Abstract

We present an overview of the properties of nonlinear guided waves and (bright and dark) spatial optical solitons in a periodic medium created by linear and nonlinear waveguides. First we consider a single layer with a cubic nonlinear response (a nonlinear slab waveguide) embedded in a periodic layered linear medium and describe nonlinear localized modes (guided waves and Bragg-like localized gap modes) and their stability. Then we study modulational instability as well as the existence and stability of discrete spatial solitons in a periodic array of identical nonlinear layers, a one-dimensional model of nonlinear photonic crystals. We emphasize both similarities to and differences from the models described by the discrete nonlinear Schrödinger equation, which is derived in the tight-binding approximation, and the coupled-mode theory, which is valid for shallow periodic modulations.

© 2002 Optical Society of America

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2001 (10)

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86, 3296–3299 (2001).
[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]

S. F. Mingaleev and Yu. S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. 86, 5474–5477 (2001).
[CrossRef] [PubMed]

S. Lan, S. Nishikawa, and O. Wada, “Leveraging deep photonic band gaps in photonic crystal impurity bands,” Appl. Phys. Lett. 78, 2101–2103 (2001).
[CrossRef]

A. A. Sukhorukov, Yu. S. Kivshar, O. Bang, and C. M. Soukoulis, “Parametric localized modes in quadratic nonlinear photonic structures,” Phys. Rev. E 63, 016615–9 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of repulsive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 63, 036612–11 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615–9 (2001).
[CrossRef]

B. Wu and Q. Niu, “Landau and dynamical instabilities of the superflow of Bose–Einstein condensates in optical lattices,” Phys. Rev. A 64, 061603–4 (2001).
[CrossRef]

P. G. Kevrekidis, A. R. Bishop, and K. O. Rasmussen, “Twisted localized modes,” Phys. Rev. E 63, 036603–6 (2001).
[CrossRef]

T. Kapitula, P. G. Kevrekidis, and B. A. Malomed, “Stability of multiple pulses in discrete systems,” Phys. Rev. E 63, 036604–8 (2001).
[CrossRef]

2000 (2)

F. Barra, P. Gaspard, and S. Rica, “Nonlinear Schrödinger flow in a periodic potential,” Phys. Rev. E 61, 5852–5863 (2000).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef] [PubMed]

1999 (2)

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
[CrossRef]

1998 (5)

Yu. S. Kivshar, A. R. Champneys, D. Cai, and A. R. Bishop, “Multiple states of intrinsic localized modes,” Phys. Rev. B 58, 5423–5428 (1998).
[CrossRef]

Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81–197 (1998).
[CrossRef]

S. Darmanyan, A. Kobyakov, E. Schmidt, and F. Lederer, “Strongly localized vectorial modes in nonlinear waveguide arrays,” Phys. Rev. E 57, 3520–3530 (1998).
[CrossRef]

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]

C. M. de Sterke, “Theory of modulational instability in fiber Bragg gratings,” J. Opt. Soc. Am. B 15, 2660–2667 (1998).
[CrossRef]

1997 (1)

E. Lidorikis, K. Busch, Q. M. Li, C. T. Chan, and C. M. Soukoulis, “Optical nonlinear response of a single nonlinear dielectric layer sandwiched between two linear dielectric structures,” Phys. Rev. B 56, 15090–15099 (1997).
[CrossRef]

1996 (3)

E. Lidorikis, Q. M. Li, and C. M. Soukoulis, “Wave propagation in nonlinear multilayer structures,” Phys. Rev. B 54, 10249–10252 (1996).
[CrossRef]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

W. Krolikowski and Yu. S. Kivshar, “Soliton-based optical switching in waveguide arrays,” J. Opt. Soc. Am. B 13, 876–887 (1996).
[CrossRef]

1995 (1)

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear-optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

1994 (1)

Yu. S. Kivshar, W. Krolikowski, and O. A. Chubykalo, “Dark solitons in discrete lattices,” Phys. Rev. E 50, 5020–5032 (1994).
[CrossRef]

1993 (4)

1992 (2)

Yu. S. Kivshar and M. Peyrard, “Modulational instabilities in discrete lattices,” Phys. Rev. A 46, 3198–3205 (1992).
[CrossRef] [PubMed]

C. Wachter, F. Lederer, L. Leine, U. Trutschel, and M. Mann, “Nonlinear Bragg reflection wave-guide,” J. Appl. Phys. 71, 3688–3692 (1992).
[CrossRef]

1989 (1)

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[CrossRef] [PubMed]

1988 (1)

1987 (1)

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical-response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

1977 (1)

A. Y. Cho, A. Yariv, and P. Yeh, “Observation of confined propagation in Bragg waveguides,” Appl. Phys. Lett. 30, 471–472 (1977).
[CrossRef]

1976 (1)

P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

Aceves, A. B.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Aitchison, J. S.

