Abstract

We introduce a novel class of parametric optical solitons supported simultaneously by two second-order nonlinear cascading processes, second-harmonic generation and sum-frequency mixing. We obtain analytical and numerical solutions for three-wave spatial solitons and show that the presence of an additional cascading mechanism can change dramatically the properties and stability of two-wave quadratic solitary waves.

© 1999 Optical Society of America

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  1. For an overview, see G. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996)Yu. S. Kivshar, in Advanced Photonics with Second-Order Optically Nonlinear Processes, A. D. Boardman, L. Pavlov, and S. Tanev, eds. (Kluwer Academic, Dordretch, The Netherlands, 1998), p. 451.
    [CrossRef]
  2. D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
    [CrossRef] [PubMed]
  3. K. Koynov and S. Saltiel, Opt. Commun. 152, 96 (1998).
    [CrossRef]
  4. See, e.g., S. Akhmanov, A. Dubovik, S. Saltiel, I. Tomov, and V. Tunkin, JETP Lett. 20, 117 (1974).
  5. A. V. Buryak and Yu. S. Kivshar, Opt. Lett. 20, 1612 (1994)Phys. Lett. A 197, 407 (1995).
    [CrossRef]
  6. O. Pfister, J. S. Wells, L. Hollberg, L. Zink, D. A. Van Baak, M. D. Levenson, and W. R. Basenberg, Opt. Lett. 22, 1211 (1997).
    [CrossRef] [PubMed]
  7. P. Baldi, C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, Electron. Lett. 31, 1350 (1995).
    [CrossRef]
  8. B. A. Malomed, D. Anderson, and M. Lisak, Opt. Commun. 126, 251 (1996).
    [CrossRef]
  9. A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
    [CrossRef] [PubMed]
  10. V. E. Zakharov and E. A. Kuznetsov, Zh. Eksp. Teor. Fiz. 113, 1892 (1998) JETP 86, 1035 (1998).
  11. See, e.g., V. I. Karpman, Phys. Lett. A 193, 355 (1994).
    [CrossRef]

1998 (2)

K. Koynov and S. Saltiel, Opt. Commun. 152, 96 (1998).
[CrossRef]

V. E. Zakharov and E. A. Kuznetsov, Zh. Eksp. Teor. Fiz. 113, 1892 (1998) JETP 86, 1035 (1998).

1997 (1)

1996 (3)

For an overview, see G. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996)Yu. S. Kivshar, in Advanced Photonics with Second-Order Optically Nonlinear Processes, A. D. Boardman, L. Pavlov, and S. Tanev, eds. (Kluwer Academic, Dordretch, The Netherlands, 1998), p. 451.
[CrossRef]

B. A. Malomed, D. Anderson, and M. Lisak, Opt. Commun. 126, 251 (1996).
[CrossRef]

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[CrossRef] [PubMed]

1995 (2)

P. Baldi, C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, Electron. Lett. 31, 1350 (1995).
[CrossRef]

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[CrossRef] [PubMed]

1994 (2)

A. V. Buryak and Yu. S. Kivshar, Opt. Lett. 20, 1612 (1994)Phys. Lett. A 197, 407 (1995).
[CrossRef]

See, e.g., V. I. Karpman, Phys. Lett. A 193, 355 (1994).
[CrossRef]

1974 (1)

See, e.g., S. Akhmanov, A. Dubovik, S. Saltiel, I. Tomov, and V. Tunkin, JETP Lett. 20, 117 (1974).

Akhmanov, S.

See, e.g., S. Akhmanov, A. Dubovik, S. Saltiel, I. Tomov, and V. Tunkin, JETP Lett. 20, 117 (1974).

Anderson, D.

B. A. Malomed, D. Anderson, and M. Lisak, Opt. Commun. 126, 251 (1996).
[CrossRef]

Baldi, P.

