Abstract

We investigate, experimentally and theoretically, light propagation in one-dimensional waveguide arrays exhibiting a saturable self-defocusing nonlinearity. We demonstrate low-intensity “discrete diffraction”, and the high-intensity formation of spatial gap solitons arising from the first band of the transmission spectrum. The waveguide arrays are fabricated by titanium in-diffusion in a photorefractive copper-doped lithium niobate crystal, and the optical nonlinearity arises from the bulk photovoltaic effect.

© 2005 Optical Society of America

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References

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  1. D. N. Christodoulides, F. Lederer, and Y. Silberberg, �??Discretizing light behavior in linear and nonlinear waveguide lattices,�?? Nature 424, 817-823 (2003).
    [CrossRef] [PubMed]
  2. D. N. Christodoulides and R. I. Joseph, �??Discrete self-focusing in nonlinear arrays of coupled waveguides,�?? Opt. Lett. 19, 794-796 (1988).
    [CrossRef]
  3. H. S. Eisenberg, Y. Silberberg, Y. Morandotti, R. Boyd, and J. S. Aitchison, �??Discrete spatial optical solitons in waveguide arrays,�?? Phys. Rev. Lett. 81, 3383-3386 (1998).
    [CrossRef]
  4. Y. S. Kivshar, �??Self-localization in arrays of defocusing waveguides,�?? Opt. Lett. 20, 1147-1149 (1993).
    [CrossRef]
  5. J. Feng, �??Alternative scheme for studying gap solitons in infinite periodic Kerr media,�?? Opt. Lett. 20, 1302-1304 (1993).
    [CrossRef]
  6. 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 (2003).
    [CrossRef] [PubMed]
  7. 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]
  8. S. Darmanyan, A. Kobyakov, and F. Lederer, �??Stability of strongly localized excitations in discrete media with cubic nonlinearity,�?? JETP 86, 682-686 (1998).
    [CrossRef]
  9. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, �??Spatial solitons in optically induced gratings,�?? Opt. Lett. 28, 710-712 (2003).
    [CrossRef] [PubMed]
  10. O. Manela, O. Cohen, G. Bartal, J. W. Fleischer, and M. Segev, �??Two-dimensional higher-band vortex lattice solitons,�?? Opt. Lett. 29, 2049-2051 (2004).
    [CrossRef] [PubMed]
  11. G. Bartal, O. Manela, O. Cohen, J. W. Fleischer, and M. Segev, �??Observation of 2nd-band vortex solitons in 2D photonic lattices,�?? submitted to Phys. Rev. Lett.
  12. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, �??Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,�?? Phys. Rev. Lett. 90, 053902 (2003).
    [CrossRef] [PubMed]
  13. O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, �??Multiband vector lattice solitons,�?? Phys. Rev. Lett. 91, 113901 (2003).
    [CrossRef] [PubMed]
  14. A. A. Sukhorukov and Y.S. Kivshar, �??Multigap discrete vector solitons,�?? Phys. Rev. Lett. 91, 113902 (2003).
    [CrossRef] [PubMed]
  15. Y. V. Kartashov, V. A. Vysloukh, L. Torner, �??Soliton trains in photonic lattices,�?? Opt. Express 12, 2831-2837 (2004).
    [CrossRef] [PubMed]
  16. A. S. Davydov, Solitons in Molecular Systems (Kluwer Academic, Dordrecht, 1991).
  17. A. H. Xie, L. van der Meer, V. Hoff, and R. H. Austin, �??Long-lived amide I vibrational modes in myoglobin,�?? Phys. Rev. Lett. 84, 5435-5438 (2000).
    [CrossRef] [PubMed]
  18. B. I. Swanson J. A. Brozik, S. P. Love, G. F. Strouse, A. P. Shreve, A. R. Bishop W.-Z. Wang, and M. I. Salkola, �??Observation of intrinsically localized modes in a discrete low-dimensional material,�?? Phys. Rev. Lett. 82, 3288-3291 (1999).
    [CrossRef]
  19. U. T. Schwartz, L. Q. English, and A. J. Sievers, �??Experimental generation and observation of intrinsic localized spin wave modes in an antiferromagnets,�?? Phys. Rev. Lett. 83, 223-226 (1999).
    [CrossRef]
  20. E. Trias, J. J. Mazo, and T. P. Orlando, �??Discrete breathers in nonlinear lattices: experimental detection in Josephson junctions,�?? Phys. Rev. Lett. 84, 741-744 (2000).
    [CrossRef] [PubMed]
  21. P. Binder, D. Abraimov, A. V. Ustinov, S. Flach, and Y. Zolotaryuk, �??Observation of breathers in Josephson ladders,�?? Phys. Rev. Lett. 84, 745-748 (2000).
    [CrossRef] [PubMed]
  22. A. Trombettoni and A. Smerzi, �??Discrete solitons and breathers with dilute Bose-Einstein condensates,�?? Phys. Rev. Lett. 86, 2353-2356 (2001).
    [CrossRef] [PubMed]
  23. B. Eiermann, Th. Anker, M. Albiez, M. Taglieber, P. Treutlein, K.-P. Marzlin, and M. K. Oberthaler, �??Bright Bose-Einstein gap solitons of atoms with repulsive interaction,�?? Phys. Rev. Lett. 92, 230401 (2004).
    [CrossRef] [PubMed]
  24. D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, �??Gap solitons in waveguide arrays,�?? Phys. Rev. Lett. 92, 093904 (2004).
    [CrossRef] [PubMed]
  25. D. N. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Y. S. Kivshar, �??Controlled generation and steering of spatial gap solitons,�?? Phys. Rev. Lett. 93, 083905 (2004).
