Abstract

Localized energy states such as two-color gap solitons are theoretically and numerically predicted in a periodic structure in the presence of a frequency doubling nonlinearity. These parametric solitons exhibit appealing features as compared to the Kerr case. Novel effects such as merging and all-optical buffering are envisaged.

© 1998 Optical Society of America

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  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]
  2. C. M. De Sterke and J. E. Sipe, in Progress in Optics XXXIII, E. Wolf ed., (Elsevier, Amsterdam, 1994), Chap. III.
  3. B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
    [CrossRef]
  4. D. Taverner, N. G. R. Broderick, D. J. Richardson, R. I. Laming, and M. Ibsen, “Nonlinear self-switching and multiple gap-soliton formation in a fiber Bragg grating,” Opt. Lett. 23, 328–330 (1998).
    [CrossRef]
  5. Y. Kivshar, “Gap solitons due to cascading,” Phys. Rev. E 51, 1613–1615 (1995).
    [CrossRef]
  6. C. Conti, S. Trillo, and G. Assanto, “Doubly resonant Bragg simultons via second-harmonic generation,” Phys. Rev. Lett. 78, 2341–2344 (1997).
    [CrossRef]
  7. H. He and P. D. Drummond, “Ideal soliton environment using parametric band gaps,” Phys. Rev. Lett. 78, 4311–4314 (1997).
    [CrossRef]
  8. T. Peschel, U. Peschel, F. Lederer, and B. Malomed, “Solitary waves in Bragg gratings with a quadratic nonlinearity,” Phys. Rev. E 55, 4730–4739 (1997).
    [CrossRef]
  9. C. Conti, S. Trillo, and G. Assanto, “Bloch functions approach for parametric gap solitons,” Opt. Lett. 22, 445–447 (1997).
    [CrossRef] [PubMed]
  10. C. Conti, S. Trillo, and G. Assanto, “Optical gap solitons via second-harmonic generation: exact solitary solutions,” Phys. Rev. E,  57, R1251–R1254 (1998).
    [CrossRef]
  11. C. Conti, S. Trillo, and G. Assanto, “Trapping of slowly moving or stationary two-color gap solitons,” Opt. Lett. 23, 334–336 (1998).
    [CrossRef]
  12. C. Conti, G. Assanto, and S. Trillo, “Read/write all-optical buffer by self trapped gap simultons,” Electron. Lett. 34, 689–690 (1998).
    [CrossRef]
  13. C. Conti, G. Assanto, and S. Trillo, “Excitation of self-transparency Bragg solitons in quadratic media,” Opt. Lett. 22, 1350–1352 (1997).
    [CrossRef]
  14. C. Conti, A. De Rossi, and S. Trillo, “Existence, bistability, and instability of Kerr-like parametric gap solitons in quadratic media,” Opt. Lett. 23, 1265–1267 (1998).
    [CrossRef]
  15. W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
    [CrossRef] [PubMed]
  16. R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138 (1996).
    [CrossRef]
  17. C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E 54, 1969–1989 (1996).
    [CrossRef]
  18. J. Söchtiget al., Electron. Lett. 31, 551–552 (1995).
    [CrossRef]
  19. Z. Weissman et al., “Second-harmonic generation in Bragg-resonant quasi-phase-matched periodically segmented waveguide,” Opt. Lett. 20, 674–676 (1995).
    [CrossRef] [PubMed]
  20. P. G. Kazansky and V. Pruneri, “Electric-field poling of quasi-phase-matched optical fibers,” J. Opt. Soc. Am. B,  14, 3170–3179 (1997).
    [CrossRef]
  21. Ch. Becker, A. Greiner, Th. Oesselke, A. Pape, W. Sohler, and H. Suche, “Integrated optical Ti:Er:LiNbO3 distribuited Bragg reflector laser with a fixed photorefractive grating,” Opt. Lett. 23, 1195–1197 (1998).
    [CrossRef]
  22. A. V. Buryak and Y. S. Kivshar, “Spatial optical solitons governed by quadratic nonlinearity,” Opt. Lett. 19, 1612–1614 (1994).
    [CrossRef] [PubMed]
  23. Guided-Wave Optoelectronic, chapter 2, T. Tamir ed., (Springer, Berlin, 1980).
  24. L. Torner, D. Mihalache, D. Mazilu, M. C. Santos, and N. N. Akhmediev, “Walking solitons in quadratic nonlinear media,” J. Opt. Soc. Am. B 15, 1476–1487 (1998).
    [CrossRef]
  25. C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, “Stability of temporal chirped solitary waves in quadratically nonlinear media,” Phys. Rev. E 55, 6155–6161 (1997).
    [CrossRef]
  26. C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, “Collision of solitary waves in media with a second-order nonlinearity,” Phys. Rev. A 52, R3444–R3447 (1995).
    [CrossRef] [PubMed]

