. E 68, 036614-5 (2003).
    [CrossRef]
  11. X. Liu, L. J. Qian, and F. W. Wise, "Generation of optical spatiotemporal solitons," Phys. Rev. Lett. 82, 4631- 4634 (1999).
    [CrossRef]
  12. X. Liu, K. Beckwitt, and F. Wise, "Two-dimensional optical spatiotemporal solitons in quadratic media," Phys. Rev. E 62, 1328-1340 (2000).
    [CrossRef]
  13. A. B. Aceves, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, "Multidimensional solitons in fiber arrays," Opt. Lett. 19, 329-331 (1994).
    [CrossRef] [PubMed]
  14. E.W. Laedke, K. H. Spatschek, and S. K. Turitsyn, "Stability of discrete solitons and quasicollapse to intrinsically localized modes," Phys. Rev. Lett. 73, 1055-1059 (1994).
    [CrossRef] [PubMed]
  15. A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, "Energy localization in nonlinear fiber arrays - collapse-effect compressor," Phys. Rev. Lett. 75, 73-76 (1995).
    [CrossRef] [PubMed]
  16. A. B. Aceves, M. Santagiustina, and C. De Angelis, "Analytical study of nonlinear-optical pulse dynamics in arrays of linearly coupled waveguides," J. Opt. Soc. Am. B 14, 1807-1815 (1997).
    [CrossRef]
  17. D. Mihalache, D. Mazilu, F. Lederer, and Yu. S. Kivshar, "Stable discrete surface light bullets," Opt. Express 15, 589-595 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-2-589.
    [CrossRef] [PubMed]
  18. D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
    [CrossRef] [PubMed]
  19. D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled wave-guides," Opt. Lett. 13, 794-796 (1988).
    [CrossRef] [PubMed]
  20. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
    [CrossRef]
  21. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of discrete solitons in optically induced real time waveguide arrays," Phys. Rev. Lett. 90, 023902-4 (2003).
    [CrossRef] [PubMed]
  22. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, "Spatial solitons in optically induced gratings," Opt. Lett. 28, 710-712 (2003).
    [CrossRef] [PubMed]
  23. F. Chen, M. Stepi’c, C. R¨uter, D. Runde, D. Kip, V. Shandarov, O. Manela, and M. Segev, "Discrete diffraction and spatial gap solitons in photovoltaic LiNbO3 waveguide arrays," Opt. Express 13, 4314-4324 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-11-4314.
    [CrossRef] [PubMed]
  24. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "Discrete propagation and spatial solitons in nematic liquid crystals," Opt. Lett. 29, 1530-1532 (2004).
    [CrossRef] [PubMed]
  25. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
    [CrossRef] [PubMed]
  26. C. R. Rosberg, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, and Yu. S. Kivshar, "Observation of nonlinear self-trapping in triangular photonic lattices," Opt. Lett. 32, 397-399 (2007).
    [CrossRef] [PubMed]
  27. A. Szameit, D. Blomer, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, A. Tunnermann, and F. Lederer, "Discrete nonlinear localization in femtosecond laser written waveguides in fused silica," Opt. Express 13, 10552-10557 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-26-10552.
    [CrossRef] [PubMed]
  28. F. Couny, F. Benabid, P. J. Roberts, M. T. Burnett, and S. A. Maier, "Identification of Blochmodes in hollow-core photonic crystal fiber cladding," Opt. Express 15, 325-338 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-2-325.
    [CrossRef] [PubMed]
  29. C. R. Rosberg, F. H. Bennet, D. N. Neshev, P. D. Rasmussen, O. Bang, W. Krolikowski, A. Bjarklev, and Yu. S. Kivshar, "Tunable diffraction and self-defocusing in liquid-filled photonic crystal fibers," Opt. Express 15, 12145-12150 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-19-12145.
    [CrossRef] [PubMed]
  30. J. Jasapara, T. H. Her, R. Bise, R. Windeler, and D. J. DiGiovanni, "Group-velocity dispersion measurements in a photonic bandgap fiber," J. Opt. Soc. Am. B 20, 1611-1615 (2003).
    [CrossRef]
  31. P. Mach, M. Dolinski, K. W. Baldwin, J. A. Rogers, C. Kerbage, R. S. Windeler, and B. J. Eggleton, "Tunable microfluidic optical fiber," Appl. Phys. Lett. 80, 4294-4296 (2002).
    [CrossRef]
  32. F. Luan, A. K. George, T. D. Hedley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. St. Russell, "All-solid photonic bandgap fiber," Opt. Lett. 29, 2369-2371 (2004).
    [CrossRef] [PubMed]
  33. T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, "Nonlinearity and disorder in fiber arrays," Phys. Rev. Lett. 93, 053901-4 (2004).
    [CrossRef] [PubMed]
  34. U. Ropke, H. Bartelt, S. Unger, K. Schuster, and J. Kobelke, "Two-dimensional high-precision fiber waveguide arrays for coherent light propagation," Opt. Express 15, 6894-6899 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-11-6894.
    [CrossRef] [PubMed]
  35. S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis," Opt. Express 8, 173-190 (2001), http://www.opticsinfobase.org/abstract.cfm?URI=oe-8-3-173.
    [CrossRef] [PubMed]
  36. A. Samoc, "Dispersion of refractive properties of solvents: chloroform, toluene, benzene, and carbon disulfide in ultraviolet, visible, and near-infrared," J. Appl. Phys. 94, 6167-6174 (2003).
    [CrossRef]
  37. D. N. Nikogosyan, Properties of Optical and Laser-Related Materials: A Handbook (Wiley, Chichester, UK, 1997).
  38. K. Okamoto, Fundamentals of optical waveguides (Academic Press, San Diego, 2000).
  39. I. P. Nikolakakos, A. Major, J. S. Aitchison, and P. W. E. Smith, "Broadband characterization of the nonlinear optical properties of common reference materials," IEEE J. Sel. Top. Quantum Electron. 10, 1164-1170 (2004).
    [CrossRef]
  40. COMSOL Multiphysics 3.3 (2007), COMSOL Inc. (http://www.comsol.com/).
  41. P. G. Kevrekidis, B. A. Malomed, and Y. B. Gaididei, "Solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity," Phys. Rev. E 66, 016609-10 (2002).
    [CrossRef]

