Abstract

We study numerically the trapping dynamics of nonlinear waves scattered by local guiding photonic centers with normal eigenmodes which are embedded in uniform nonlinear Kerr waveguides. The linear and nonlinear scattering from a local defect may be treated from either a wave optics approach or a ray optics approach. The former provides a better understanding of the wave dynamics while the latter enables one to perform quasi-analytical estimates of the extent of trapping in a given structure. In the presence of a single-site multi-mode local scattering center, power may localize in a certain normal mode of the center, or periodically oscillate between different normal modes. The degree of trapping and mode of localization can be controlled as function of both the input power and the angle of incidence. With multi-site local scattering centers, the trapping dynamics are strongly dependent on the degree of coupling between the neighboring sites. In the scattering by a local multi-site defect with strongly coupled adjacent sites, two possibilities for nonlinear trapping arise. At intermediate nonlinear powers, periodic tunneling of power between adjacent sites and their normal modes is observed. At highly nonlinear powers, but still within experimental feasibility, the radiation can become strongly localized in a single site. The scattering and trapping dynamics are also described in the context of nonlinear Fabry-Perot etalons, as a function of the local defect’s refractive index.

© 2008 Optical Society of America

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  1. See, for example:P. P. Banerjee, Nonlinear Optics: Theory, Numerical Modeling and Applications (Taylor and Francis, London, 2007).
  2. Z. V. Vardeny and M. Raikh, "Photonics: Light localized on the lattice," Nature 446, 37-38 (2007).
    [CrossRef] [PubMed]
  3. D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides," Opt. Lett. 13, 794-796 (1988).
    [CrossRef] [PubMed]
  4. 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]
  5. F. Geniet and J. Leon, "Energy transmission in the forbidden band gap of a nonlinear chain," Phys. Rev. Lett. 89, 134102 (2002).
    [CrossRef] [PubMed]
  6. D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, "Gap solitons in waveguide arrays," Phys. Rev. Lett. 92, 093904 (2004).
    [CrossRef] [PubMed]
  7. K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache, "Discrete surface solitons," Opt. Lett. 30, 2466-2468 (2005).
    [CrossRef] [PubMed]
  8. S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, "Observation of discrete surface solitons," Phys. Rev. Lett. 96, 063901 (2006).
    [CrossRef] [PubMed]
  9. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Bragg-type soliton mirror," Opt. Express 14, 1576-1581 (2006).
    [CrossRef] [PubMed]
  10. X. D. Cao and B. A. Malomed, "Soliton-defect collisions in the nonlinear Schrodinger equation," Phys. Lett. A 206, 177-182 (1995).
    [CrossRef]
  11. A. E. Miroshnichenko, S. Flach, and B. Malomed, "Resonant scattering of solitons," Chaos 13, 874-879 (2003).
    [CrossRef] [PubMed]
  12. H. Sakaguchi and M. Tamura, "Scattering and trapping of nonlinear schrodinger solitons in external potentials," J. Phys. Soc. Jpn. 73, 503-506 (2004).
    [CrossRef]
  13. R. H. Goodman, P. J. Holmes, and M. I. Weinstein, "Strong NLS soliton-defect interactions," Physica D 192, 215-248 (2004).
    [CrossRef]
  14. K. T. Stoychev, M. T. Primatarowa, and R. S. Kamburova, "Resonant scattering of nonlinear Schrdinger solitons from potential wells," Phys. Rev. E 70, 066622 (2004).
    [CrossRef]
  15. H. Sakaguchi and M. Tamura, "Scattering of solitons and dark solitons by potential walls in the nonlinear schrodinger equation," J. Phys. Soc. Jpn. 74, 292-298 (2005).
    [CrossRef]
  16. B. Piette, W. J. Zakrzewski, and J. Brand, "Scattering of topological solitons on holes and barriers," J. Phys. A 38, 1040310412 (2005).
    [CrossRef]
  17. J. Garnier and F. Kh. Abdullaev, "Transmission of matter-wave solitons through nonlinear traps and barriers," Phys. Rev. A 74, 013604 (2006).
    [CrossRef]
  18. C. Lee and J. Brand, "Enhanced quantum reflection of matter-wave solitons," Europhys. Lett. 73, 321-327 (2006).
    [CrossRef]
  19. K. Rapedius, D. Witthaut, and H. J. Korsch, "Analytical study of resonant transport of Bose-Einstein condensates," Phys. Rev. A 73, 033608 (2006).
    [CrossRef]
  20. Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, "Nonlinear scattering and trapping by local photonic potentials," Phys. Rev. Lett. 99, 133901 (2007).
    [CrossRef] [PubMed]
  21. Y. Shavit, Y. Linzon, S. Bar-Ad, R. Morandotti, M. Volatier, V. Aimez, and R. Ares, "Power-dependent switching of nonlinear trapping by local photonic potentials," Opt. Lett. 33, 1056-1058 (2008).
    [CrossRef] [PubMed]
  22. D. Mandelik, Y. Lahini, and Y. Silberberg, "Nonlinearly induced relaxation to the ground state in a two-level system," Phys. Rev. Lett. 95, 073902 (2005).
    [CrossRef] [PubMed]
  23. A. Soffer and M. I. Weinstein, "Theory of nonlinear dispersive waves and selection of the ground state," Phys. Rev. Lett. 95, 213905 (2005).
    [CrossRef] [PubMed]
  24. J. Meier, J. Hudock, D. 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]
  25. Y. Linzon, Y. Shavit, M. Elazar, R. Morandotti, M. Volatier-Ravat, V. Aimez, R. Ares, and S. Bar-Ad, "Single beam mapping of nonlinear phase shift profiles in planar waveguides with an embedded mirror," Opt. Express 15, 12068-12075 (2007).
    [CrossRef] [PubMed]
  26. J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, "The nonlinear optical properties of AlGaAs at the half band gap," IEEE J. Quant. Electron. 33, 341-348 (1997).
    [CrossRef]