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86, 3296–3299 (2001).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
[CrossRef]

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]

Bang, O.

A. A. Sukhorukov, Yu. S. Kivshar, O. Bang, and C. M. Soukoulis, “Parametric localized modes in quadratic nonlinear photonic structures,” Phys. Rev. E 63, 016615–9 (2001).
[CrossRef]

Barra, F.

F. Barra, P. Gaspard, and S. Rica, “Nonlinear Schrödinger flow in a periodic potential,” Phys. Rev. E 61, 5852–5863 (2000).
[CrossRef]

Bishop, A. R.

P. G. Kevrekidis, A. R. Bishop, and K. O. Rasmussen, “Twisted localized modes,” Phys. Rev. E 63, 036603–6 (2001).
[CrossRef]

Yu. S. Kivshar, A. R. Champneys, D. Cai, and A. R. Bishop, “Multiple states of intrinsic localized modes,” Phys. Rev. B 58, 5423–5428 (1998).
[CrossRef]

Bloemer, M. J.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear-optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

Botez, D.

Bowden, C. M.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear-optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

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]

Bronski, J. C.

J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of repulsive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 63, 036612–11 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615–9 (2001).
[CrossRef]

Busch, K.

E. Lidorikis, K. Busch, Q. M. Li, C. T. Chan, and C. M. Soukoulis, “Optical nonlinear response of a single nonlinear dielectric layer sandwiched between two linear dielectric structures,” Phys. Rev. B 56, 15090–15099 (1997).
[CrossRef]

Cai, D.

Yu. S. Kivshar, A. R. Champneys, D. Cai, and A. R. Bishop, “Multiple states of intrinsic localized modes,” Phys. Rev. B 58, 5423–5428 (1998).
[CrossRef]

Campbell, D. K.

Yu. S. Kivshar and D. K. Campbell, “Peierls–Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077–3081 (1993).
[CrossRef]

Carr, L. D.

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615–9 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of repulsive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 63, 036612–11 (2001).
[CrossRef]

Carretero-Gonzalez, R.

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615–9 (2001).
[CrossRef]

Champneys, A. R.

Yu. S. Kivshar, A. R. Champneys, D. Cai, and A. R. Bishop, “Multiple states of intrinsic localized modes,” Phys. Rev. B 58, 5423–5428 (1998).
[CrossRef]

Chan, C. T.

E. Lidorikis, K. Busch, Q. M. Li, C. T. Chan, and C. M. Soukoulis, “Optical nonlinear response of a single nonlinear dielectric layer sandwiched between two linear dielectric structures,” Phys. Rev. B 56, 15090–15099 (1997).
[CrossRef]

Chen, W.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical-response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Cho, A. Y.

A. Y. Cho, A. Yariv, and P. Yeh, “Observation of confined propagation in Bragg waveguides,” Appl. Phys. Lett. 30, 471–472 (1977).
[CrossRef]

Christodoulides, D. N.

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[CrossRef] [PubMed]

D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794–796 (1988).
[CrossRef] [PubMed]

Chubykalo, O. A.

Yu. S. Kivshar, W. Krolikowski, and O. A. Chubykalo, “Dark solitons in discrete lattices,” Phys. Rev. E 50, 5020–5032 (1994).
[CrossRef]

Darmanyan, S.

S. Darmanyan, A. Kobyakov, E. Schmidt, and F. Lederer, “Strongly localized vectorial modes in nonlinear waveguide arrays,” Phys. Rev. E 57, 3520–3530 (1998).
[CrossRef]

De Angelis, C.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

de Sterke, C. M.

Deconinck, B.