    [CrossRef] [PubMed]
  26. M. Segev, G. C. Valley, B. Crosignani, P. D. Porto, and A. Yariv, �??Steady-state spatial screening solitons in photorefractive materials with external applied field,�?? Phys. Rev. Lett. 73, 3211-3214 (1994).
    [CrossRef] [PubMed]
  27. N. K. Efremidis S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, �??Discrete solitons in photorefractive optically induced photonic lattices,�?? Phys. Rev. E 66, 046602 (2002).
    [CrossRef]
  28. J. Meier, J. Hudock, D. N. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, �??Discrete vector solitons in Kerr nonlinear waveguide arrays,�?? Phys. Rev. Lett. 91, 143907 (2003).
    [CrossRef] [PubMed]
  29. S. Orlov, A. Yariv, and M. Segev, �??Nonlinear self-phase matching of optical second harmonic generation in lithium niobate,�?? Appl. Phys. Lett. 68, 1610-1612 (1996).
    [CrossRef]
  30. M. Segev, B. Crosignani, A. Yariv, and B. Fischer, �??Spatial solitons in photorefractive media,�?? Phys. Rev. Lett. 68, 923-926 (1992).
    [CrossRef] [PubMed]
  31. M. Segev, B. Crosignani, P. DiPorto, G. C. Valley, and A. Yariv, �??Steady state spatial screening-solitons in photorefractive media with external applied field,�?? Phys. Rev. Lett. 73, 3211-3214 (1994).
    [CrossRef] [PubMed]
  32. G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. Fejer, and M. Bashaw, �??Bright and dark photovoltaic spatial solitons,�?? Phys. Rev. A 50, R4457-R4460 (1994).
    [CrossRef] [PubMed]
  33. G. Duree, J. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, and R. R. Neurgaonkar, �??Observation of self-trapping of an optical beam due to the photorefractive effect,�?? Phys. Rev. Lett. 71, 533-536 (1993).
    [CrossRef] [PubMed]
  34. M. Taya, M. Bashaw, M. Fejer, M. Segev, and G. C. Valley, �??Observation of dark photovoltaic spatial solitons,�?? Phys. Rev. A 52, 3095-3100 (1995).
    [CrossRef] [PubMed]
  35. Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, "Self-trapping of an optical vortex by use of the bulk photovoltaic effect," Phys. Rev. Lett. 78, 2948-2951 (1997).
    [CrossRef]
  36. J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, �??Observation of vortex-ring �??discrete�?? solitons in 2D photonic lattices,�?? Phys. Rev. Lett. 92, 123904 (2004).
    [CrossRef] [PubMed]
  37. D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, �??Observation of discrete vortex solitons in optically induced photonic lattices,�?? Phys. Rev. Lett. 92, 123903 (2004).
    [CrossRef] [PubMed]
  38. O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, �??Observation of random-phase lattice solitons,�?? Nature 433, 500-503 (2005).
    [CrossRef] [PubMed]
  39. H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, �??Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,�?? Phys. Rev. Lett. 92, 123902 (2004).
    [CrossRef] [PubMed]
  40. Z. Chen, H. Martin, E. D. Eugenieva, J. Xu, and A. Bezryadina, �??Anisotropic enhancement of discrete diffraction and formation of two-dimensional discrete-soliton trains,�?? Phys. Rev. Lett. 92, 143902 (2004).
    [CrossRef] [PubMed]
  41. J. Yang, I. Makasynk, A. Bezryadina, and Z. Chen, �??Dipole solitons in optically-induced two-dimensional photonic lattices,�?? Opt. Lett. 29, 1662-1664 (2004).
    [CrossRef] [PubMed]
  42. Z. Chen, A. Bezryadina, I. Makasynk, and J. Yang, �??Observation of two-dimensional vector lattice solitons,�?? Opt. Lett. 29, 1656-1658 (2004).
    [CrossRef] [PubMed]
  43. M. Stepic, D. Kip, Lj. Hadzievski, and A. Maluckov, �??One-dimensional bright discrete solitons in media with saturable nonlinearity,�?? Phys. Rev. E 69, 066618 (2004).
    [CrossRef]
  44. K. Peithmann, J. Hukriede, K. Buse, and E. Krätzig: �??Photorefractive properties of lithium niobate volume crystals doped by copper diffusion,�?? Phys. Rev. B 61, 4615-4620 (2000).
    [CrossRef]
  45. H. Yoshida, �??Construction of higher order sympletic integrators,�?? Phys. Lett. A 150, 262-269 (1990).
    [CrossRef]
  46. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison: �??Diffraction management,�?? Phys. Rev. Lett. 85, 1863-1866 (2000).
    [CrossRef] [PubMed]
  47. T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tünnermann, �??Discrete diffraction in two-dimensional arrays of coupled waveguides in silica,�?? Opt. Lett. 29, 468-470 (2004).
    [CrossRef] [PubMed]
  48. The long response time is a result of the Cu doping and the low photoconductivity of our sample. Very recently we have fabricated samples with Fe doping where the response time can be shortened to about 100 s.
  49. M. Segev, G. C. Valley, M. C. Bashaw, M. Taya, and M. M. Fejer, �??Photovoltaic spatial solitons,�?? J. Opt. Soc. Am. B 14, 1772-1781 (1997).
    [CrossRef]