P. Baldi, C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, Electron. Lett. 31, 1350 (1995).
[CrossRef]

Basenberg, W. R.

Buryak, A. V.

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[CrossRef] [PubMed]

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[CrossRef] [PubMed]

A. V. Buryak and Yu. S. Kivshar, Opt. Lett. 20, 1612 (1994)Phys. Lett. A 197, 407 (1995).
[CrossRef]

De Micheli, M. P.

P. Baldi, C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, Electron. Lett. 31, 1350 (1995).
[CrossRef]

Delacourt, D.

P. Baldi, C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, Electron. Lett. 31, 1350 (1995).
[CrossRef]

Dubovik, A.

See, e.g., S. Akhmanov, A. Dubovik, S. Saltiel, I. Tomov, and V. Tunkin, JETP Lett. 20, 117 (1974).

Hagan, D. J.

For an overview, see G. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996)Yu. S. Kivshar, in Advanced Photonics with Second-Order Optically Nonlinear Processes, A. D. Boardman, L. Pavlov, and S. Tanev, eds. (Kluwer Academic, Dordretch, The Netherlands, 1998), p. 451.
[CrossRef]

Hollberg, L.

Karpman, V. I.

See, e.g., V. I. Karpman, Phys. Lett. A 193, 355 (1994).
[CrossRef]

Kivshar, Yu. S.

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[CrossRef] [PubMed]

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[CrossRef] [PubMed]

A. V. Buryak and Yu. S. Kivshar, Opt. Lett. 20, 1612 (1994)Phys. Lett. A 197, 407 (1995).
[CrossRef]

Koynov, K.

K. Koynov and S. Saltiel, Opt. Commun. 152, 96 (1998).
[CrossRef]

Kuznetsov, E. A.

V. E. Zakharov and E. A. Kuznetsov, Zh. Eksp. Teor. Fiz. 113, 1892 (1998) JETP 86, 1035 (1998).

Levenson, M. D.

Lisak, M.

B. A. Malomed, D. Anderson, and M. Lisak, Opt. Commun. 126, 251 (1996).
[CrossRef]

Malomed, B. A.

B. A. Malomed, D. Anderson, and M. Lisak, Opt. Commun. 126, 251 (1996).
[CrossRef]

Ostrowsky, D. B.

P. Baldi, C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, Electron. Lett. 31, 1350 (1995).
[CrossRef]

Papuchon, M.

P. Baldi, C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, Electron. Lett. 31, 1350 (1995).
[CrossRef]

Pelinovsky, D. E.

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[CrossRef] [PubMed]

Pfister, O.

Saltiel, S.

K. Koynov and S. Saltiel, Opt. Commun. 152, 96 (1998).
[CrossRef]

See, e.g., S. Akhmanov, A. Dubovik, S. Saltiel, I. Tomov, and V. Tunkin, JETP Lett. 20, 117 (1974).

Stegeman, G.

For an overview, see G. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996)Yu. S. Kivshar, in Advanced Photonics with Second-Order Optically Nonlinear Processes, A. D. Boardman, L. Pavlov, and S. Tanev, eds. (Kluwer Academic, Dordretch, The Netherlands, 1998), p. 451.
[CrossRef]

Stegeman, G. I.

P. Baldi, C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, Electron. Lett. 31, 1350 (1995).
[CrossRef]

Tomov, I.

See, e.g., S. Akhmanov, A. Dubovik, S. Saltiel, I. Tomov, and V. Tunkin, JETP Lett. 20, 117 (1974).

Torner, L.

For an overview, see G. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996)Yu. S. Kivshar, in Advanced Photonics with Second-Order Optically Nonlinear Processes, A. D. Boardman, L. Pavlov, and S. Tanev, eds. (Kluwer Academic, Dordretch, The Netherlands, 1998), p. 451.
[CrossRef]

Trevino-Palacios, C. G.

P. Baldi, C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, Electron. Lett. 31, 1350 (1995).
[CrossRef]

Trillo, S.