2008 (1)

2007 (3)

Y. Linzon, Y. Shavit, M. Elazar, R. Morandotti, M. Volatier-Ravat, V. Aimez, R. Ares, and S. Bar-Ad, "Single beam mapping of nonlinear phase shift profiles in planar waveguides with an embedded mirror," Opt. Express 15, 12068-12075 (2007).
[CrossRef] [PubMed]

Z. V. Vardeny and M. Raikh, "Photonics: Light localized on the lattice," Nature 446, 37-38 (2007).
[CrossRef] [PubMed]

Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, "Nonlinear scattering and trapping by local photonic potentials," Phys. Rev. Lett. 99, 133901 (2007).
[CrossRef] [PubMed]

2006 (5)

J. Garnier and F. Kh. Abdullaev, "Transmission of matter-wave solitons through nonlinear traps and barriers," Phys. Rev. A 74, 013604 (2006).
[CrossRef]

C. Lee and J. Brand, "Enhanced quantum reflection of matter-wave solitons," Europhys. Lett. 73, 321-327 (2006).
[CrossRef]

K. Rapedius, D. Witthaut, and H. J. Korsch, "Analytical study of resonant transport of Bose-Einstein condensates," Phys. Rev. A 73, 033608 (2006).
[CrossRef]

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, "Observation of discrete surface solitons," Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Bragg-type soliton mirror," Opt. Express 14, 1576-1581 (2006).
[CrossRef] [PubMed]

2005 (5)

K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache, "Discrete surface solitons," Opt. Lett. 30, 2466-2468 (2005).
[CrossRef] [PubMed]

H. Sakaguchi and M. Tamura, "Scattering of solitons and dark solitons by potential walls in the nonlinear schrodinger equation," J. Phys. Soc. Jpn. 74, 292-298 (2005).
[CrossRef]

