Abstract

We demonstrate that an array of discrete waveguides on a slab substrate, both featuring χ2 nonlinearity, supports stable solitons composed of discrete and continuous components. Two classes of fundamental composite soliton are identified: ones consisting of a discrete fundamental-frequency (FF) component in the waveguide array, coupled to a continuous second-harmonic (SH) component in the slab waveguide, and solitons with an inverted FF/SH structure. Twisted bound states of the fundamental solitons are found, too. In contrast with the usual systems, the intersite-centered fundamental solitons and bound states with the twisted continuous components are stable over almost the entire domain of their existence.

© 2006 Optical Society of America

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  1. K. Hayata and M. Koshiba, Phys. Rev. Lett. 71, 3275 (1993).
    [Crossref] [PubMed]
  2. W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
    [Crossref] [PubMed]
  3. C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, Prog. Opt. 41, 483 (2000).
    [Crossref]
  4. A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, Phys. Rep. 370, 63 (2002).
    [Crossref]
  5. O. Bang, P. L. Christiansen, and C. B. Clausen, Phys. Rev. E 56, 7257 (1997).
    [Crossref]
  6. T. Peschel, U. Peschel, and F. Lederer, Phys. Rev. E 57, 1127 (1998).
    [Crossref]
  7. B. A. Malomed, P. G. Kevrekidis, D. J. Frantzeskakis, H. E. Nistazakis, and A. N. Yannacopoulos, Phys. Rev. E 65, 056606 (2002).
    [Crossref]
  8. Y. V. Kartashov, L. Torner, and V. A. Vysloukh, Opt. Lett. 29, 1117 (2004).
    [Crossref] [PubMed]
  9. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 29, 1399 (2004).
    [Crossref] [PubMed]
  10. R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
    [Crossref] [PubMed]
  11. D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
    [Crossref] [PubMed]
  12. C. B. Clausen and L. Torner, Opt. Lett. 24, 7 (1999).
    [Crossref]
  13. T. Peschel, U. Peschel, and F. Lederer, Opt. Lett. 28, 102 (2003).
    [Crossref] [PubMed]
  14. R. Schiek, Y. Baek, G. I. Stegeman, and W. Sohler, Opt. Quantum Electron. 30, 861 (1998).
    [Crossref]
  15. R. A. Vicencio, M. I. Molina, and Yu. S. Kivshar, Phys. Rev. E 70, 026602 (2004).
    [Crossref]
  16. D. Christodoulides and R. Joseph, Opt. Lett. 13, 794 (1988).
    [Crossref] [PubMed]
  17. R. Vallée and G. He, Opt. Commun. 126, 293 (1996).
    [Crossref]
  18. R. Vallée and G. He, J. Lightwave Technol. 11, 1196 (1993).
    [Crossref]
  19. M. G. Vakhitov and A. A. Kolokolov, Radiophys. Quantum Electron. 16, 783 (1973).
    [Crossref]
  20. L. Torner, D. Mazilu, and D. Mihalache, Phys. Rev. Lett. 77, 2455 (1996).
    [Crossref] [PubMed]

2004 (4)

Y. V. Kartashov, L. Torner, and V. A. Vysloukh, Opt. Lett. 29, 1117 (2004).
[Crossref] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 29, 1399 (2004).
[Crossref] [PubMed]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[Crossref] [PubMed]

R. A. Vicencio, M. I. Molina, and Yu. S. Kivshar, Phys. Rev. E 70, 026602 (2004).
[Crossref]

2003 (2)

T. Peschel, U. Peschel, and F. Lederer, Opt. Lett. 28, 102 (2003).
[Crossref] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[Crossref] [PubMed]

2002 (2)

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, Phys. Rep. 370, 63 (2002).
[Crossref]

B. A. Malomed, P. G. Kevrekidis, D. J. Frantzeskakis, H. E. Nistazakis, and A. N. Yannacopoulos, Phys. Rev. E 65, 056606 (2002).
[Crossref]

2000 (1)

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, Prog. Opt. 41, 483 (2000).
[Crossref]