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615–9 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of repulsive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 63, 036612–11 (2001).
[CrossRef]

Dowling, J. P.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear-optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

Eisenberg, H. S.

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86, 3296–3299 (2001).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
[CrossRef]

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]

Gaspard, P.

F. Barra, P. Gaspard, and S. Rica, “Nonlinear Schrödinger flow in a periodic potential,” Phys. Rev. E 61, 5852–5863 (2000).
[CrossRef]

Grebel, H.

Joseph, R. I.

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[CrossRef] [PubMed]

D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794–796 (1988).
[CrossRef] [PubMed]

Kapitula, T.

T. Kapitula, P. G. Kevrekidis, and B. A. Malomed, “Stability of multiple pulses in discrete systems,” Phys. Rev. E 63, 036604–8 (2001).
[CrossRef]

Kevrekidis, P. G.

T. Kapitula, P. G. Kevrekidis, and B. A. Malomed, “Stability of multiple pulses in discrete systems,” Phys. Rev. E 63, 036604–8 (2001).
[CrossRef]

P. G. Kevrekidis, A. R. Bishop, and K. O. Rasmussen, “Twisted localized modes,” Phys. Rev. E 63, 036603–6 (2001).
[CrossRef]

Kivshar, Yu. S.

S. F. Mingaleev and Yu. S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. 86, 5474–5477 (2001).
[CrossRef] [PubMed]

A. A. Sukhorukov, Yu. S. Kivshar, O. Bang, and C. M. Soukoulis, “Parametric localized modes in quadratic nonlinear photonic structures,” Phys. Rev. E 63, 016615–9 (2001).
[CrossRef]

Yu. S. Kivshar, A. R. Champneys, D. Cai, and A. R. Bishop, “Multiple states of intrinsic localized modes,” Phys. Rev. B 58, 5423–5428 (1998).
[CrossRef]

Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81–197 (1998).
[CrossRef]

W. Krolikowski and Yu. S. Kivshar, “Soliton-based optical switching in waveguide arrays,” J. Opt. Soc. Am. B 13, 876–887 (1996).
[CrossRef]

Yu. S. Kivshar, W. Krolikowski, and O. A. Chubykalo, “Dark solitons in discrete lattices,” Phys. Rev. E 50, 5020–5032 (1994).
[CrossRef]

Yu. S. Kivshar, “Self-localization in arrays of defocusing wave-guides,” Opt. Lett. 18, 1147–1149 (1993).
[CrossRef]

Yu. S. Kivshar and D. K. Campbell, “Peierls–Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077–3081 (1993).
[CrossRef]

Yu. S. Kivshar and M. Peyrard, “Modulational instabilities in discrete lattices,” Phys. Rev. A 46, 3198–3205 (1992).
[CrossRef] [PubMed]

Kobyakov, A.

S. Darmanyan, A. Kobyakov, E. Schmidt, and F. Lederer, “Strongly localized vectorial modes in nonlinear waveguide arrays,” Phys. Rev. E 57, 3520–3530 (1998).
[CrossRef]

Krolikowski, W.

W. Krolikowski and Yu. S. Kivshar, “Soliton-based optical switching in waveguide arrays,” J. Opt. Soc. Am. B 13, 876–887 (1996).
[CrossRef]

Yu. S. Kivshar, W. Krolikowski, and O. A. Chubykalo, “Dark solitons in discrete lattices,” Phys. Rev. E 50, 5020–5032 (1994).
[CrossRef]

Kutz, J. N.

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615–9 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of repulsive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 63, 036612–11 (2001).
[CrossRef]

Lan, S.

S. Lan, S. Nishikawa, and O. Wada, “Leveraging deep photonic band gaps in photonic crystal impurity bands,” Appl. Phys. Lett. 78, 2101–2103 (2001).
[CrossRef]

Lederer, F.

S. Darmanyan, A. Kobyakov, E. Schmidt, and F. Lederer, “Strongly localized vectorial modes in nonlinear waveguide arrays,” Phys. Rev. E 57, 3520–3530 (1998).
[CrossRef]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

C. Wachter, F. Lederer, L. Leine, U. Trutschel, and M. Mann, “Nonlinear Bragg reflection wave-guide,” J. Appl. Phys. 71, 3688–3692 (1992).
[CrossRef]

Leine, L.