B. Piette, W. J. Zakrzewski, and J. Brand, "Scattering of topological solitons on holes and barriers," J. Phys. A 38, 1040310412 (2005).
[CrossRef]

D. Mandelik, Y. Lahini, and Y. Silberberg, "Nonlinearly induced relaxation to the ground state in a two-level system," Phys. Rev. Lett. 95, 073902 (2005).
[CrossRef] [PubMed]

A. Soffer and M. I. Weinstein, "Theory of nonlinear dispersive waves and selection of the ground state," Phys. Rev. Lett. 95, 213905 (2005).
[CrossRef] [PubMed]

2004 (4)

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

H. Sakaguchi and M. Tamura, "Scattering and trapping of nonlinear schrodinger solitons in external potentials," J. Phys. Soc. Jpn. 73, 503-506 (2004).
[CrossRef]

R. H. Goodman, P. J. Holmes, and M. I. Weinstein, "Strong NLS soliton-defect interactions," Physica D 192, 215-248 (2004).
[CrossRef]

K. T. Stoychev, M. T. Primatarowa, and R. S. Kamburova, "Resonant scattering of nonlinear Schrdinger solitons from potential wells," Phys. Rev. E 70, 066622 (2004).
[CrossRef]

2003 (2)

A. E. Miroshnichenko, S. Flach, and B. Malomed, "Resonant scattering of solitons," Chaos 13, 874-879 (2003).
[CrossRef] [PubMed]

J. Meier, J. Hudock, D. 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]

2002 (1)

F. Geniet and J. Leon, "Energy transmission in the forbidden band gap of a nonlinear chain," Phys. Rev. Lett. 89, 134102 (2002).
[CrossRef] [PubMed]

1998 (1)

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 (1)

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, "The nonlinear optical properties of AlGaAs at the half band gap," IEEE J. Quant. Electron. 33, 341-348 (1997).
[CrossRef]

1995 (1)

X. D. Cao and B. A. Malomed, "Soliton-defect collisions in the nonlinear Schrodinger equation," Phys. Lett. A 206, 177-182 (1995).
[CrossRef]

1988 (1)

Abdullaev, F. Kh.

J. Garnier and F. Kh. Abdullaev, "Transmission of matter-wave solitons through nonlinear traps and barriers," Phys. Rev. A 74, 013604 (2006).
[CrossRef]

Aimez, V.

Aitchison, J. S.

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

J. Meier, J. Hudock, D. 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]

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. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, "The nonlinear optical properties of AlGaAs at the half band gap," IEEE J. Quant. Electron. 33, 341-348 (1997).
[CrossRef]

Ares, R.

Bar-Ad, S.

Boyd, A. R.

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]

Brand, J.

C. Lee and J. Brand, "Enhanced quantum reflection of matter-wave solitons," Europhys. Lett. 73, 321-327 (2006).
[CrossRef]

B. Piette, W. J. Zakrzewski, and J. Brand, "Scattering of topological solitons on holes and barriers," J. Phys. A 38, 1040310412 (2005).
[CrossRef]

Cao, X. D.

X. D. Cao and B. A. Malomed, "Soliton-defect collisions in the nonlinear Schrodinger equation," Phys. Lett. A 206, 177-182 (1995).
[CrossRef]

Christodoulides, D.

J. Meier, J. Hudock, D. 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]

Christodoulides, D. N.

Eisenberg, H. S.

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]

Elazar, M.

Flach, S.

A. E. Miroshnichenko, S. Flach, and B. Malomed, "Resonant scattering of solitons," Chaos 13, 874-879 (2003).
[CrossRef] [PubMed]

Garnier, J.

J. Garnier and F. Kh. Abdullaev, "Transmission of matter-wave solitons through nonlinear traps and barriers," Phys. Rev. A 74, 013604 (2006).
[CrossRef]

Geniet, F.

F. Geniet and J. Leon, "Energy transmission in the forbidden band gap of a nonlinear chain," Phys. Rev. Lett. 89, 134102 (2002).
[CrossRef] [PubMed]

Goodman, R. H.