1999 (1)

1998 (2)

T. Peschel, U. Peschel, and F. Lederer, Phys. Rev. E 57, 1127 (1998).
[Crossref]

R. Schiek, Y. Baek, G. I. Stegeman, and W. Sohler, Opt. Quantum Electron. 30, 861 (1998).
[Crossref]

1997 (1)

O. Bang, P. L. Christiansen, and C. B. Clausen, Phys. Rev. E 56, 7257 (1997).
[Crossref]

1996 (2)

R. Vallée and G. He, Opt. Commun. 126, 293 (1996).
[Crossref]

L. Torner, D. Mazilu, and D. Mihalache, Phys. Rev. Lett. 77, 2455 (1996).
[Crossref] [PubMed]

1995 (1)

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[Crossref] [PubMed]

1993 (2)

K. Hayata and M. Koshiba, Phys. Rev. Lett. 71, 3275 (1993).
[Crossref] [PubMed]

R. Vallée and G. He, J. Lightwave Technol. 11, 1196 (1993).
[Crossref]

1988 (1)

1973 (1)

M. G. Vakhitov and A. A. Kolokolov, Radiophys. Quantum Electron. 16, 783 (1973).
[Crossref]

Baek, Y.

R. Schiek, Y. Baek, G. I. Stegeman, and W. Sohler, Opt. Quantum Electron. 30, 861 (1998).
[Crossref]

Bang, O.

O. Bang, P. L. Christiansen, and C. B. Clausen, Phys. Rev. E 56, 7257 (1997).
[Crossref]

Buryak, A. V.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, Phys. Rep. 370, 63 (2002).
[Crossref]

Christiansen, P. L.

O. Bang, P. L. Christiansen, and C. B. Clausen, Phys. Rev. E 56, 7257 (1997).
[Crossref]

Christodoulides, D.

Christodoulides, D. N.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[Crossref] [PubMed]

Clausen, C. B.

C. B. Clausen and L. Torner, Opt. Lett. 24, 7 (1999).
[Crossref]

O. Bang, P. L. Christiansen, and C. B. Clausen, Phys. Rev. E 56, 7257 (1997).
[Crossref]

Di Trapani, P.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, Phys. Rep. 370, 63 (2002).
[Crossref]

Etrich, C.

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, Prog. Opt. 41, 483 (2000).
[Crossref]

Frantzeskakis, D. J.

B. A. Malomed, P. G. Kevrekidis, D. J. Frantzeskakis, H. E. Nistazakis, and A. N. Yannacopoulos, Phys. Rev. E 65, 056606 (2002).
[Crossref]

Hagan, D. J.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[Crossref] [PubMed]

Hayata, K.

K. Hayata and M. Koshiba, Phys. Rev. Lett. 71, 3275 (1993).
[Crossref] [PubMed]

He, G.

R. Vallée and G. He, Opt. Commun. 126, 293 (1996).
[Crossref]

R. Vallée and G. He, J. Lightwave Technol. 11, 1196 (1993).
[Crossref]

Iwanow, R.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[Crossref] [PubMed]

Joseph, R.

Kartashov, Y. V.

Kevrekidis, P. G.

B. A. Malomed, P. G. Kevrekidis, D. J. Frantzeskakis, H. E. Nistazakis, and A. N. Yannacopoulos, Phys. Rev. E 65, 056606 (2002).
[Crossref]

Kivshar, Yu. S.

R. A. Vicencio, M. I. Molina, and Yu. S. Kivshar, Phys. Rev. E 70, 026602 (2004).
[Crossref]

Kolokolov, A. A.

M. G. Vakhitov and A. A. Kolokolov, Radiophys. Quantum Electron. 16, 783 (1973).
[Crossref]

Koshiba, M.

K. Hayata and M. Koshiba, Phys. Rev. Lett. 71, 3275 (1993).
[Crossref] [PubMed]

Lederer, F.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[Crossref] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[Crossref] [PubMed]

T. Peschel, U. Peschel, and F. Lederer, Opt. Lett. 28, 102 (2003).
[Crossref] [PubMed]

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, Prog. Opt. 41, 483 (2000).
[Crossref]

T. Peschel, U. Peschel, and F. Lederer, Phys. Rev. E 57, 1127 (1998).
[Crossref]

Malomed, B. A.