1998 (7)

D. Taverner, N. G. R. Broderick, D. J. Richardson, R. I. Laming, and M. Ibsen, “Nonlinear self-switching and multiple gap-soliton formation in a fiber Bragg grating,” Opt. Lett. 23, 328–330 (1998).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, “Optical gap solitons via second-harmonic generation: exact solitary solutions,” Phys. Rev. E,  57, R1251–R1254 (1998).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, “Trapping of slowly moving or stationary two-color gap solitons,” Opt. Lett. 23, 334–336 (1998).
[CrossRef]

C. Conti, G. Assanto, and S. Trillo, “Read/write all-optical buffer by self trapped gap simultons,” Electron. Lett. 34, 689–690 (1998).
[CrossRef]

C. Conti, A. De Rossi, and S. Trillo, “Existence, bistability, and instability of Kerr-like parametric gap solitons in quadratic media,” Opt. Lett. 23, 1265–1267 (1998).
[CrossRef]

Ch. Becker, A. Greiner, Th. Oesselke, A. Pape, W. Sohler, and H. Suche, “Integrated optical Ti:Er:LiNbO3 distribuited Bragg reflector laser with a fixed photorefractive grating,” Opt. Lett. 23, 1195–1197 (1998).
[CrossRef]

L. Torner, D. Mihalache, D. Mazilu, M. C. Santos, and N. N. Akhmediev, “Walking solitons in quadratic nonlinear media,” J. Opt. Soc. Am. B 15, 1476–1487 (1998).
[CrossRef]

1997 (8)

C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, “Stability of temporal chirped solitary waves in quadratically nonlinear media,” Phys. Rev. E 55, 6155–6161 (1997).
[CrossRef]

P. G. Kazansky and V. Pruneri, “Electric-field poling of quasi-phase-matched optical fibers,” J. Opt. Soc. Am. B,  14, 3170–3179 (1997).
[CrossRef]

C. Conti, G. Assanto, and S. Trillo, “Excitation of self-transparency Bragg solitons in quadratic media,” Opt. Lett. 22, 1350–1352 (1997).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, “Doubly resonant Bragg simultons via second-harmonic generation,” Phys. Rev. Lett. 78, 2341–2344 (1997).
[CrossRef]

H. He and P. D. Drummond, “Ideal soliton environment using parametric band gaps,” Phys. Rev. Lett. 78, 4311–4314 (1997).
[CrossRef]

T. Peschel, U. Peschel, F. Lederer, and B. Malomed, “Solitary waves in Bragg gratings with a quadratic nonlinearity,” Phys. Rev. E 55, 4730–4739 (1997).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, “Bloch functions approach for parametric gap solitons,” Opt. Lett. 22, 445–447 (1997).
[CrossRef] [PubMed]

1996 (2)

R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138 (1996).
[CrossRef]

C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E 54, 1969–1989 (1996).
[CrossRef]

1995 (5)

J. Söchtiget al., Electron. Lett. 31, 551–552 (1995).
[CrossRef]

Z. Weissman et al., “Second-harmonic generation in Bragg-resonant quasi-phase-matched periodically segmented waveguide,” Opt. Lett. 20, 674–676 (1995).
[CrossRef] [PubMed]

Y. Kivshar, “Gap solitons due to cascading,” Phys. Rev. E 51, 1613–1615 (1995).
[CrossRef]

C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, “Collision of solitary waves in media with a second-order nonlinearity,” Phys. Rev. A 52, R3444–R3447 (1995).
[CrossRef] [PubMed]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

1994 (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]

Akhmediev, N. N.