R. H. Goodman, P. J. Holmes, and M. I. Weinstein, "Strong NLS soliton-defect interactions," Physica D 192, 215-248 (2004).
[CrossRef]

Hache, A.

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, "Observation of discrete surface solitons," Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef] [PubMed]

K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache, "Discrete surface solitons," Opt. Lett. 30, 2466-2468 (2005).
[CrossRef] [PubMed]

Holmes, P. J.

R. H. Goodman, P. J. Holmes, and M. I. Weinstein, "Strong NLS soliton-defect interactions," Physica D 192, 215-248 (2004).
[CrossRef]

Hudock, J.

J. Meier, J. Hudock, D. 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]

Hutchings, D. C.

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, "The nonlinear optical properties of AlGaAs at the half band gap," IEEE J. Quant. Electron. 33, 341-348 (1997).
[CrossRef]

Joseph, R. I.

Kamburova, R. S.

K. T. Stoychev, M. T. Primatarowa, and R. S. Kamburova, "Resonant scattering of nonlinear Schrdinger solitons from potential wells," Phys. Rev. E 70, 066622 (2004).
[CrossRef]

Kang, J. U.

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, "The nonlinear optical properties of AlGaAs at the half band gap," IEEE J. Quant. Electron. 33, 341-348 (1997).
[CrossRef]

Kartashov, Y. V.

Korsch, H. J.

K. Rapedius, D. Witthaut, and H. J. Korsch, "Analytical study of resonant transport of Bose-Einstein condensates," Phys. Rev. A 73, 033608 (2006).
[CrossRef]

Lahini, Y.

D. Mandelik, Y. Lahini, and Y. Silberberg, "Nonlinearly induced relaxation to the ground state in a two-level system," Phys. Rev. Lett. 95, 073902 (2005).
[CrossRef] [PubMed]

Lee, C.

C. Lee and J. Brand, "Enhanced quantum reflection of matter-wave solitons," Europhys. Lett. 73, 321-327 (2006).
[CrossRef]

Leon, J.

F. Geniet and J. Leon, "Energy transmission in the forbidden band gap of a nonlinear chain," Phys. Rev. Lett. 89, 134102 (2002).
[CrossRef] [PubMed]

Linzon, Y.

Makris, K. G.

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, "Observation of discrete surface solitons," Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef] [PubMed]

K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache, "Discrete surface solitons," Opt. Lett. 30, 2466-2468 (2005).
[CrossRef] [PubMed]

Malomed, B.

A. E. Miroshnichenko, S. Flach, and B. Malomed, "Resonant scattering of solitons," Chaos 13, 874-879 (2003).
[CrossRef] [PubMed]

Malomed, B. A.

X. D. Cao and B. A. Malomed, "Soliton-defect collisions in the nonlinear Schrodinger equation," Phys. Lett. A 206, 177-182 (1995).
[CrossRef]

Mandelik, D.

D. Mandelik, Y. Lahini, and Y. Silberberg, "Nonlinearly induced relaxation to the ground state in a two-level system," Phys. Rev. Lett. 95, 073902 (2005).
[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]

Meier, J.

J. Meier, J. Hudock, D. 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]

Miroshnichenko, A. E.

A. E. Miroshnichenko, S. Flach, and B. Malomed, "Resonant scattering of solitons," Chaos 13, 874-879 (2003).
[CrossRef] [PubMed]

Morandotti, R.