2007

2005

2004

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, "Nonlinearity and disorder in fiber arrays," Phys. Rev. Lett. 93, 053901-4 (2004).
[CrossRef] [PubMed]

A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "Discrete propagation and spatial solitons in nematic liquid crystals," Opt. Lett. 29, 1530-1532 (2004).
[CrossRef] [PubMed]

F. Luan, A. K. George, T. D. Hedley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. St. Russell, "All-solid photonic bandgap fiber," Opt. Lett. 29, 2369-2371 (2004).
[CrossRef] [PubMed]

I. P. Nikolakakos, A. Major, J. S. Aitchison, and P. W. E. Smith, "Broadband characterization of the nonlinear optical properties of common reference materials," IEEE J. Sel. Top. Quantum Electron. 10, 1164-1170 (2004).
[CrossRef]

2003

D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, "Spatial solitons in optically induced gratings," Opt. Lett. 28, 710-712 (2003).
[CrossRef] [PubMed]

J. Jasapara, T. H. Her, R. Bise, R. Windeler, and D. J. DiGiovanni, "Group-velocity dispersion measurements in a photonic bandgap fiber," J. Opt. Soc. Am. B 20, 1611-1615 (2003).
[CrossRef]

A. Samoc, "Dispersion of refractive properties of solvents: chloroform, toluene, benzene, and carbon disulfide in ultraviolet, visible, and near-infrared," J. Appl. Phys. 94, 6167-6174 (2003).
[CrossRef]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of discrete solitons in optically induced real time waveguide arrays," Phys. Rev. Lett. 90, 023902-4 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

D. Cheskis, S. Bar Ad, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, and D. Ross, "Strong spatiotemporal localization in a silica nonlinear waveguide array," Phys. Rev. Lett. 91, 223901-4 (2003).
[CrossRef] [PubMed]

N. I. Nikolov, D. Neshev, O. Bang, and W. Z. Krolikowski, "Quadratic solitons as nonlocal solitons," Phys. Rev. E 68, 036614-5 (2003).
[CrossRef]