Appl. Phys. Lett. (1)

S. Orlov, A. Yariv, and M. Segev, �??Nonlinear self-phase matching of optical second harmonic generation in lithium niobate,�?? Appl. Phys. Lett. 68, 1610-1612 (1996).
[CrossRef]

J. Opt. Soc. Am. B (1)

JETP (1)

S. Darmanyan, A. Kobyakov, and F. Lederer, �??Stability of strongly localized excitations in discrete media with cubic nonlinearity,�?? JETP 86, 682-686 (1998).
[CrossRef]

Nature (3)

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]

O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, �??Observation of random-phase lattice solitons,�?? Nature 433, 500-503 (2005).
[CrossRef] [PubMed]

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

Opt. Express (1)

Opt. Lett. (8)

Z. Chen, A. Bezryadina, I. Makasynk, and J. Yang, �??Observation of two-dimensional vector lattice solitons,�?? Opt. Lett. 29, 1656-1658 (2004).
[CrossRef] [PubMed]

O. Manela, O. Cohen, G. Bartal, J. W. Fleischer, and M. Segev, �??Two-dimensional higher-band vortex lattice solitons,�?? Opt. Lett. 29, 2049-2051 (2004).
[CrossRef] [PubMed]

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

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tünnermann, �??Discrete diffraction in two-dimensional arrays of coupled waveguides in silica,�?? Opt. Lett. 29, 468-470 (2004).
[CrossRef] [PubMed]

D. N. Christodoulides and R. I. Joseph, �??Discrete self-focusing in nonlinear arrays of coupled waveguides,�?? Opt. Lett. 19, 794-796 (1988).
[CrossRef]

J. Yang, I. Makasynk, A. Bezryadina, and Z. Chen, �??Dipole solitons in optically-induced two-dimensional photonic lattices,�?? Opt. Lett. 29, 1662-1664 (2004).
[CrossRef] [PubMed]

Y. S. Kivshar, �??Self-localization in arrays of defocusing waveguides,�?? Opt. Lett. 20, 1147-1149 (1993).
[CrossRef]

J. Feng, �??Alternative scheme for studying gap solitons in infinite periodic Kerr media,�?? Opt. Lett. 20, 1302-1304 (1993).
[CrossRef]

Phys. Lett. A (1)

H. Yoshida, �??Construction of higher order sympletic integrators,�?? Phys. Lett. A 150, 262-269 (1990).
[CrossRef]

Phys. Rev. A (2)

G. C. Valley, M. Segev, B. Crosignani, A. Yariv, M. Fejer, and M. Bashaw, �??Bright and dark photovoltaic spatial solitons,�?? Phys. Rev. A 50, R4457-R4460 (1994).
[CrossRef] [PubMed]

M. Taya, M. Bashaw, M. Fejer, M. Segev, and G. C. Valley, �??Observation of dark photovoltaic spatial solitons,�?? Phys. Rev. A 52, 3095-3100 (1995).
[CrossRef] [PubMed]