Y. Shavit, Y. Linzon, S. Bar-Ad, R. Morandotti, M. Volatier, V. Aimez, and R. Ares, "Power-dependent switching of nonlinear trapping by local photonic potentials," Opt. Lett. 33, 1056-1058 (2008).
[CrossRef] [PubMed]

Y. Linzon, Y. Shavit, M. Elazar, R. Morandotti, M. Volatier-Ravat, V. Aimez, R. Ares, and S. Bar-Ad, "Single beam mapping of nonlinear phase shift profiles in planar waveguides with an embedded mirror," Opt. Express 15, 12068-12075 (2007).
[CrossRef] [PubMed]

Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, "Nonlinear scattering and trapping by local photonic potentials," Phys. Rev. Lett. 99, 133901 (2007).
[CrossRef] [PubMed]

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, "Observation of discrete surface solitons," Phys. Rev. Lett. 96, 063901 (2006).
[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]

J. Meier, J. Hudock, D. 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]

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]

Piette, B.

B. Piette, W. J. Zakrzewski, and J. Brand, "Scattering of topological solitons on holes and barriers," J. Phys. A 38, 1040310412 (2005).
[CrossRef]

Primatarowa, M. T.

K. T. Stoychev, M. T. Primatarowa, and R. S. Kamburova, "Resonant scattering of nonlinear Schrdinger solitons from potential wells," Phys. Rev. E 70, 066622 (2004).
[CrossRef]

Raikh, M.

Z. V. Vardeny and M. Raikh, "Photonics: Light localized on the lattice," Nature 446, 37-38 (2007).
[CrossRef] [PubMed]

Rapedius, K.

K. Rapedius, D. Witthaut, and H. J. Korsch, "Analytical study of resonant transport of Bose-Einstein condensates," Phys. Rev. A 73, 033608 (2006).
[CrossRef]

Sakaguchi, H.

H. Sakaguchi and M. Tamura, "Scattering of solitons and dark solitons by potential walls in the nonlinear schrodinger equation," J. Phys. Soc. Jpn. 74, 292-298 (2005).
[CrossRef]

H. Sakaguchi and M. Tamura, "Scattering and trapping of nonlinear schrodinger solitons in external potentials," J. Phys. Soc. Jpn. 73, 503-506 (2004).
[CrossRef]

Salamo, G.

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, "Observation of discrete surface solitons," Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef] [PubMed]

Shavit, Y.

Silberberg, Y.

D. Mandelik, Y. Lahini, and Y. Silberberg, "Nonlinearly induced relaxation to the ground state in a two-level system," Phys. Rev. Lett. 95, 073902 (2005).
[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]

J. Meier, J. Hudock, D. 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]

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]

Soffer, A.

A. Soffer and M. I. Weinstein, "Theory of nonlinear dispersive waves and selection of the ground state," Phys. Rev. Lett. 95, 213905 (2005).
[CrossRef] [PubMed]

Sorel, M.

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, "Observation of discrete surface solitons," Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef] [PubMed]

Stegeman, G.

J. Meier, J. Hudock, D. 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]

Stegeman, G. I.

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, "Observation of discrete surface solitons," Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef] [PubMed]

K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache, "Discrete surface solitons," Opt. Lett. 30, 2466-2468 (2005).
[CrossRef] [PubMed]

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, "The nonlinear optical properties of AlGaAs at the half band gap," IEEE J. Quant. Electron. 33, 341-348 (1997).
[CrossRef]

Stoychev, K. T.

K. T. Stoychev, M. T. Primatarowa, and R. S. Kamburova, "Resonant scattering of nonlinear Schrdinger solitons from potential wells," Phys. Rev. E 70, 066622 (2004).
[CrossRef]

Suntsov, S.

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, "Observation of discrete surface solitons," Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef] [PubMed]

K. G. Makris, S. Suntsov, D. N. Christodoulides, G. I. Stegeman, and A. Hache, "Discrete surface solitons," Opt. Lett. 30, 2466-2468 (2005).
[CrossRef] [PubMed]

Tamura, M.

H. Sakaguchi and M. Tamura, "Scattering of solitons and dark solitons by potential walls in the nonlinear schrodinger equation," J. Phys. Soc. Jpn. 74, 292-298 (2005).
[CrossRef]

H. Sakaguchi and M. Tamura, "Scattering and trapping of nonlinear schrodinger solitons in external potentials," J. Phys. Soc. Jpn. 73, 503-506 (2004).
[CrossRef]

Torner, L.