2002

F. Wise and P. Di Trapani, "The hunt for light bullets - spatiotemporal solitons," Opt. Photon. News 13, 28-32 (2002).
[CrossRef]

P. Mach, M. Dolinski, K. W. Baldwin, J. A. Rogers, C. Kerbage, R. S. Windeler, and B. J. Eggleton, "Tunable microfluidic optical fiber," Appl. Phys. Lett. 80, 4294-4296 (2002).
[CrossRef]

P. G. Kevrekidis, B. A. Malomed, and Y. B. Gaididei, "Solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity," Phys. Rev. E 66, 016609-10 (2002).
[CrossRef]

2001

S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis," Opt. Express 8, 173-190 (2001), http://www.opticsinfobase.org/abstract.cfm?URI=oe-8-3-173.
[CrossRef] [PubMed]

H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar Ad, D. Ross, and J. S. Aitchison, "Kerr spatiotemporal self-focusing in a planar glass waveguide," Phys. Rev. Lett. 87, 043902-4 (2001).
[CrossRef] [PubMed]

2000

X. Liu, K. Beckwitt, and F. Wise, "Two-dimensional optical spatiotemporal solitons in quadratic media," Phys. Rev. E 62, 1328-1340 (2000).
[CrossRef]

1999

X. Liu, L. J. Qian, and F. W. Wise, "Generation of optical spatiotemporal solitons," Phys. Rev. Lett. 82, 4631- 4634 (1999).
[CrossRef]

G. I. Stegeman and M. Segev, "Optical spatial solitons and their interactions: Universality and diversity," Science 286, 1518-1523 (1999).
[CrossRef] [PubMed]

1998

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

1997

1995

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, "Energy localization in nonlinear fiber arrays - collapse-effect compressor," Phys. Rev. Lett. 75, 73-76 (1995).
[CrossRef] [PubMed]

1994

E.W. Laedke, K. H. Spatschek, and S. K. Turitsyn, "Stability of discrete solitons and quasicollapse to intrinsically localized modes," Phys. Rev. Lett. 73, 1055-1059 (1994).
[CrossRef] [PubMed]

A. B. Aceves, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, "Multidimensional solitons in fiber arrays," Opt. Lett. 19, 329-331 (1994).
[CrossRef] [PubMed]

1991

A. B. Blagoeva, S. G. Dinev, A. A. Dreischuh, and A. Naidenov, "Light bullets formation in a bulk media," IEEE J. Quantum Electron. QE-27, 2060 (1991).
[CrossRef]

1990

1988

Appl. Phys. Lett.

P. Mach, M. Dolinski, K. W. Baldwin, J. A. Rogers, C. Kerbage, R. S. Windeler, and B. J. Eggleton, "Tunable microfluidic optical fiber," Appl. Phys. Lett. 80, 4294-4296 (2002).
[CrossRef]

IEEE J. Quantum Electron.

A. B. Blagoeva, S. G. Dinev, A. A. Dreischuh, and A. Naidenov, "Light bullets formation in a bulk media," IEEE J. Quantum Electron. QE-27, 2060 (1991).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

I. P. Nikolakakos, A. Major, J. S. Aitchison, and P. W. E. Smith, "Broadband characterization of the nonlinear optical properties of common reference materials," IEEE J. Sel. Top. Quantum Electron. 10, 1164-1170 (2004).
[CrossRef]

J. Appl. Phys.

A. Samoc, "Dispersion of refractive properties of solvents: chloroform, toluene, benzene, and carbon disulfide in ultraviolet, visible, and near-infrared," J. Appl. Phys. 94, 6167-6174 (2003).
[CrossRef]

J. Opt. B: Quantum Semicl. Opt.

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, "Spatiotemporal optical solitons," J. Opt. B: Quantum Semicl. Opt. 7, R53-R72 (2005).
[CrossRef]

J. Opt. Soc. Am. B

Nature

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

Opt. Express

S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis," Opt. Express 8, 173-190 (2001), http://www.opticsinfobase.org/abstract.cfm?URI=oe-8-3-173.
[CrossRef] [PubMed]

