Abstract

A symbiotic pair of dark (second-harmonic) and bright (fundamental frequency) solitons sustain each other and travel locked in second-harmonic generation dominated by group-velocity difference or spatial walk-off. The conditions of existence of the soliton pair are determined, and its stability in the presence of dispersion is discussed.

© 1996 Optical Society of America

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  1. Y. N. Karamzin, A. P. Sukhorukov, JETP Lett. 20, 339 (1974 ); Sov. Phys. JETP 41, 414 (1976).
  2. K. Hayata, M. Koshiba, Phys. Rev. Lett. 20, 3275 (1993 ); Phys. Rev. A 50, 675 (1994).
    [CrossRef] [PubMed]
  3. M. J. Werner, P. D. Drummond, J. Opt. Soc. Am. B 10, 2390 (1993 ); Opt. Lett. 19, 813 (1994).
    [CrossRef] [PubMed]
  4. M. A. Karpierz, M. Sypek, Opt. Commun. 110, 75 (1994).
    [CrossRef]
  5. A. V. Buryak, S. Kivshar Yu, Opt. Lett. 19, 1612 (1994 ); Phys. Rev. 51, R41 (1995 ); Phys. Lett. A 197, 407 (1995 ); Phys. Rev. Lett. 75, 591 (1995).
    [CrossRef] [PubMed]
  6. C. R. Menyuk, R. Schiek, L. Torner, J. Opt. Soc. B 11, 2434 (1994 ); L. Torner, C. R. Menyuk, G. Stegeman, Opt. Lett. 19, 1615 (1994); L. Torner, Opt. Commun. 114, 136 (1995).
    [CrossRef] [PubMed]
  7. S. Trillo, P. Ferro, Opt. Lett. 20, 438 (1995 ); Phys. Rev. E 51, 4994 (1995).
    [CrossRef] [PubMed]
  8. W. E. Torruelas, A. Wang, C. R. Manyuk, Phys. Rev. Lett. 74, 5036 (1995).
    [CrossRef]
  9. D. J. Kaup, Stud. Appl. Math. 59, 25 (1978).
  10. J. A. Armstrong, S. S. Jha, N. S. Shiren, IEEE J. Quantum Electron. QE-6, 123 (1970 ); K. Nozaki, T. Taniuti, J. Phys. Soc. Jpn. 34, 796 (1973).
    [CrossRef]
  11. S. Trillo, S. Wabnitz, R. Chisari, G. Cappellini, Opt. Lett. 17, 637 (1992).
    [CrossRef] [PubMed]
  12. J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 6, 1918 (1962).
    [CrossRef]
  13. S. Trillo, S. Wabnitz, E. M. Wright, G. I. Stegeman, Opt. Lett. 13, 871 (1989).
    [CrossRef]

1995 (2)

W. E. Torruelas, A. Wang, C. R. Manyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef]

S. Trillo, P. Ferro, Opt. Lett. 20, 438 (1995 ); Phys. Rev. E 51, 4994 (1995).
[CrossRef] [PubMed]

1994 (3)

A. V. Buryak, S. Kivshar Yu, Opt. Lett. 19, 1612 (1994 ); Phys. Rev. 51, R41 (1995 ); Phys. Lett. A 197, 407 (1995 ); Phys. Rev. Lett. 75, 591 (1995).
[CrossRef] [PubMed]

C. R. Menyuk, R. Schiek, L. Torner, J. Opt. Soc. B 11, 2434 (1994 ); L. Torner, C. R. Menyuk, G. Stegeman, Opt. Lett. 19, 1615 (1994); L. Torner, Opt. Commun. 114, 136 (1995).
[CrossRef] [PubMed]

M. A. Karpierz, M. Sypek, Opt. Commun. 110, 75 (1994).
[CrossRef]

1993 (2)

K. Hayata, M. Koshiba, Phys. Rev. Lett. 20, 3275 (1993 ); Phys. Rev. A 50, 675 (1994).
[CrossRef] [PubMed]

M. J. Werner, P. D. Drummond, J. Opt. Soc. Am. B 10, 2390 (1993 ); Opt. Lett. 19, 813 (1994).
[CrossRef] [PubMed]

1992 (1)

1989 (1)

1978 (1)

D. J. Kaup, Stud. Appl. Math. 59, 25 (1978).

1974 (1)

Y. N. Karamzin, A. P. Sukhorukov, JETP Lett. 20, 339 (1974 ); Sov. Phys. JETP 41, 414 (1976).

1970 (1)