B. A. Malomed, P. G. Kevrekidis, D. J. Frantzeskakis, H. E. Nistazakis, and A. N. Yannacopoulos, Phys. Rev. E 65, 056606 (2002).
[Crossref]

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, Prog. Opt. 41, 483 (2000).
[Crossref]

Mazilu, D.

L. Torner, D. Mazilu, and D. Mihalache, Phys. Rev. Lett. 77, 2455 (1996).
[Crossref] [PubMed]

Menyuk, C. R.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[Crossref] [PubMed]

Mihalache, D.

L. Torner, D. Mazilu, and D. Mihalache, Phys. Rev. Lett. 77, 2455 (1996).
[Crossref] [PubMed]

Min, Y.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[Crossref] [PubMed]

Molina, M. I.

R. A. Vicencio, M. I. Molina, and Yu. S. Kivshar, Phys. Rev. E 70, 026602 (2004).
[Crossref]

Nistazakis, H. E.

B. A. Malomed, P. G. Kevrekidis, D. J. Frantzeskakis, H. E. Nistazakis, and A. N. Yannacopoulos, Phys. Rev. E 65, 056606 (2002).
[Crossref]

Pertsch, T.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[Crossref] [PubMed]

Peschel, T.

T. Peschel, U. Peschel, and F. Lederer, Opt. Lett. 28, 102 (2003).
[Crossref] [PubMed]

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, Prog. Opt. 41, 483 (2000).
[Crossref]

T. Peschel, U. Peschel, and F. Lederer, Phys. Rev. E 57, 1127 (1998).
[Crossref]

Peschel, U.

T. Peschel, U. Peschel, and F. Lederer, Opt. Lett. 28, 102 (2003).
[Crossref] [PubMed]

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, Prog. Opt. 41, 483 (2000).
[Crossref]

T. Peschel, U. Peschel, and F. Lederer, Phys. Rev. E 57, 1127 (1998).
[Crossref]

Schiek, R.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[Crossref] [PubMed]

R. Schiek, Y. Baek, G. I. Stegeman, and W. Sohler, Opt. Quantum Electron. 30, 861 (1998).
[Crossref]

Silberberg, Y.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[Crossref] [PubMed]

Skryabin, D. V.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, Phys. Rep. 370, 63 (2002).
[Crossref]

Sohler, W.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[Crossref] [PubMed]

R. Schiek, Y. Baek, G. I. Stegeman, and W. Sohler, Opt. Quantum Electron. 30, 861 (1998).
[Crossref]

Stegeman, G. I.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[Crossref] [PubMed]

R. Schiek, Y. Baek, G. I. Stegeman, and W. Sohler, Opt. Quantum Electron. 30, 861 (1998).
[Crossref]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[Crossref] [PubMed]

Torner, L.

Y. V. Kartashov, L. Torner, and V. A. Vysloukh, Opt. Lett. 29, 1117 (2004).
[Crossref] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 29, 1399 (2004).
[Crossref] [PubMed]

C. B. Clausen and L. Torner, Opt. Lett. 24, 7 (1999).
[Crossref]

L. Torner, D. Mazilu, and D. Mihalache, Phys. Rev. Lett. 77, 2455 (1996).
[Crossref] [PubMed]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (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, Phys. Rev. Lett. 74, 5036 (1995).
[Crossref] [PubMed]

Trillo, S.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, Phys. Rep. 370, 63 (2002).
[Crossref]

Vakhitov, M. G.

M. G. Vakhitov and A. A. Kolokolov, Radiophys. Quantum Electron. 16, 783 (1973).
[Crossref]

Vallée, R.

R. Vallée and G. He, Opt. Commun. 126, 293 (1996).
[Crossref]

R. Vallée and G. He, J. Lightwave Technol. 11, 1196 (1993).
[Crossref]

VanStryland, E. W.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[Crossref] [PubMed]

Vicencio, R. A.

