Abstract

We investigate the stability of bright solitary waves formed by the mutual trapping of the fundamental and second-harmonic waves propagating in a nonlinear quadratic medium. First, we find that most of the solitary waves are stable under propagation. Second, we study the evolution of the solitary waves in the presence of small linear absorption. At phase matching the adiabatic evolution of the solitary waves obeys an amplitude × width-squared rule.

© 1995 Optical Society of America

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References

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  1. Yu. N. Karamzin, A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976).
  2. R. Schiek, J. Opt. Soc. Am. B 10, 1848 (1993); M. J. Werner, P. D. Drummond, J. Opt. Soc. Am. B 10, 2390 (1993); K. Hayata, M. Koshiba, Phys. Rev. Lett. 71, 3275 (1993).
    [CrossRef]
  3. C. R. Menyuk, R. Schiek, L. Torner, J. Opt. Soc. Am. B 11, 2434 (1994).
    [CrossRef]
  4. L. Torner, C. R. Menyuk, G. I. Stegeman, Opt. Lett. 19, 1615 (1994) ;J. Opt. Soc. Am. B 12, 889 (1995).
    [CrossRef] [PubMed]
  5. W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. VanStryland, G. I. Stegeman, L. Torner, C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
    [CrossRef] [PubMed]
  6. R. Schiek, Y. Baek, G. I. Stegeman, “One-dimensional spatial solitons due to cascaded second-order nonlinearities in planar waveguides,” submitted toPhys. Rev. Lett.
    [PubMed]
  7. A. V. Buryak, Y. S. Kivshar, Opt. Lett. 19, 1612 (1994); Phys. Lett. A 197, 407 (1995); L. Torner, Opt. Commun. 114, 136 (1995).
    [CrossRef] [PubMed]
  8. N. Akhmediev, A. Buryak, J. M. Soto-Crespo, Opt. Commun. 112, 278 (1994); N. Akhmediev, A. Ankiewicz, Phys. Rev. Lett. 70, 2395 (1993).
    [CrossRef]
  9. E. A. Kuznetsov, A. M. Rubenchik, V. E. Zakharov, Phys. Rep. 142, 103 (1986); V. E. Zakharov, E. A. Kuznetzov, Sov. Phys. JETP 39, 285 (1975).
    [CrossRef]
  10. The boundedness of the (3 + 1) version of the Hamiltonian was shown byA. A. Kanashov, A. M. Rubenchik, Physica D 4, 122 (1981); the same method was used later for the (2 + 1) case byL. Torner, C. R. Menyuk, W. E. Torruellas, G. I. Stegeman, Opt. Lett. 20, 13 (1995).
    [CrossRef]

1995 (1)

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

1994 (4)

1993 (1)

1986 (1)

E. A. Kuznetsov, A. M. Rubenchik, V. E. Zakharov, Phys. Rep. 142, 103 (1986); V. E. Zakharov, E. A. Kuznetzov, Sov. Phys. JETP 39, 285 (1975).
[CrossRef]

1981 (1)

The boundedness of the (3 + 1) version of the Hamiltonian was shown byA. A. Kanashov, A. M. Rubenchik, Physica D 4, 122 (1981); the same method was used later for the (2 + 1) case byL. Torner, C. R. Menyuk, W. E. Torruellas, G. I. Stegeman, Opt. Lett. 20, 13 (1995).
[CrossRef]

1976 (1)

Yu. N. Karamzin, A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976).

Akhmediev, N.

N. Akhmediev, A. Buryak, J. M. Soto-Crespo, Opt. Commun. 112, 278 (1994); N. Akhmediev, A. Ankiewicz, Phys. Rev. Lett. 70, 2395 (1993).
[CrossRef]

Baek, Y.

R. Schiek, Y. Baek, G. I. Stegeman, “One-dimensional spatial solitons due to cascaded second-order nonlinearities in planar waveguides,” submitted toPhys. Rev. Lett.
[PubMed]

Buryak, A.

N. Akhmediev, A. Buryak, J. M. Soto-Crespo, Opt. Commun. 112, 278 (1994); N. Akhmediev, A. Ankiewicz, Phys. Rev. Lett. 70, 2395 (1993).
[CrossRef]

Buryak, A. V.

Hagan, D. J.

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

Kanashov, A. A.

The boundedness of the (3 + 1) version of the Hamiltonian was shown byA. A. Kanashov, A. M. Rubenchik, Physica D 4, 122 (1981); the same method was used later for the (2 + 1) case byL. Torner, C. R. Menyuk, W. E. Torruellas, G. I. Stegeman, Opt. Lett. 20, 13 (1995).
[CrossRef]

Karamzin, Yu. N.

Yu. N. Karamzin, A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976).

Kivshar, Y. S.

Kuznetsov, E. A.

E. A. Kuznetsov, A. M. Rubenchik, V. E. Zakharov, Phys. Rep. 142, 103 (1986); V. E. Zakharov, E. A. Kuznetzov, Sov. Phys. JETP 39, 285 (1975).
[CrossRef]

Menyuk, C. R.

Rubenchik, A. M.