Assanto, G.

C. Conti, S. Trillo, and G. Assanto, “Optical gap solitons via second-harmonic generation: exact solitary solutions,” Phys. Rev. E,  57, R1251–R1254 (1998).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, “Trapping of slowly moving or stationary two-color gap solitons,” Opt. Lett. 23, 334–336 (1998).
[CrossRef]

C. Conti, G. Assanto, and S. Trillo, “Read/write all-optical buffer by self trapped gap simultons,” Electron. Lett. 34, 689–690 (1998).
[CrossRef]

C. Conti, G. Assanto, and S. Trillo, “Excitation of self-transparency Bragg solitons in quadratic media,” Opt. Lett. 22, 1350–1352 (1997).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, “Bloch functions approach for parametric gap solitons,” Opt. Lett. 22, 445–447 (1997).
[CrossRef] [PubMed]

C. Conti, S. Trillo, and G. Assanto, “Doubly resonant Bragg simultons via second-harmonic generation,” Phys. Rev. Lett. 78, 2341–2344 (1997).
[CrossRef]

Baek, Y.

R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138 (1996).
[CrossRef]

Becker, Ch.

Ch. Becker, A. Greiner, Th. Oesselke, A. Pape, W. Sohler, and H. Suche, “Integrated optical Ti:Er:LiNbO3 distribuited Bragg reflector laser with a fixed photorefractive grating,” Opt. Lett. 23, 1195–1197 (1998).
[CrossRef]

Broderick, N. G. R.

Buryak, A. V.

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]

Conti, C.

C. Conti, S. Trillo, and G. Assanto, “Optical gap solitons via second-harmonic generation: exact solitary solutions,” Phys. Rev. E,  57, R1251–R1254 (1998).
[CrossRef]

C. Conti, G. Assanto, and S. Trillo, “Read/write all-optical buffer by self trapped gap simultons,” Electron. Lett. 34, 689–690 (1998).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, “Trapping of slowly moving or stationary two-color gap solitons,” Opt. Lett. 23, 334–336 (1998).
[CrossRef]

C. Conti, A. De Rossi, and S. Trillo, “Existence, bistability, and instability of Kerr-like parametric gap solitons in quadratic media,” Opt. Lett. 23, 1265–1267 (1998).
[CrossRef]

C. Conti, G. Assanto, and S. Trillo, “Excitation of self-transparency Bragg solitons in quadratic media,” Opt. Lett. 22, 1350–1352 (1997).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, “Bloch functions approach for parametric gap solitons,” Opt. Lett. 22, 445–447 (1997).
[CrossRef] [PubMed]

C. Conti, S. Trillo, and G. Assanto, “Doubly resonant Bragg simultons via second-harmonic generation,” Phys. Rev. Lett. 78, 2341–2344 (1997).
[CrossRef]

De Rossi, A.

de Sterke, C. M.

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E 54, 1969–1989 (1996).
[CrossRef]

C. M. De Sterke and J. E. Sipe, in Progress in Optics XXXIII, E. Wolf ed., (Elsevier, Amsterdam, 1994), Chap. III.

Drummond, P. D.

H. He and P. D. Drummond, “Ideal soliton environment using parametric band gaps,” Phys. Rev. Lett. 78, 4311–4314 (1997).
[CrossRef]

Eggleton, B. J.

Etrich, C.

C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, “Stability of temporal chirped solitary waves in quadratically nonlinear media,” Phys. Rev. E 55, 6155–6161 (1997).
[CrossRef]

C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, “Collision of solitary waves in media with a second-order nonlinearity,” Phys. Rev. A 52, R3444–R3447 (1995).
[CrossRef] [PubMed]

Greiner, A.