Vardeny, Z. V.

Z. V. Vardeny and M. Raikh, "Photonics: Light localized on the lattice," Nature 446, 37-38 (2007).
[CrossRef] [PubMed]

Villeneuve, A.

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, "The nonlinear optical properties of AlGaAs at the half band gap," IEEE J. Quant. Electron. 33, 341-348 (1997).
[CrossRef]

Volatier, M.

Y. Shavit, Y. Linzon, S. Bar-Ad, R. Morandotti, M. Volatier, V. Aimez, and R. Ares, "Power-dependent switching of nonlinear trapping by local photonic potentials," Opt. Lett. 33, 1056-1058 (2008).
[CrossRef] [PubMed]

Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, "Nonlinear scattering and trapping by local photonic potentials," Phys. Rev. Lett. 99, 133901 (2007).
[CrossRef] [PubMed]

Volatier-Ravat, M.

Vysloukh, V. A.

Weinstein, M. I.

A. Soffer and M. I. Weinstein, "Theory of nonlinear dispersive waves and selection of the ground state," Phys. Rev. Lett. 95, 213905 (2005).
[CrossRef] [PubMed]

R. H. Goodman, P. J. Holmes, and M. I. Weinstein, "Strong NLS soliton-defect interactions," Physica D 192, 215-248 (2004).
[CrossRef]

Witthaut, D.

K. Rapedius, D. Witthaut, and H. J. Korsch, "Analytical study of resonant transport of Bose-Einstein condensates," Phys. Rev. A 73, 033608 (2006).
[CrossRef]

Yang, H.

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, "Observation of discrete surface solitons," Phys. Rev. Lett. 96, 063901 (2006).
[CrossRef] [PubMed]

Zakrzewski, W. J.

B. Piette, W. J. Zakrzewski, and J. Brand, "Scattering of topological solitons on holes and barriers," J. Phys. A 38, 1040310412 (2005).
[CrossRef]

Chaos (1)

A. E. Miroshnichenko, S. Flach, and B. Malomed, "Resonant scattering of solitons," Chaos 13, 874-879 (2003).
[CrossRef] [PubMed]

Europhys. Lett. (1)

C. Lee and J. Brand, "Enhanced quantum reflection of matter-wave solitons," Europhys. Lett. 73, 321-327 (2006).
[CrossRef]

IEEE J. Quant. Electron. (1)

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, "The nonlinear optical properties of AlGaAs at the half band gap," IEEE J. Quant. Electron. 33, 341-348 (1997).
[CrossRef]

J. Phys. A (1)

B. Piette, W. J. Zakrzewski, and J. Brand, "Scattering of topological solitons on holes and barriers," J. Phys. A 38, 1040310412 (2005).
[CrossRef]

J. Phys. Soc. Jpn. (2)

H. Sakaguchi and M. Tamura, "Scattering and trapping of nonlinear schrodinger solitons in external potentials," J. Phys. Soc. Jpn. 73, 503-506 (2004).
[CrossRef]

H. Sakaguchi and M. Tamura, "Scattering of solitons and dark solitons by potential walls in the nonlinear schrodinger equation," J. Phys. Soc. Jpn. 74, 292-298 (2005).
[CrossRef]

Nature (1)

Z. V. Vardeny and M. Raikh, "Photonics: Light localized on the lattice," Nature 446, 37-38 (2007).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (3)

Phys. Lett. A (1)

X. D. Cao and B. A. Malomed, "Soliton-defect collisions in the nonlinear Schrodinger equation," Phys. Lett. A 206, 177-182 (1995).
[CrossRef]

Phys. Rev. A (2)

J. Garnier and F. Kh. Abdullaev, "Transmission of matter-wave solitons through nonlinear traps and barriers," Phys. Rev. A 74, 013604 (2006).
[CrossRef]