F. Chen, M. Stepi’c, C. R¨uter, D. Runde, D. Kip, V. Shandarov, O. Manela, and M. Segev, "Discrete diffraction and spatial gap solitons in photovoltaic LiNbO3 waveguide arrays," Opt. Express 13, 4314-4324 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-11-4314.
[CrossRef] [PubMed]

A. Szameit, D. Blomer, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, A. Tunnermann, and F. Lederer, "Discrete nonlinear localization in femtosecond laser written waveguides in fused silica," Opt. Express 13, 10552-10557 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-26-10552.
[CrossRef] [PubMed]

F. Couny, F. Benabid, P. J. Roberts, M. T. Burnett, and S. A. Maier, "Identification of Blochmodes in hollow-core photonic crystal fiber cladding," Opt. Express 15, 325-338 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-2-325.
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, F. Lederer, and Yu. S. Kivshar, "Stable discrete surface light bullets," Opt. Express 15, 589-595 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-2-589.
[CrossRef] [PubMed]

U. Ropke, H. Bartelt, S. Unger, K. Schuster, and J. Kobelke, "Two-dimensional high-precision fiber waveguide arrays for coherent light propagation," Opt. Express 15, 6894-6899 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-11-6894.
[CrossRef] [PubMed]

C. R. Rosberg, F. H. Bennet, D. N. Neshev, P. D. Rasmussen, O. Bang, W. Krolikowski, A. Bjarklev, and Yu. S. Kivshar, "Tunable diffraction and self-defocusing in liquid-filled photonic crystal fibers," Opt. Express 15, 12145-12150 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-19-12145.
[CrossRef] [PubMed]

Opt. Lett.

Opt. Photon. News

F. Wise and P. Di Trapani, "The hunt for light bullets - spatiotemporal solitons," Opt. Photon. News 13, 28-32 (2002).
[CrossRef]

Phys. Rev. E

N. I. Nikolov, D. Neshev, O. Bang, and W. Z. Krolikowski, "Quadratic solitons as nonlocal solitons," Phys. Rev. E 68, 036614-5 (2003).
[CrossRef]

X. Liu, K. Beckwitt, and F. Wise, "Two-dimensional optical spatiotemporal solitons in quadratic media," Phys. Rev. E 62, 1328-1340 (2000).
[CrossRef]

P. G. Kevrekidis, B. A. Malomed, and Y. B. Gaididei, "Solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity," Phys. Rev. E 66, 016609-10 (2002).
[CrossRef]

Phys. Rev. Lett.

X. Liu, L. J. Qian, and F. W. Wise, "Generation of optical spatiotemporal solitons," Phys. Rev. Lett. 82, 4631- 4634 (1999).
[CrossRef]

H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar Ad, D. Ross, and J. S. Aitchison, "Kerr spatiotemporal self-focusing in a planar glass waveguide," Phys. Rev. Lett. 87, 043902-4 (2001).
[CrossRef] [PubMed]

D. Cheskis, S. Bar Ad, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, and D. Ross, "Strong spatiotemporal localization in a silica nonlinear waveguide array," Phys. Rev. Lett. 91, 223901-4 (2003).
[CrossRef] [PubMed]

E.W. Laedke, K. H. Spatschek, and S. K. Turitsyn, "Stability of discrete solitons and quasicollapse to intrinsically localized modes," Phys. Rev. Lett. 73, 1055-1059 (1994).
[CrossRef] [PubMed]

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, "Energy localization in nonlinear fiber arrays - collapse-effect compressor," Phys. Rev. Lett. 75, 73-76 (1995).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of discrete solitons in optically induced real time waveguide arrays," Phys. Rev. Lett. 90, 023902-4 (2003).
[CrossRef] [PubMed]

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tunnermann, and F. Lederer, "Nonlinearity and disorder in fiber arrays," Phys. Rev. Lett. 93, 053901-4 (2004).
[CrossRef] [PubMed]

Science

G. I. Stegeman and M. Segev, "Optical spatial solitons and their interactions: Universality and diversity," Science 286, 1518-1523 (1999).
[CrossRef] [PubMed]

Other

G. P. Agrawal, Nonlinear Fiber Optics, fourth ed. (Academic Press, New York, 2007).

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003).