Ch. Becker, A. Greiner, Th. Oesselke, A. Pape, W. Sohler, and H. Suche, “Integrated optical Ti:Er:LiNbO3 distribuited Bragg reflector laser with a fixed photorefractive grating,” Opt. Lett. 23, 1195–1197 (1998).
[CrossRef]

Hagan, D. J.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

He, H.

H. He and P. D. Drummond, “Ideal soliton environment using parametric band gaps,” Phys. Rev. Lett. 78, 4311–4314 (1997).
[CrossRef]

Ibsen, M.

Kazansky, P. G.

Kivshar, Y.

Y. Kivshar, “Gap solitons due to cascading,” Phys. Rev. E 51, 1613–1615 (1995).
[CrossRef]

Kivshar, Y. S.

Laming, R. I.

Lederer, F.

T. Peschel, U. Peschel, F. Lederer, and B. Malomed, “Solitary waves in Bragg gratings with a quadratic nonlinearity,” Phys. Rev. E 55, 4730–4739 (1997).
[CrossRef]

C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, “Stability of temporal chirped solitary waves in quadratically nonlinear media,” Phys. Rev. E 55, 6155–6161 (1997).
[CrossRef]

C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, “Collision of solitary waves in media with a second-order nonlinearity,” Phys. Rev. A 52, R3444–R3447 (1995).
[CrossRef] [PubMed]

Malomed, B.

T. Peschel, U. Peschel, F. Lederer, and B. Malomed, “Solitary waves in Bragg gratings with a quadratic nonlinearity,” Phys. Rev. E 55, 4730–4739 (1997).
[CrossRef]

Malomed, B. A.

C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, “Stability of temporal chirped solitary waves in quadratically nonlinear media,” Phys. Rev. E 55, 6155–6161 (1997).
[CrossRef]

C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, “Collision of solitary waves in media with a second-order nonlinearity,” Phys. Rev. A 52, R3444–R3447 (1995).
[CrossRef] [PubMed]

Mazilu, D.

Menyuk, C. R.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Mihalache, D.

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]

Oesselke, Th.

Ch. Becker, A. Greiner, Th. Oesselke, A. Pape, W. Sohler, and H. Suche, “Integrated optical Ti:Er:LiNbO3 distribuited Bragg reflector laser with a fixed photorefractive grating,” Opt. Lett. 23, 1195–1197 (1998).
[CrossRef]

Pape, A.

Ch. Becker, A. Greiner, Th. Oesselke, A. Pape, W. Sohler, and H. Suche, “Integrated optical Ti:Er:LiNbO3 distribuited Bragg reflector laser with a fixed photorefractive grating,” Opt. Lett. 23, 1195–1197 (1998).
[CrossRef]

Peschel, T.

T. Peschel, U. Peschel, F. Lederer, and B. Malomed, “Solitary waves in Bragg gratings with a quadratic nonlinearity,” Phys. Rev. E 55, 4730–4739 (1997).
[CrossRef]

Peschel, U.

T. Peschel, U. Peschel, F. Lederer, and B. Malomed, “Solitary waves in Bragg gratings with a quadratic nonlinearity,” Phys. Rev. E 55, 4730–4739 (1997).
[CrossRef]

C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, “Stability of temporal chirped solitary waves in quadratically nonlinear media,” Phys. Rev. E 55, 6155–6161 (1997).
[CrossRef]

C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, “Collision of solitary waves in media with a second-order nonlinearity,” Phys. Rev. A 52, R3444–R3447 (1995).
[CrossRef] [PubMed]

Pruneri, V.

Richardson, D. J.

Salinas, D. G.

C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E 54, 1969–1989 (1996).
[CrossRef]

Santos, M. C.

Schiek, R.

R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138 (1996).
[CrossRef]

Sipe, J. E.

C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E 54, 1969–1989 (1996).
[CrossRef]

C. M. De Sterke and J. E. Sipe, in Progress in Optics XXXIII, E. Wolf ed., (Elsevier, Amsterdam, 1994), Chap. III.

Slusher, R. E.

Söchtig, J.