R. A. Vicencio, M. I. Molina, and Yu. S. Kivshar, Phys. Rev. E 70, 026602 (2004).
[Crossref]

Vysloukh, V. A.

Wang, Z.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[Crossref] [PubMed]

Yannacopoulos, A. N.

B. A. Malomed, P. G. Kevrekidis, D. J. Frantzeskakis, H. E. Nistazakis, and A. N. Yannacopoulos, Phys. Rev. E 65, 056606 (2002).
[Crossref]

J. Lightwave Technol. (1)

R. Vallée and G. He, J. Lightwave Technol. 11, 1196 (1993).
[Crossref]

Nature (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[Crossref] [PubMed]

Opt. Commun. (1)

R. Vallée and G. He, Opt. Commun. 126, 293 (1996).
[Crossref]

Opt. Lett. (5)

Opt. Quantum Electron. (1)

R. Schiek, Y. Baek, G. I. Stegeman, and W. Sohler, Opt. Quantum Electron. 30, 861 (1998).
[Crossref]

Phys. Rep. (1)

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, Phys. Rep. 370, 63 (2002).
[Crossref]

Phys. Rev. E (4)

O. Bang, P. L. Christiansen, and C. B. Clausen, Phys. Rev. E 56, 7257 (1997).
[Crossref]

T. Peschel, U. Peschel, and F. Lederer, Phys. Rev. E 57, 1127 (1998).
[Crossref]

B. A. Malomed, P. G. Kevrekidis, D. J. Frantzeskakis, H. E. Nistazakis, and A. N. Yannacopoulos, Phys. Rev. E 65, 056606 (2002).
[Crossref]

R. A. Vicencio, M. I. Molina, and Yu. S. Kivshar, Phys. Rev. E 70, 026602 (2004).
[Crossref]

Phys. Rev. Lett. (4)

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[Crossref] [PubMed]

K. Hayata and M. Koshiba, Phys. Rev. Lett. 71, 3275 (1993).
[Crossref] [PubMed]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[Crossref] [PubMed]

L. Torner, D. Mazilu, and D. Mihalache, Phys. Rev. Lett. 77, 2455 (1996).
[Crossref] [PubMed]

Prog. Opt. (1)

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, Prog. Opt. 41, 483 (2000).
[Crossref]

Radiophys. Quantum Electron. (1)

M. G. Vakhitov and A. A. Kolokolov, Radiophys. Quantum Electron. 16, 783 (1973).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1

Schematic of the proposed setting.

Fig. 2
Fig. 2

Field profiles of composite solitons. Here and in Fig. 3 top and bottom panels correspond to solitons of types A and B, respectively, whereas left and right panels display odd and even solitons. Vertical lines designate the location of discrete waveguides. The wavevector mismatch is β = 0 , and the soliton wave numbers are λ = 3 and λ = 1.5 in cases A and B, respectively.

Fig. 3
Fig. 3

Total power P of the semidiscrete solitons versus the wavenumber λ.

Fig. 4
Fig. 4

Top panels, field profiles of twisted composite solitons of types A (left) and B (right). In both cases, the wave-vector mismatch and the soliton’s wavenumber are β = 0 and λ = 3.6 . The bottom panel shows the total power P versus λ, the inset enlarging the vicinity of the cutoff point (stability border) on the curve for β = 0 .

Equations (4)

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i d ϕ n d ζ + ρ ( ϕ n 1 + ϕ n + 1 ) + ϕ n * Ψ ( ζ , n ) = 0 ,
i Ψ ζ + 1 2 2 Ψ η 2 + β Ψ + 1 2 n ϕ n 2 δ ( η n ) = 0 ,
i Φ ζ + 1 2 2 Φ η 2 + Φ * n ψ n δ ( η n ) = 0 ,
i d ψ n d ζ + β ψ n + ρ ( ψ n 1 + ψ n + 1 ) + 1 2 Φ 2 ( ζ , n ) = 0 .

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