Phys. Rev. B (1)

K. Peithmann, J. Hukriede, K. Buse, and E. Krätzig: �??Photorefractive properties of lithium niobate volume crystals doped by copper diffusion,�?? Phys. Rev. B 61, 4615-4620 (2000).
[CrossRef]

Phys. Rev. E (2)

N. K. Efremidis S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, �??Discrete solitons in photorefractive optically induced photonic lattices,�?? Phys. Rev. E 66, 046602 (2002).
[CrossRef]

M. Stepic, D. Kip, Lj. Hadzievski, and A. Maluckov, �??One-dimensional bright discrete solitons in media with saturable nonlinearity,�?? Phys. Rev. E 69, 066618 (2004).
[CrossRef]

Phys. Rev. Lett. (26)

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 (2003).
[CrossRef] [PubMed]

G. Bartal, O. Manela, O. Cohen, J. W. Fleischer, and M. Segev, �??Observation of 2nd-band vortex solitons in 2D photonic lattices,�?? submitted to Phys. Rev. Lett.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, �??Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,�?? Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, �??Multiband vector lattice solitons,�?? Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef] [PubMed]

A. A. Sukhorukov and Y.S. Kivshar, �??Multigap discrete vector solitons,�?? Phys. Rev. Lett. 91, 113902 (2003).
[CrossRef] [PubMed]

G. Duree, J. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, and R. R. Neurgaonkar, �??Observation of self-trapping of an optical beam due to the photorefractive effect,�?? Phys. Rev. Lett. 71, 533-536 (1993).
[CrossRef] [PubMed]

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, �??Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,�?? Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef] [PubMed]

Z. Chen, H. Martin, E. D. Eugenieva, J. Xu, and A. Bezryadina, �??Anisotropic enhancement of discrete diffraction and formation of two-dimensional discrete-soliton trains,�?? Phys. Rev. Lett. 92, 143902 (2004).
[CrossRef] [PubMed]

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

J. Meier, J. Hudock, D. N. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, �??Discrete vector solitons in Kerr nonlinear waveguide arrays,�?? Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef] [PubMed]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, �??Spatial solitons in photorefractive media,�?? Phys. Rev. Lett. 68, 923-926 (1992).
[CrossRef] [PubMed]

M. Segev, B. Crosignani, P. DiPorto, G. C. Valley, and A. Yariv, �??Steady state spatial screening-solitons in photorefractive media with external applied field,�?? Phys. Rev. Lett. 73, 3211-3214 (1994).
[CrossRef] [PubMed]

Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, "Self-trapping of an optical vortex by use of the bulk photovoltaic effect," Phys. Rev. Lett. 78, 2948-2951 (1997).
[CrossRef]

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

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, �??Observation of discrete vortex solitons in optically induced photonic lattices,�?? Phys. Rev. Lett. 92, 123903 (2004).
[CrossRef] [PubMed]

A. H. Xie, L. van der Meer, V. Hoff, and R. H. Austin, �??Long-lived amide I vibrational modes in myoglobin,�?? Phys. Rev. Lett. 84, 5435-5438 (2000).
[CrossRef] [PubMed]

B. I. Swanson J. A. Brozik, S. P. Love, G. F. Strouse, A. P. Shreve, A. R. Bishop W.-Z. Wang, and M. I. Salkola, �??Observation of intrinsically localized modes in a discrete low-dimensional material,�?? Phys. Rev. Lett. 82, 3288-3291 (1999).
[CrossRef]

U. T. Schwartz, L. Q. English, and A. J. Sievers, �??Experimental generation and observation of intrinsic localized spin wave modes in an antiferromagnets,�?? Phys. Rev. Lett. 83, 223-226 (1999).
[CrossRef]

E. Trias, J. J. Mazo, and T. P. Orlando, �??Discrete breathers in nonlinear lattices: experimental detection in Josephson junctions,�?? Phys. Rev. Lett. 84, 741-744 (2000).
[CrossRef] [PubMed]

P. Binder, D. Abraimov, A. V. Ustinov, S. Flach, and Y. Zolotaryuk, �??Observation of breathers in Josephson ladders,�?? Phys. Rev. Lett. 84, 745-748 (2000).
[CrossRef] [PubMed]

A. Trombettoni and A. Smerzi, �??Discrete solitons and breathers with dilute Bose-Einstein condensates,�?? Phys. Rev. Lett. 86, 2353-2356 (2001).
[CrossRef] [PubMed]