J. Söchtiget al., Electron. Lett. 31, 551–552 (1995).
[CrossRef]

Sohler, W.

Ch. Becker, A. Greiner, Th. Oesselke, A. Pape, W. Sohler, and H. Suche, “Integrated optical Ti:Er:LiNbO3 distribuited Bragg reflector laser with a fixed photorefractive grating,” Opt. Lett. 23, 1195–1197 (1998).
[CrossRef]

Stegeman, G. I.

R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138 (1996).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Suche, H.

Ch. Becker, A. Greiner, Th. Oesselke, A. Pape, W. Sohler, and H. Suche, “Integrated optical Ti:Er:LiNbO3 distribuited Bragg reflector laser with a fixed photorefractive grating,” Opt. Lett. 23, 1195–1197 (1998).
[CrossRef]

Taverner, D.

Torner, L.

L. Torner, D. Mihalache, D. Mazilu, M. C. Santos, and N. N. Akhmediev, “Walking solitons in quadratic nonlinear media,” J. Opt. Soc. Am. B 15, 1476–1487 (1998).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Torruellas, W. E.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Trillo, S.

C. Conti, A. De Rossi, and S. Trillo, “Existence, bistability, and instability of Kerr-like parametric gap solitons in quadratic media,” Opt. Lett. 23, 1265–1267 (1998).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, “Optical gap solitons via second-harmonic generation: exact solitary solutions,” Phys. Rev. E,  57, R1251–R1254 (1998).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, “Trapping of slowly moving or stationary two-color gap solitons,” Opt. Lett. 23, 334–336 (1998).
[CrossRef]

C. Conti, G. Assanto, and S. Trillo, “Read/write all-optical buffer by self trapped gap simultons,” Electron. Lett. 34, 689–690 (1998).
[CrossRef]

C. Conti, G. Assanto, and S. Trillo, “Excitation of self-transparency Bragg solitons in quadratic media,” Opt. Lett. 22, 1350–1352 (1997).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, “Bloch functions approach for parametric gap solitons,” Opt. Lett. 22, 445–447 (1997).
[CrossRef] [PubMed]

C. Conti, S. Trillo, and G. Assanto, “Doubly resonant Bragg simultons via second-harmonic generation,” Phys. Rev. Lett. 78, 2341–2344 (1997).
[CrossRef]

VanStryland, E. W.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Wang, Z.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Weissman, Z.

Electron. Lett. (2)

C. Conti, G. Assanto, and S. Trillo, “Read/write all-optical buffer by self trapped gap simultons,” Electron. Lett. 34, 689–690 (1998).
[CrossRef]

J. Söchtiget al., Electron. Lett. 31, 551–552 (1995).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Lett. (8)

Phys. Rev. A (1)

C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, “Collision of solitary waves in media with a second-order nonlinearity,” Phys. Rev. A 52, R3444–R3447 (1995).
[CrossRef] [PubMed]

Phys. Rev. E (6)

C. Etrich, U. Peschel, F. Lederer, and B. A. Malomed, “Stability of temporal chirped solitary waves in quadratically nonlinear media,” Phys. Rev. E 55, 6155–6161 (1997).
[CrossRef]

T. Peschel, U. Peschel, F. Lederer, and B. Malomed, “Solitary waves in Bragg gratings with a quadratic nonlinearity,” Phys. Rev. E 55, 4730–4739 (1997).
[CrossRef]

R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138 (1996).
[CrossRef]

C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E 54, 1969–1989 (1996).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, “Optical gap solitons via second-harmonic generation: exact solitary solutions,” Phys. Rev. E,  57, R1251–R1254 (1998).
[CrossRef]

Y. Kivshar, “Gap solitons due to cascading,” Phys. Rev. E 51, 1613–1615 (1995).
[CrossRef]

Phys. Rev. Lett. (4)

C. Conti, S. Trillo, and G. Assanto, “Doubly resonant Bragg simultons via second-harmonic generation,” Phys. Rev. Lett. 78, 2341–2344 (1997).
[CrossRef]

H. He and P. D. Drummond, “Ideal soliton environment using parametric band gaps,” Phys. Rev. Lett. 78, 4311–4314 (1997).
[CrossRef]

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

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Other (2)

C. M. De Sterke and J. E. Sipe, in Progress in Optics XXXIII, E. Wolf ed., (Elsevier, Amsterdam, 1994), Chap. III.

Guided-Wave Optoelectronic, chapter 2, T. Tamir ed., (Springer, Berlin, 1980).