C. Wachter, F. Lederer, L. Leine, U. Trutschel, and M. Mann, “Nonlinear Bragg reflection wave-guide,” J. Appl. Phys. 71, 3688–3692 (1992).
[CrossRef]

Li, Q. M.

E. Lidorikis, K. Busch, Q. M. Li, C. T. Chan, and C. M. Soukoulis, “Optical nonlinear response of a single nonlinear dielectric layer sandwiched between two linear dielectric structures,” Phys. Rev. B 56, 15090–15099 (1997).
[CrossRef]

E. Lidorikis, Q. M. Li, and C. M. Soukoulis, “Wave propagation in nonlinear multilayer structures,” Phys. Rev. B 54, 10249–10252 (1996).
[CrossRef]

Lidorikis, E.

E. Lidorikis, K. Busch, Q. M. Li, C. T. Chan, and C. M. Soukoulis, “Optical nonlinear response of a single nonlinear dielectric layer sandwiched between two linear dielectric structures,” Phys. Rev. B 56, 15090–15099 (1997).
[CrossRef]

E. Lidorikis, Q. M. Li, and C. M. Soukoulis, “Wave propagation in nonlinear multilayer structures,” Phys. Rev. B 54, 10249–10252 (1996).
[CrossRef]

Luther-Davies, B.

Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81–197 (1998).
[CrossRef]

Malomed, B. A.

T. Kapitula, P. G. Kevrekidis, and B. A. Malomed, “Stability of multiple pulses in discrete systems,” Phys. Rev. E 63, 036604–8 (2001).
[CrossRef]

Mann, M.

C. Wachter, F. Lederer, L. Leine, U. Trutschel, and M. Mann, “Nonlinear Bragg reflection wave-guide,” J. Appl. Phys. 71, 3688–3692 (1992).
[CrossRef]

Mills, D. L.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical-response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Mingaleev, S. F.

S. F. Mingaleev and Yu. S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. 86, 5474–5477 (2001).
[CrossRef] [PubMed]

Morandotti, R.

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86, 3296–3299 (2001).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
[CrossRef]

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]

Muschall, R.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Nabiev, R. F.

Nishikawa, S.

S. Lan, S. Nishikawa, and O. Wada, “Leveraging deep photonic band gaps in photonic crystal impurity bands,” Appl. Phys. Lett. 78, 2101–2103 (2001).
[CrossRef]

Niu, Q.

B. Wu and Q. Niu, “Landau and dynamical instabilities of the superflow of Bose–Einstein condensates in optical lattices,” Phys. Rev. A 64, 061603–4 (2001).
[CrossRef]

Peschel, T.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Peschel, U.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
[CrossRef]

Peyrard, M.

Yu. S. Kivshar and M. Peyrard, “Modulational instabilities in discrete lattices,” Phys. Rev. A 46, 3198–3205 (1992).
[CrossRef] [PubMed]

Promislow, K.

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615–9 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of repulsive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 63, 036612–11 (2001).
[CrossRef]

Rasmussen, K. O.

P. G. Kevrekidis, A. R. Bishop, and K. O. Rasmussen, “Twisted localized modes,” Phys. Rev. E 63, 036603–6 (2001).
[CrossRef]

Rica, S.

F. Barra, P. Gaspard, and S. Rica, “Nonlinear Schrödinger flow in a periodic potential,” Phys. Rev. E 61, 5852–5863 (2000).
[CrossRef]

Scalora, M.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear-optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

Schmidt, E.

S. Darmanyan, A. Kobyakov, E. Schmidt, and F. Lederer, “Strongly localized vectorial modes in nonlinear waveguide arrays,” Phys. Rev. E 57, 3520–3530 (1998).
[CrossRef]

Silberberg, Y.

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86, 3296–3299 (2001).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
[CrossRef]

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]

Smerzi, A.

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

Sorel, M.

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86, 3296–3299 (2001).
[CrossRef] [PubMed]

Soukoulis, C. M.