K. Rapedius, D. Witthaut, and H. J. Korsch, "Analytical study of resonant transport of Bose-Einstein condensates," Phys. Rev. A 73, 033608 (2006).
[CrossRef]

Phys. Rev. E (1)

K. T. Stoychev, M. T. Primatarowa, and R. S. Kamburova, "Resonant scattering of nonlinear Schrdinger solitons from potential wells," Phys. Rev. E 70, 066622 (2004).
[CrossRef]

Phys. Rev. Lett. (8)

Y. Linzon, R. Morandotti, M. Volatier, V. Aimez, R. Ares, and S. Bar-Ad, "Nonlinear scattering and trapping by local photonic potentials," Phys. Rev. Lett. 99, 133901 (2007).
[CrossRef] [PubMed]

D. Mandelik, Y. Lahini, and Y. Silberberg, "Nonlinearly induced relaxation to the ground state in a two-level system," Phys. Rev. Lett. 95, 073902 (2005).
[CrossRef] [PubMed]

A. Soffer and M. I. Weinstein, "Theory of nonlinear dispersive waves and selection of the ground state," Phys. Rev. Lett. 95, 213905 (2005).
[CrossRef] [PubMed]

J. Meier, J. Hudock, D. 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]

S. Suntsov, K. G. Makris, D. N. Christodoulides, G. I. Stegeman, A. Hache, R. Morandotti, H. Yang, G. Salamo, and M. Sorel, "Observation of discrete surface solitons," Phys. Rev. Lett. 96, 063901 (2006).
[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]

F. Geniet and J. Leon, "Energy transmission in the forbidden band gap of a nonlinear chain," Phys. Rev. Lett. 89, 134102 (2002).
[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]

Physica D (1)

R. H. Goodman, P. J. Holmes, and M. I. Weinstein, "Strong NLS soliton-defect interactions," Physica D 192, 215-248 (2004).
[CrossRef]

Other (1)

See, for example:P. P. Banerjee, Nonlinear Optics: Theory, Numerical Modeling and Applications (Taylor and Francis, London, 2007).

Cited By

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Figures (7)

Fig. 1.
Fig. 1.

Possible refractive index profiles (top) and their corresponding linear propagation spectra (bottom). An electric field of the form E(x, z) ~ A(x, z)eiβz is assumed, where β is the propagation constant. (a) Periodic modulations (top) which extend over a wide transverse region Δx define waveguide arrays. These structures are characterized by Band-Gap transmission spectra (bottom). kx denotes the beam’s transverse Bloch wave number. Localizations induced by the Kerr nonlinearity constitute new bound states, which form inside gaps of the linear system, either in the semi-infinite gap (above Band-1) upon direct excitation of the array (blue), or in a band gap (between bands) upon side excitation of the array (red). (b),(c) Local modulations (top) which are applied at a localized region Δx define scattering centers for side excitations (red), and may have linear normal modes for direct excitations (blue). Such structures are characterized by plane-wave transmission spectra (bottom) which plot the transmission coefficient T as a function of the propagation constant. Shaded green areas denotes gaps, the β values of which correspond to waves which cannot propagate in the structure. Dashed vertical lines denote normal modes of the defect.

Fig. 2.
Fig. 2.

(a)–(c): Wave optics approach. Shaded purple regions indicate spatial frequency content of a wave packet; (a) Top - A side-coupled excitation scattering by a local defect: at z=0 the beam is launched towards the defect with a transverse wave number kx =ksin θ. Upon its arrival at the interface (z=zs ) the beam is scattered by the local defect. Asymptotically (as z→∞) there remain transmitted (T) and reflected (R) wave packets. Bottom - T(β) spectrum of the single-site, two-mode defect, in which scattering states exist in β<k 0 ncl and normal modes in k 0 ncl <β<k 0 nd . (b) A beam which is coupled to Mode No. 2 has a double-hump intensity profile (top), and is localized around β=β 2 in the T(β) spectrum (bottom). (c) A beam which is coupled to Mode No. 1 has a single-hump intensity profile (top), and is localized around β=β 1 in the T(β) spectrum (bottom); (d)–(e): Ray optics approach; (d) In the linear regime, a ray incident from the left at an angle α refracts at an angle β and refracts again at the same angle γ=α. (e) In the nonlinear regime, the larger refractive index at the defect interface induces an increase in the angle β, such that γ>α. If the angle β becomes larger than the critical angle βc , this ray will trap inside the defect.