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Figures (5)

Fig. 1.
Fig. 1.

Twin bandgap structure with the definition of the Bragg frequencies ωoj , and upper and lower gap edge frequencies ωuj and ωlj , respectively.

Fig. 2.
Fig. 2.

Injection of a FF pulse in a finite nonlinear grating (κ 2 = 1, v 2 = 0.5): Excitation of a two-color GS propagating at low velocity in the LB-LB case; Here we show the contour levels of the total FF (a) and SH (b) intensity, respectively.

Fig. 3.
Fig. 3.

Formation of a stationary gap solitons via inelastic collision of two counterpropagating low-speed solitons: (a) FF; (b) generated SH.

Fig. 4.
Fig. 4.

Interrogation process of the GS formed as in Fig. 3 (same parameters), by means of launching a new moving GS.

Fig. 5.
Fig. 5.

Long range dynamics of excitation of slow GSs in a singly resonant semi-infinite grating from Eqs. (22) with δ 1 = -0.7, δ 1 = 2, κ 2 = 0 and v 2 = 0.5.

Equations (67)

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H t = 1 μ 0 E z ; E t = 1 ε 0 n 2 ( z ) ( H z + P N L t ) ,
A ± ( z , t ) = 1 2 n ( z ) n 0 [ E ( z , t ) ± μ 0 ε 0 H ( z , t ) n ( z ) ] ,
i n ¯ ( z ) · A t = M ¯ · A + B ,
M ¯ [ i c z i c n ' ( z ) 2 n ( z ) i c n ' ( z ) 2 n ( z ) i c z ] ,
B ± = i 2 ε 0 n 0 n ( z ) P N L t .
P N L ( z , t ) = ε 0 χ E 2 ( z , t ) .
M ¯ · Ψ μ = ω μ n ¯ · Ψ μ .
Ψ j μ · n ¯ · Ψ j ' μ ' = 0 L Ψ j μ · n ¯ · Ψ j ' μ ' d z = N δ j j ' δ μ μ ' .
A = A 1 e i ω o 1 t + A 2 e i ω o 2 t + c . c . ,
A j = f j u Ψ j u + f j l Ψ j l + p u , l f j p Ψ j p f j h Ψ j h ,
ω o j n ¯ · A j ¯ + i η n ¯ · A ¯ j t 1 = M ¯ ( 0 ) · A ¯ j i η V ¯ · A ¯ j z 1 + B ¯ j
i f j u t Δ j 2 f j u V j f j l z + ϑ j u = 0 ,
i f j l t Δ j 2 f j l V j f j u z + ϑ j l = 0 ,
α r p q n 0 χ 0 d ϕ r , 1 ( z ) ϕ p , 1 ( z ) ϕ q , 2 ( z ) d z ,
α p q r n 0 χ 0 d ϕ p , 1 ( z ) ϕ q , 1 ( z ) ϕ r , 2 ( z ) d z .
E j + = 1 2 ( f j u i f j l ) ; E j = 1 2 ( f j u i f j l ) ,