A. A. Sukhorukov, Yu. S. Kivshar, O. Bang, and C. M. Soukoulis, “Parametric localized modes in quadratic nonlinear photonic structures,” Phys. Rev. E 63, 016615–9 (2001).
[CrossRef]

E. Lidorikis, K. Busch, Q. M. Li, C. T. Chan, and C. M. Soukoulis, “Optical nonlinear response of a single nonlinear dielectric layer sandwiched between two linear dielectric structures,” Phys. Rev. B 56, 15090–15099 (1997).
[CrossRef]

E. Lidorikis, Q. M. Li, and C. M. Soukoulis, “Wave propagation in nonlinear multilayer structures,” Phys. Rev. B 54, 10249–10252 (1996).
[CrossRef]

Sukhorukov, A. A.

A. A. Sukhorukov, Yu. S. Kivshar, O. Bang, and C. M. Soukoulis, “Parametric localized modes in quadratic nonlinear photonic structures,” Phys. Rev. E 63, 016615–9 (2001).
[CrossRef]

Tocci, M. D.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear-optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

Trillo, S.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Trombettoni, A.

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

Trutschel, U.

C. Wachter, F. Lederer, L. Leine, U. Trutschel, and M. Mann, “Nonlinear Bragg reflection wave-guide,” J. Appl. Phys. 71, 3688–3692 (1992).
[CrossRef]

Wabnitz, S.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Wachter, C.

C. Wachter, F. Lederer, L. Leine, U. Trutschel, and M. Mann, “Nonlinear Bragg reflection wave-guide,” J. Appl. Phys. 71, 3688–3692 (1992).
[CrossRef]

Wada, O.

S. Lan, S. Nishikawa, and O. Wada, “Leveraging deep photonic band gaps in photonic crystal impurity bands,” Appl. Phys. Lett. 78, 2101–2103 (2001).
[CrossRef]

Wu, B.

B. Wu and Q. Niu, “Landau and dynamical instabilities of the superflow of Bose–Einstein condensates in optical lattices,” Phys. Rev. A 64, 061603–4 (2001).
[CrossRef]

Yariv, A.

A. Y. Cho, A. Yariv, and P. Yeh, “Observation of confined propagation in Bragg waveguides,” Appl. Phys. Lett. 30, 471–472 (1977).
[CrossRef]

P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

Yeh, P.

R. F. Nabiev, P. Yeh, and D. Botez, “Spatial gap solitons in periodic nonlinear structures,” Opt. Lett. 18, 1612–1614 (1993).
[CrossRef] [PubMed]

A. Y. Cho, A. Yariv, and P. Yeh, “Observation of confined propagation in Bragg waveguides,” Appl. Phys. Lett. 30, 471–472 (1977).
[CrossRef]

P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

Zhong, W.

Appl. Phys. Lett. (4)

A. Y. Cho, A. Yariv, and P. Yeh, “Observation of confined propagation in Bragg waveguides,” Appl. Phys. Lett. 30, 471–472 (1977).
[CrossRef]

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear-optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

S. Lan, S. Nishikawa, and O. Wada, “Leveraging deep photonic band gaps in photonic crystal impurity bands,” Appl. Phys. Lett. 78, 2101–2103 (2001).
[CrossRef]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
[CrossRef]

J. Appl. Phys. (1)

C. Wachter, F. Lederer, L. Leine, U. Trutschel, and M. Mann, “Nonlinear Bragg reflection wave-guide,” J. Appl. Phys. 71, 3688–3692 (1992).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

Opt. Lett. (4)

Phys. Rep. (1)

Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81–197 (1998).
[CrossRef]

Phys. Rev. A (2)

B. Wu and Q. Niu, “Landau and dynamical instabilities of the superflow of Bose–Einstein condensates in optical lattices,” Phys. Rev. A 64, 061603–4 (2001).
[CrossRef]

Yu. S. Kivshar and M. Peyrard, “Modulational instabilities in discrete lattices,” Phys. Rev. A 46, 3198–3205 (1992).
[CrossRef] [PubMed]

Phys. Rev. B (3)

Yu. S. Kivshar, A. R. Champneys, D. Cai, and A. R. Bishop, “Multiple states of intrinsic localized modes,” Phys. Rev. B 58, 5423–5428 (1998).
[CrossRef]

E. Lidorikis, K. Busch, Q. M. Li, C. T. Chan, and C. M. Soukoulis, “Optical nonlinear response of a single nonlinear dielectric layer sandwiched between two linear dielectric structures,” Phys. Rev. B 56, 15090–15099 (1997).
[CrossRef]