B. Eiermann, Th. Anker, M. Albiez, M. Taglieber, P. Treutlein, K.-P. Marzlin, and M. K. Oberthaler, �??Bright Bose-Einstein gap solitons of atoms with repulsive interaction,�?? Phys. Rev. Lett. 92, 230401 (2004).
[CrossRef] [PubMed]

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, �??Gap solitons in waveguide arrays,�?? Phys. Rev. Lett. 92, 093904 (2004).
[CrossRef] [PubMed]

D. N. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Y. S. Kivshar, �??Controlled generation and steering of spatial gap solitons,�?? Phys. Rev. Lett. 93, 083905 (2004).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. D. Porto, and A. Yariv, �??Steady-state spatial screening solitons in photorefractive materials with external applied field,�?? Phys. Rev. Lett. 73, 3211-3214 (1994).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison: �??Diffraction management,�?? Phys. Rev. Lett. 85, 1863-1866 (2000).
[CrossRef] [PubMed]

Other (2)

The long response time is a result of the Cu doping and the low photoconductivity of our sample. Very recently we have fabricated samples with Fe doping where the response time can be shortened to about 100 s.

A. S. Davydov, Solitons in Molecular Systems (Kluwer Academic, Dordrecht, 1991).

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Figures (8)

Fig. 1.
Fig. 1.

Refractive index profiles n(z) of a LiNbO3 waveguide array at two different depths of the in-diffused structures: at the depth of maximum field amplitude of the modes (dash-dotted line) and at a depth where the amplitude has dropped by a factor 1/e (solid line). The substrate refractive index is 2.242 (dotted line at the bottom).

Fig. 2.
Fig. 2.

Experimental set-up: P, polarizer; λ/2, half-wave plate; M 1,2, mirrors; GP, thin glass plate; CL, cylindrical lens; L 1,2, microscope lenses; WA, waveguide array; CCD, CCD camera. The light source is a 514.5 nm wavelength argon ion laser.

Fig. 3.
Fig. 3.

Experimental (a) and theoretical (b,c) results showing discrete diffraction of light in a LiNbO3 waveguide array, when a single input channel is excited. The upper part in (a) shows the intensity distribution at the output of the waveguide array, as photographed with a CCD camera. The two lower parts (b) and (c) show the simulated propagation of a beam in a waveguide array under the same parameters, at the “depth” of the maximum intensity of each individual mode.

Fig. 4.
Fig. 4.

Band-gap diagram of the waveguide array, relating the propagation constant β to the Bloch wave number Kz as described in section 2. The value “0” corresponds to a plane wave propagating in the substrate. The shaded regions represent the gaps where light propagation is forbidden. The black dot at the edge of the first band indicates the propagation constant of the gap soliton (shown in Fig. 7). Increasing the optical intensity creates a negative defect in the periodic structure, thereby localizing the corresponding Bloch wave by pushing its propagation constant β down into the gap, thus converting it from an extended Bloch wave into a self-localized state: a gap soliton.

Fig. 5.
Fig. 5.

Probing diffraction in the periodic waveguide array by varying the angle of incidence of a four-channel input beam, from Kz =0 (solid line), to nearly diffraction-free propagation at Kz ≈±π/2Λ(dotted and dashed lines).

Fig. 6.
Fig. 6.

Formation of a gap soliton in a 1D LiNbO3 waveguide array. The figure shows a line scan of the light intensity distribution measured by a CCD at the output facet of the array, where the dotted, dashed and solid lines represent the intensity profile at times t≈0, t=45 min, and t=160 min, respectively.

Fig. 7.
Fig. 7.

Calculated wavefunction and propagation dynamic of a spatial gap soliton in our setting. Left and middle panels: amplitude and intensity of a gap soliton (solid line) plotted on the background of the waveguide structure with the light-induced (negative) defect the soliton creates (dotted lines). Right panel: simulated stable and stationary propagation of the gap soliton in the waveguide array. The intensity profile of the soliton shown propagating in the right panel corresponds to the intensity of the soliton shown on the left panel.

Fig. 8.
Fig. 8.

Numerical results for the nonlinear propagation of a Gaussian beam with a π phase jump at its center (a dipole): a), b), the beam is launched normal to the waveguide array, and c), d), the same input beam is launched with a tilt of 2 mrad into the array. In both cases the input power is 16 µW and the input beam covers about half a lattice constant (FWHM of the Gaussian beam of 4.2 µm).

Equations (3)

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i d E d y + 1 2 k d 2 E d z 2 + k ( n ( z ) + Δ n n ) E = 0 .
Δ n = 1 2 n 3 r E p v I I + I d ,
1 2 k d 2 U d z 2 + i K z k d U d z K z 2 2 k U + k n ( z ) n U = β U .

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