f i = Θ E j [ f j u f j l ] = [ f j 1 f j 2 ] = [ 1 1 i i ] [ E j 1 E j 2 ] ,
i ( E j + t + V j E j + z ) + V j Γ j E j + τ j + = 0 ,
i ( E j t - V j E j z ) + V j Γ j E j + + τ j = 0 ,
[ τ j + τ j ] [ τ j 1 τ j 2 ] 1 2 [ 1 i 1 i ] [ ϑ j 1 ϑ j 2 ] ,
τ 1 j = ω o 1 2 exp ( iδωt ) H j h k ( E 1 h ) * E 2 k ; τ 2 j = ω o 1 2 exp ( + iδωt ) H h k j E 1 h E 1 k ;
H j h k ( Θ j m ) ( Θ p h ) * Θ q k α m p q ; H h k j ( Θ j m ) Θ p h Θ q k α p q m .
H j h k = { ( 1 ) j [ ( 1 ) h + k α 11 1 + α 12 2 ] + ( 1 ) h α 21 2 ( 1 ) k α 22 1 } +
+ i { ( 1 ) h + k α 21 1 α 22 2 + ( 1 ) j [ ( 1 ) h α 11 2 ( 1 ) k α 12 2 ] } ,
H h k j = { ( 1 ) j [ ( 1 ) h + k α 11 1 α 22 1 ] + ( 1 ) h α 21 2 + ( 1 ) h α 12 2 } +
+ i { ( 1 ) h + k α 11 2 + α 22 2 + ( 1 ) j [ ( 1 ) k α 21 1 + ( 1 ) h α 12 1 ] } .
n ( z ) = n B + n 1 sin ( 2 π d z ) + n 2 sin ( 4 π d z )
ϕ u j ( z ) = N j cos ( j π z d ) ,
ϕ l j ( z ) = N j sin ( j π z d ) ,
α u l l = α l u l = α l l u = α u u u = 0 ,
α l l l = α u l u = α l u u = α u u l = d 4 n 0 χ N L N 1 2 N 2 α .
H 1 11 = H 2 22 = 4 i α ; H 11 1 = H 22 1 = 4 ,
i ( E 1 + t + V 1 E 1 + z ) + V 1 Γ 1 E 1 2 i α ω o 1 ( E 1 + ) * E 2 + e i δ ω t = 0 ,
i ( E 1 t V 1 E 1 z ) + V 1 Γ 1 E 1 + 2 i α ω o 1 ( E 1 ) * E 2 e i δ ω t = 0 ,
i ( E 2 + t + V 2 E 2 + z ) + V 2 Γ 2 E 2 + 2 i α ω o 1 ( E 1 + ) 2 e i δ ω t = 0 ,
i ( E 2 t V 2 E 2 z ) + V 2 Γ 2 E 2 + + 2 i α ω o 1 ( E 1 ) 2 e i δ ω t = 0 .
i ( ± w 1 , ξ ± + w 1 , τ ± v 1 ) + κ 1 w 1 + ( w 1 ± ) * w 2 ± e i ω τ = 0 ,
i ( ± w 2 , ξ ± + w 2 , τ ± v 2 ) + κ 2 w 2 + ( w 1 ± ) 2 2 e i ω τ = 0 ,
i ( ± u 1 , ξ ± + u 1 , τ ± v 1 ) + δ 1 u 1 ± + κ 1 u 1 + ( u 1 ± ) * u 2 ± = 0 ,
i ( ± u 2 , ξ ± + u 2 , τ ± v 2 ) + δ 2 u 2 ± + κ 2 u 2 + ( u 1 ± ) 2 2 = 0 ,
σ 0 = [ 1 0 0 1 ] σ 1 = [ 0 i i 0 ] σ 3 = [ 1 0 0 1 ] ,
L m ϕ m + κ m σ 1 ϕ m + N m = 0 ,
N 1 [ ( w 1 + ) * w 2 + ( w 1 ) * w 2 ] e i ω τ ; N 2 [ ( w 1 + ) 2 2 ( w 1 ) 2 2 ] e i ω τ .