E. Lidorikis, Q. M. Li, and C. M. Soukoulis, “Wave propagation in nonlinear multilayer structures,” Phys. Rev. B 54, 10249–10252 (1996).
[CrossRef]

Phys. Rev. E (10)

A. A. Sukhorukov, Yu. S. Kivshar, O. Bang, and C. M. Soukoulis, “Parametric localized modes in quadratic nonlinear photonic structures,” Phys. Rev. E 63, 016615–9 (2001).
[CrossRef]

Yu. S. Kivshar and D. K. Campbell, “Peierls–Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077–3081 (1993).
[CrossRef]

J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of repulsive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 63, 036612–11 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615–9 (2001).
[CrossRef]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

S. Darmanyan, A. Kobyakov, E. Schmidt, and F. Lederer, “Strongly localized vectorial modes in nonlinear waveguide arrays,” Phys. Rev. E 57, 3520–3530 (1998).
[CrossRef]

Yu. S. Kivshar, W. Krolikowski, and O. A. Chubykalo, “Dark solitons in discrete lattices,” Phys. Rev. E 50, 5020–5032 (1994).
[CrossRef]

F. Barra, P. Gaspard, and S. Rica, “Nonlinear Schrödinger flow in a periodic potential,” Phys. Rev. E 61, 5852–5863 (2000).
[CrossRef]

P. G. Kevrekidis, A. R. Bishop, and K. O. Rasmussen, “Twisted localized modes,” Phys. Rev. E 63, 036603–6 (2001).
[CrossRef]

T. Kapitula, P. G. Kevrekidis, and B. A. Malomed, “Stability of multiple pulses in discrete systems,” Phys. Rev. E 63, 036604–8 (2001).
[CrossRef]

Phys. Rev. Lett. (8)

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical-response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[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]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86, 3296–3299 (2001).
[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]

S. F. Mingaleev and Yu. S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. 86, 5474–5477 (2001).
[CrossRef] [PubMed]

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B. Luther-Davies and G. I. Stegeman, “Materials for spatial solitons,” in Spatial Optical Solitons, S. Trillo and W. E. Torruellas, eds. (Springer-Verlag, New York, 2001), pp. 19–35.

Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin, “Stationary waves in non-linear, periodically modulated media with higher group retardation,” Zh. Tekh. Fiz. 51, 902–907 (1981) [Tech. Phys. 26, 541–544 (1981)].

C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1994), Vol. XXXIII, pp. 203–260.

F. Lederer, S. Darmanyan, and A. Kobyakov, “Discrete solitons,” in Spatial Optical Solitons, S. Trillo and W. E. Torruellas, eds. (Springer-Verlag, New York, 2001), pp. 269–292.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1992).

S. Darmanyan, A. Kobyakov, and F. Lederer, “Stability of strongly localized excitations in discrete media with cubic nonlinearity,” Zh. Eksp. Teor. Fiz. 113, 1253–1260 (1998) [JETP 86, 682–686 (1998)].
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Yu. S. Kivshar and A. A. Sukhorukov, “Stability of spatial optical solitons,” in Spatial Optical Solitons, S. Trillo and W. E. Torruellas, eds. (Springer-Verlag, New York, 2001), pp. 211–245.

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Figures (13)

Fig. 1
Fig. 1

Characteristic dependence of ζ on β for localized states. Waveguiding (white areas, ζ>0) and antiwaveguiding (dotted area, ζ<0) localization regimes inside the band gaps are shown. The lattice parameters are h=1 and ν(x)=0 for n-1/2<x/h<n and ν(x)=30 for n<x/h<n+1/2, where n is an integer, α=0.5, and γ=1.

Fig. 2
Fig. 2

Top, power versus propagation constant for the nonlinear localized states: solid curves, stable; dashed curve, unstable; dotted curve, oscillatory unstable. Middle, real (dotted curves), and imaginary (solid curves) parts of the eigenvalues associated with wave instability. Waveguiding (white areas) and antiwaveguiding (dotted area) localization regimes inside the band gaps are shown. Bottom, localized states that correspond to points marked a and b at the top. The lattice parameters are the same as in Fig. 1.

Fig. 3
Fig. 3

Example of the resonance that occurs between an internal mode of the localized state and a bandgap edge, leading to oscillatory instability.