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[CrossRef] [PubMed]

Tunkin, V.

See, e.g., S. Akhmanov, A. Dubovik, S. Saltiel, I. Tomov, and V. Tunkin, JETP Lett. 20, 117 (1974).

Van Baak, D. A.

Wells, J. S.

Zakharov, V. E.

V. E. Zakharov and E. A. Kuznetsov, Zh. Eksp. Teor. Fiz. 113, 1892 (1998) JETP 86, 1035 (1998).

Zink, L.

Electron. Lett. (1)

P. Baldi, C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, Electron. Lett. 31, 1350 (1995).
[CrossRef]

JETP Lett. (1)

See, e.g., S. Akhmanov, A. Dubovik, S. Saltiel, I. Tomov, and V. Tunkin, JETP Lett. 20, 117 (1974).

Opt. Commun. (2)

B. A. Malomed, D. Anderson, and M. Lisak, Opt. Commun. 126, 251 (1996).
[CrossRef]

K. Koynov and S. Saltiel, Opt. Commun. 152, 96 (1998).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

For an overview, see G. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996)Yu. S. Kivshar, in Advanced Photonics with Second-Order Optically Nonlinear Processes, A. D. Boardman, L. Pavlov, and S. Tanev, eds. (Kluwer Academic, Dordretch, The Netherlands, 1998), p. 451.
[CrossRef]

Phys. Lett. A (1)

See, e.g., V. I. Karpman, Phys. Lett. A 193, 355 (1994).
[CrossRef]

Phys. Rev. Lett. (2)

D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, Phys. Rev. Lett. 75, 591 (1995).
[CrossRef] [PubMed]

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[CrossRef] [PubMed]

Zh. Eksp. Teor. Fiz. (1)

V. E. Zakharov and E. A. Kuznetsov, Zh. Eksp. Teor. Fiz. 113, 1892 (1998) JETP 86, 1035 (1998).

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Figures (4)

Fig. 1
Fig. 1

Examples of three-wave solitary waves of Eqs.  (3) for (a) α=0.05, α1=5, and (b) α=5, α1=0.35.

Fig. 2
Fig. 2

Two branches of multistep cascading solitons, shown as the total soliton power P versus α1 for α=1 and χ=1. The filled circles show the analytical solutions. The lower branch approaches a family of two-wave quadratic solitons (for α1), shown by the dashed line.

Fig. 3
Fig. 3

Peak intensity ratios for the family of three-wave solitary waves χ=1 at α1=-10 and α1=10. The dashed curve shows the asymptotic limit of large α1 corresponding to the two-wave quadratic solitons.

Fig. 4
Fig. 4

Examples of (a) a three-wave quasi-soliton α=1,α1=-5 and (b) its instability-induced long-term evolution shown for the peak intensities α=1,α1=-10. (c), (d) Evolution of the fundamental harmonic of a two-wave soliton owing to an unseeded unmatched SFM process with (c) α1=+8 and (d) α1=-8.

Equations (5)

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2ik1A1z+2A1x2+χ1A3A2*exp-iΔk3z+χ2A2A1*exp-iΔk2z=0,4ik1A2z+2A2x2+χ4A3A1*exp-iΔk3z+χ5A12expiΔk2z=0,6ik1A3z+2A3x2+χ3A2A1expiΔk3z=0,
A1=2βk1χ2χ5expiβzw,A2=2βk1χ2exp2iβz+iΔk2zv,A3=2χ2βk1χ1χ5exp3iβz+iΔkzu,
iwz+2wx2-w+w*v+v*u=0,2ivz+2vx2-αv+12w2+w*u=0,3iuz+2ux2-α1u+χvw=0,
iwz+2wx2-w+w*v+ϵχv2w=0,2ivz+2vx2-αv+w22+ϵχw2v=0.
P=-+dxw2+4v2+9χu2,

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