E. A. Kuznetsov, A. M. Rubenchik, V. E. Zakharov, Phys. Rep. 142, 103 (1986); V. E. Zakharov, E. A. Kuznetzov, Sov. Phys. JETP 39, 285 (1975).
[CrossRef]

The boundedness of the (3 + 1) version of the Hamiltonian was shown byA. A. Kanashov, A. M. Rubenchik, Physica D 4, 122 (1981); the same method was used later for the (2 + 1) case byL. Torner, C. R. Menyuk, W. E. Torruellas, G. I. Stegeman, Opt. Lett. 20, 13 (1995).
[CrossRef]

Schiek, R.

Soto-Crespo, J. M.

N. Akhmediev, A. Buryak, J. M. Soto-Crespo, Opt. Commun. 112, 278 (1994); N. Akhmediev, A. Ankiewicz, Phys. Rev. Lett. 70, 2395 (1993).
[CrossRef]

Stegeman, G. I.

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

L. Torner, C. R. Menyuk, G. I. Stegeman, Opt. Lett. 19, 1615 (1994) ;J. Opt. Soc. Am. B 12, 889 (1995).
[CrossRef] [PubMed]

R. Schiek, Y. Baek, G. I. Stegeman, “One-dimensional spatial solitons due to cascaded second-order nonlinearities in planar waveguides,” submitted toPhys. Rev. Lett.
[PubMed]

Sukhorukov, A. P.

Yu. N. Karamzin, A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976).

Torner, L.

Torruellas, W. E.

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

VanStryland, E. W.

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

Wang, Z.

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

Zakharov, V. E.

E. A. Kuznetsov, A. M. Rubenchik, V. E. Zakharov, Phys. Rep. 142, 103 (1986); V. E. Zakharov, E. A. Kuznetzov, Sov. Phys. JETP 39, 285 (1975).
[CrossRef]

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

Opt. Commun. (1)

N. Akhmediev, A. Buryak, J. M. Soto-Crespo, Opt. Commun. 112, 278 (1994); N. Akhmediev, A. Ankiewicz, Phys. Rev. Lett. 70, 2395 (1993).
[CrossRef]

Opt. Lett. (2)

Phys. Rep. (1)

E. A. Kuznetsov, A. M. Rubenchik, V. E. Zakharov, Phys. Rep. 142, 103 (1986); V. E. Zakharov, E. A. Kuznetzov, Sov. Phys. JETP 39, 285 (1975).
[CrossRef]

Phys. Rev. Lett. (1)

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

Physica D (1)

The boundedness of the (3 + 1) version of the Hamiltonian was shown byA. A. Kanashov, A. M. Rubenchik, Physica D 4, 122 (1981); the same method was used later for the (2 + 1) case byL. Torner, C. R. Menyuk, W. E. Torruellas, G. I. Stegeman, Opt. Lett. 20, 13 (1995).
[CrossRef]

Sov. Phys. JETP (1)

Yu. N. Karamzin, A. P. Sukhorukov, Sov. Phys. JETP 41, 414 (1976).

Other (1)

R. Schiek, Y. Baek, G. I. Stegeman, “One-dimensional spatial solitons due to cascaded second-order nonlinearities in planar waveguides,” submitted toPhys. Rev. Lett.
[PubMed]

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Figures (3)

Fig. 1
Fig. 1

Hamiltonian and wave-number shift κ1 versus energy flow for the stationary solitary wave solutions. (a), (c) Curves for phase matching (β = 0) and positive phase mismatch. (b), (d) Curves for negative phase mismatch. Continuous curves, stable solitary waves; dashed curves, unstable solutions.

Fig. 2
Fig. 2

Detail of the solitary-wave evolution with small linear absorption at exact phase matching: (a) fundamental wave, (b) second harmonic. Dashed curves, input beams; continuous curves, waves after propagating 40 diffraction lengths (ξ = 20). The input beams are the stationary solution with energy flow I = 18 and the absorption coefficients Γ1 = Γ2 = 0.03.

Fig. 3
Fig. 3

Width of the stationary solitary waves as a function of their amplitude: (a) positive phase mismatch, (b) negative phase mismatch. Only the stable branch has been plotted.

Equations (4)

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i a 1 ξ - r 2 2 a 1 s 2 + a 1 * a 2 exp ( - i β ξ ) = - i Γ 1 2 a 1 , i a 2 ξ - α 2 2 a 2 s 2 - i δ a 2 s + a 1 2 exp ( i β ξ ) = - i Γ 2 2 a 2 ,
I = { A 1 2 + A 2 2 } d s ,
H = - 1 2 { r | A 1 s | 2 + α 2 | A 2 s | 2 - β A 2 2 + i δ 2 ( A 2 A 2 * s - A 2 * A 2 s ) + ( A 1 * 2 A 2 + A 1 2 A 2 * ) } d s ,
r 2 d 2 U 1 d s 2 + κ 1 U 1 - U 1 U 2 = 0 , α 2 d 2 U 2 d s 2 + ( 2 κ 1 + β ) U 2 - U 1 2 = 0.

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