Fig. 4
Fig. 4

Top, power versus propagation constant. (a)–(c) Localized mode profiles for α=-5 and γ=+1. Notation is the same as in Fig. 2.

Fig. 5
Fig. 5

Characteristic dependencies of parameters η and ξ on propagation constant β. White areas mark bandgaps. The lattice parameters are h=0.5 and α=10.

Fig. 6
Fig. 6

Development of modulational instability in a self-focusing medium for a slightly perturbed unstaggered BW solution with I00.44. The lattice parameters are α=3, h=0.5, and γ=+1.

Fig. 7
Fig. 7

Modulationally unstable staggered BW modes (shaded area) in a self-focusing medium, shown as intensity (I0) versus lattice parameter α (at h=0.5 and γ=+1). Dashed curve, analytical approximation (16) for the low-intensity instability threshold.

Fig. 8
Fig. 8

Unstable modulation frequencies (shaded area) versus intensity I0 of the staggered BW modes (α=3 and h=0.5). Dark, solid curve, instability frequency with the largest growth rate.

Fig. 9
Fig. 9

Development of the instability-induced period-doubling modulations. The initial profile corresponds to a slightly perturbed staggered solution with I029.87. The parameters are the same as in Fig. 8.

Fig. 10
Fig. 10

Top, power versus propagation constant for odd (dark curve) and even (light curve) localized modes in a self-focusing (γ=+1) regime: solid curve stable; dashed curve, unstable. Bottom, profiles of the localized modes that correspond to marked points a and b at the top. The lattice parameters are the same as in Fig. 5.

Fig. 11
Fig. 11

Top, power versus propagation constant in the self-defocusing (γ=-1) regime. Notation is the same as in Fig. 10; dotted curve, oscillatory unstable modes.

Fig. 12
Fig. 12

Top, complementary power versus propagation constant for odd (dark curve) and even (lighter curve) dark localized solitons in a self-focusing (γ=+1) regime. Notation is the same as in Fig. 10.

Fig. 13
Fig. 13

Top, complementary power versus propagation constant in the self-defocusing (γ=-1) regime. Notation is the same as in Fig. 12.

Equations (29)

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i EZ+D 2EX2+(X)E+g(X)|E|2E=0,
i ψz+2ψx2+F(I; x)ψ=0,
P=-+|ψ(x, z)|2dx
ψ(x, z)=u(x;β)exp(iβz),
-βu+d2udx2+F(I; x)u=0.
F(I; x)=ν(x)+δ(x)G(I),
ub(x;β)=a(x;β)exp[-μ(x;β)x]+b(x;β)exp[+μ(x;β)x],
T(x;β)1r(x;β)=τ(β)1r(x;β),
G0G(I0)=ζ(β),
ζ±=μ± (1-r±)(1+r±);
ψ(x, t)=[u(x)+v(x)exp(iΓz)+w*(x)exp(-iΓ*z)]exp(iβz),
-(β+Γ)v+d2vdx2+ν(x)v+δ(x)[G1v
+(G1-G0)w]=0,
-(β-Γ)w+d2wdx2+ν(x)w+δ(x)[G1w
+(G1-G0)v]=0,
Y(Γ)=[G1-ζ(β+Γ)][G1-ζ(β-Γ)]-(G1-G0)2.
F(I; x)=n(α+γI)δ(x-hn),
ηUn+(Un-1+Un+1)+χ|Un|2Un=0.
η=-2 cosh(hμ)+αξ,
ξ=sinh(hμ)/μ,μ=β.
I0(β)=-[2 cos K+η(β)]γξ(β).
[η(β+Γ)+2γξ(β+Γ)I0+2 cos(q+K)]×[η(β-Γ)+2γξ(β-Γ)I0+2 cos(q-K)]
=γ2ξ(β+Γ)ξ(β-Γ)I02.
γI0(cr)α+22hα3/2/π+O(α2).
-(β+Γ)v+d2vdx2+n[(α+2γun2)v+γun2w]
×δ(x-hn)=0,
-(β-Γ)w+d2wdx2+n[(α+2γun2)w+γun2v]
×δ(x-hn)=0,
Pc=limn+-nh+nh[|u(x+2nh)|2-|u(x)|2]dx,

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