Fig. 3.
Fig. 3.

Normalized beam intensity propagation maps I(x, z) (0<z<15 mm,-0.2<x<0.2 mm) in the presence of a single-site defect, whose index gradient is Δn=4×10-3 (relative index step of 0.12%) and its width is Δx=6 µm. This defect supports 2 normal modes. The input Gaussian beam is of width w=20 µm and is centered at xc =-40 µm. The insets in panels (b),(d) and (f) show close ups of the intensity distribution within the defect. The excitation angles are (a),(b) θ=0.5°; (c),(d) θ=1.1°; (e),(f) θ=1.75°.

Fig. 4.
Fig. 4.

Propagation of (a) the relative fraction of power inside the defect, and (b) position of the power distribution’s center of mass, corresponding to the simulations of Fig. 3.

Fig. 5.
Fig. 5.

(a) Refractive index profile of the 3-site defect (top), and the corresponding 3-layer AlGaAs sample cross section (bottom). (b)–(e) Normalized beam intensity propagation (0<z<15 mm, -0.3<x<0.3 mm) in the presence of a 3-site defect, with parameters Δn=3.3×10-3 (0.1% index step), d=18 µm, w=4 µm and Δx=62 µm. The input Gaussian beam is of width w=40 µm, centered at xc =-40 µm and tilted at an angle of θ=1.34°. The input power densities are (b) 10 W; (c) 1.8 kW; (d) 3.4 kW; (e) 3.5 kW.

Fig. 6.
Fig. 6.

Propagation of (a) the relative power fraction inside the entire defect, and (b)–(e) the power fraction found within each site, corresponding to the simulations of Fig. 5.

Fig. 7.
Fig. 7.

(a) Illustration of a Fabry-Perot etalon, consisting of two parallel surfaces of reflectance ℜ coupled from the side. (b)–(d) Transmission (red triangle), reflection (green square), and trapping (blue circle) curves as a function of the index ratio n=nd /ncl , with input beam parameters w=20 µm, xc =-70 µm and θ=0.7°; The output observation plane is at Z=15.4 mm. (b) Low-power input (I 0=10 W) and (c) high-power input (I 0=2.5 kW) with a single-site scatterer (Δx=18 µm); (d) high power input (I 0=2.5 kW) with a 3-site scatterer [Δx=62 µm, d=18 µm and w=4 µm, see Fig. 5(a)].

Tables (1)

Tables Icon

Table 1. Physical parameters used in the simulations

Equations (7)

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2 i k 0 E ( x , z ) z + 2 E ( x , z ) x 2 + k 0 2 n 2 ( x ) E ( x , z ) + k 0 2 n 2 n 0 E ( x , z ) 2 E ( x , z ) = 0
I ( z = 0 ) = I 0 · exp [ ( x x c 2 ) w 2 + i φ 0 ( x ) ]
sin β c = n cl n d
sin α = n d n cl + Δ n nl sin β > n d n cl + Δ n nl · n cl n d
n ( x ) = { n cl x > Δ x 2 , x < Δ x 2 n d x < Δ x 2
n ( x ) = { n cl x < Δ x 2 , x > Δ x 2 , d 2 < x < d 2 + w , d 2 w < x < d 2 n d x < d 2 , d 2 + w < x < 3 d 2 + w , 3 d 2 w < x < d 2 w
T = ( 1 ) 2 1 + 2 2 cos [ 2 k 0 n Δ x cos θ ]

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