ϕ m ( ξ j , τ j ; j 0 ) = n = 0 η n ϕ m ( n ) ( ξ j , τ j ; j 0 ) =
= η 2 ϕ m ( 2 ) ( ξ j , τ j ; j 0 ) + η 3 ϕ m ( 3 ) ( ξ j , τ j ; j 0 ) =
= [ η 2 a m ( ξ j , τ j ; j 1 ) f m ( ± ) + η 3 b m ( ξ j , τ j ; j 1 ) f m ( ) ] exp ( i Q m ξ 0 i Ω m ( ± ) τ 0 ) ,
f m ( + ) ( Q m ) = 1 2 κ m 2 + Q m 2 [ κ m κ m 2 + Q m 2 Q m κ m 2 + Q m 2 Q m ] ,
f m ( ) ( Q m ) = 1 2 κ m 2 + Q m 2 [ κ m 2 + Q m 2 Q m κ m κ m 2 + Q m 2 Q m ] .
b m = i 2 κ m κ m 2 + Q m 2 a m ξ 1 ,
i a 1 τ 2 + i Ω 1 a 1 ξ 2 + Ω 1 2 2 a 1 ξ 1 2 + v 1 χ a 1 * a 2 e i ( Δ Q ¯ ξ 1 ΔΩ ¯ τ 1 ) = 0 ,
i a 2 τ 2 + i Ω 2 a 2 ξ 2 + Ω 2 2 2 a 2 ξ 1 2 + v 2 χ a 1 2 e i ( Δ Q ¯ ξ 1 ΔΩ ¯ τ 1 ) = 0 ,
i a 1 τ 1 + i Ω 2 a 1 ξ 1 = 0 ; i a 2 τ 1 + i Ω 2 a 2 ξ 1 = 0 ,
χ = χ ( Q 1,2 ; κ 1,2 ) [ f 1 + ( ± ) ] 2 f 2 + ( ± ) + [ f 1 ( ± ) ] 2 f 2 ( ± ) ,
Ω m = d Ω m ± d Q m = ± v m Q m κ m 2 + Q m 2 ; Ω m = d 2 Ω m ± d Q m 2
[ u m + u m ] = [ 𝜜 m f m ( ± ) + i 2 ρ m κ m κ m 2 + Q m 2 𝜜 m ξ f m ( ) ] exp ( i Q m ξ i Ω m ( ± ) τ + i δ m v m τ ) ,
i 𝜜 1 τ + i Ω 1 𝜜 1 ξ + Ω 1 2 2 𝜜 1 ξ 2 + v 1 χ 𝜜 1 * 𝜜 2 e i ( Δ Q ξ ΔΩ τ ) = 0 ,
i 𝜜 2 τ + i Ω 2 𝜜 2 ξ + Ω 2 2 2 𝜜 2 ξ 2 + v 2 χ 𝜜 1 2 e i ( Δ Q ξ ΔΩ τ ) = 0 .
𝜜 1 ( ξ , τ ) = Ω 1 2 v 1 v 2 u 1 [ ( ξ Ω 1 τ ) , τ Ω 1 ] ,
𝜜 2 ( ξ , τ ) = Ω 1 v 1 u 2 [ ( ξ Ω 1 τ ) , τ Ω 1 ] exp { i [ Δ Q ξ ( Ω 2 Δ Q 2 / 2 Δ Q Ω 2 ) τ ] } .
i u 1 σ ρ 1 2 2 u 1 s 2 + χ u 1 * u 2 e i β τ = 0 ,
i u 2 σ ρ 2 2 γ 2 u 2 s 2 i δ W u 2 s + χ u 1 2 2 e i β τ = 0 ,
i u 1 τ ρ 1 2 2 u 1 ξ 2 + v 1 2 1 + ρ 2 2 u 1 * u 2 e i β τ = 0 ,
i u 2 τ ρ 2 2 v 2 κ 2 2 u 2 ξ 2 + v 2 2 1 + ρ 2 2 u 1 2 2 e i β τ = 0 ,
[ u m + u m ] = A m { 1 ρ m U m ( ζ ) + 1 2 ρ m 1 [ m V U m ( ζ ) + i ζ U m ( ζ ) ] } e i m V ξ
( i τ ± i ξ ) w 1 ± + w 1 + σ w ± 2 w ± = 0
i τ a + sgn ( δ 1 ) 2 ξ 2 a + σ a 2 a = 0 ,
w 1 + w 1 = 1 2 1 sgn ( δ 1 ) a ( ξ , τ ) exp [ i sgn ( δ 1 ) τ ] .

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