D. N. Nikogosyan, Properties of Optical and Laser-Related Materials: A Handbook (Wiley, Chichester, UK, 1997).

K. Okamoto, Fundamentals of optical waveguides (Academic Press, San Diego, 2000).

COMSOL Multiphysics 3.3 (2007), COMSOL Inc. (http://www.comsol.com/).

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Figures (5)

Fig. 1.
Fig. 1.

(a) Real space hexagonal lattice and (b) the corresponding reciprocal space lattice. The triangle ΓMK defines the irreducible Brillouin zone. (c) Allowed modes in an array with d/Λ=0.5. The refractive index inside the holes is 1.59, and the refractive index of the background material is 1.44. The shaded areas indicate bands where modes are allowed, and the horizontal line indicates the refractive index of the solid material. The vertical line marks where an individual capillary becomes single-mode. (d) Photonic band structure for a constant normalized wavelength of λ/Λ=0.5 and d/Λ=0.5. (e)–(g) Intensities of the E-fields corresponding to the modes at the top and bottom of the first band and the top of the second band also for a normalized wavelength of λ/Λ=0.5. The corresponding phases are shown in (h)–(i).

Fig. 2.
Fig. 2.

(a) Normalized geometrical dispersion of the fundamental mode of an isolated waveguide. (b) and (c) show the normalized dispersion of the Bloch mode corresponding to the top and bottom of the first band respectively, and (d) shows the normalized dispersion of the mode at the top of the second band. d/Λ=0.5 for (b), (c) and (d). Note that in (c) and (d) the dispersion is only shown when the fundamental bandgap exists.

Fig. 3.
Fig. 3.

(a) Chromatic dispersion of Bloch modes corresponding to the top of the first (solid blue line) and second band (dashed red line), and the dispersion of the corresponding defect modes (blue and red points) for ΔnNL =10-3. (b) Chromatic dispersion of Bloch mode corresponding to the bottom of first band (blue line), and the dispersion of the corresponding defect mode for ΔnNL =-10-3 (blue points). The dimensions of the structures are (a) d/Λ=0.45, Λ=10µm, and (b) d/Λ=0.60, Λ=3.5µm.

Fig. 4.
Fig. 4.

Normalized effective area for defect mode in a 2D hexagonal structure. The index the silica background has been set to 1.44, and the index in the holes is 1.59. (a)d/Λ=0.45 and defect Δn=+10-3. The solid line shows the results for the defect mode in the semiinfinite bandgap above the first band, and the dotted line shows the results for the defect mode in the fundamental bandgap just above the edge of the second band. Insets: Intensities and phases of the defect modes at λ/Λ=0.15. (b)d/Λ=0.60 and defect Δn=-10-3. The insets show the intensity and phase of the defect mode at λ/Λ=0.43.

Fig. 5.
Fig. 5.

Coupling length for different index contrasts for a structure with d/Λ=0.5. The refractive index of the background is fixed at 1.44, corresponding to silica at λ 0=1.5µm. The black dots show the result of the coupled mode approximation for the first band.

Tables (1)

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Table 1. Parameters for refractive index formulas given in Eqs. (4–5).

Equations (8)

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× 1 ε ( r ) × H ω ( r ) = ω 2 c 2 H ω ( r ) ,
H k , β , m ( r , t ) = u k , m ( r ) exp [ i ( k · r + β z ω m ( k , β ) t ) ] ,
D = ω 2 2 π c v g 2 d v g d ω ,
n SiO 2 2 = 1 + j = 1 3 a j λ 2 λ 2 b j ,
n CS 2 = A 1 + A 2 λ 2 + A 3 λ 4 + A 4 λ 6 + A 5 λ 8 ,
i dA mn dz + C ( A m + 1 , n + A m , n + 1 + A m 1 , n + 1 + A m 1 , n + A m , n 1 + A m + 1 , n 1 ) = 0 ,
β = 2 C { cos [ 2 π k 1 ] + cos [ 2 π k 2 ] + cos [ 2 π ( k 1 k 2 ) ] } ,
L c = Λ max k x , k y k x , k y β .

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