J. A. Armstrong, S. S. Jha, N. S. Shiren, IEEE J. Quantum Electron. QE-6, 123 (1970 ); K. Nozaki, T. Taniuti, J. Phys. Soc. Jpn. 34, 796 (1973).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 6, 1918 (1962).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, S. S. Jha, N. S. Shiren, IEEE J. Quantum Electron. QE-6, 123 (1970 ); K. Nozaki, T. Taniuti, J. Phys. Soc. Jpn. 34, 796 (1973).
[CrossRef]

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 6, 1918 (1962).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 6, 1918 (1962).
[CrossRef]

Buryak, A. V.

Cappellini, G.

Chisari, R.

Drummond, P. D.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 6, 1918 (1962).
[CrossRef]

Ferro, P.

Hayata, K.

K. Hayata, M. Koshiba, Phys. Rev. Lett. 20, 3275 (1993 ); Phys. Rev. A 50, 675 (1994).
[CrossRef] [PubMed]

Jha, S. S.

J. A. Armstrong, S. S. Jha, N. S. Shiren, IEEE J. Quantum Electron. QE-6, 123 (1970 ); K. Nozaki, T. Taniuti, J. Phys. Soc. Jpn. 34, 796 (1973).
[CrossRef]

Karamzin, Y. N.

Y. N. Karamzin, A. P. Sukhorukov, JETP Lett. 20, 339 (1974 ); Sov. Phys. JETP 41, 414 (1976).

Karpierz, M. A.

M. A. Karpierz, M. Sypek, Opt. Commun. 110, 75 (1994).
[CrossRef]

Kaup, D. J.

D. J. Kaup, Stud. Appl. Math. 59, 25 (1978).

Koshiba, M.

K. Hayata, M. Koshiba, Phys. Rev. Lett. 20, 3275 (1993 ); Phys. Rev. A 50, 675 (1994).
[CrossRef] [PubMed]

Manyuk, C. R.

W. E. Torruelas, A. Wang, C. R. Manyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef]

Menyuk, C. R.

C. R. Menyuk, R. Schiek, L. Torner, J. Opt. Soc. B 11, 2434 (1994 ); L. Torner, C. R. Menyuk, G. Stegeman, Opt. Lett. 19, 1615 (1994); L. Torner, Opt. Commun. 114, 136 (1995).
[CrossRef] [PubMed]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 6, 1918 (1962).
[CrossRef]

Schiek, R.

C. R. Menyuk, R. Schiek, L. Torner, J. Opt. Soc. B 11, 2434 (1994 ); L. Torner, C. R. Menyuk, G. Stegeman, Opt. Lett. 19, 1615 (1994); L. Torner, Opt. Commun. 114, 136 (1995).
[CrossRef] [PubMed]

Shiren, N. S.

J. A. Armstrong, S. S. Jha, N. S. Shiren, IEEE J. Quantum Electron. QE-6, 123 (1970 ); K. Nozaki, T. Taniuti, J. Phys. Soc. Jpn. 34, 796 (1973).
[CrossRef]

Stegeman, G. I.

Sukhorukov, A. P.

Y. N. Karamzin, A. P. Sukhorukov, JETP Lett. 20, 339 (1974 ); Sov. Phys. JETP 41, 414 (1976).

Sypek, M.

M. A. Karpierz, M. Sypek, Opt. Commun. 110, 75 (1994).
[CrossRef]

Torner, L.

C. R. Menyuk, R. Schiek, L. Torner, J. Opt. Soc. B 11, 2434 (1994 ); L. Torner, C. R. Menyuk, G. Stegeman, Opt. Lett. 19, 1615 (1994); L. Torner, Opt. Commun. 114, 136 (1995).
[CrossRef] [PubMed]

Torruelas, W. E.

W. E. Torruelas, A. Wang, C. R. Manyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef]

Trillo, S.

Wabnitz, S.

Wang, A.

W. E. Torruelas, A. Wang, C. R. Manyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef]

Werner, M. J.

Wright, E. M.

Yu, S. Kivshar

IEEE J. Quantum Electron. (1)

J. A. Armstrong, S. S. Jha, N. S. Shiren, IEEE J. Quantum Electron. QE-6, 123 (1970 ); K. Nozaki, T. Taniuti, J. Phys. Soc. Jpn. 34, 796 (1973).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Opt. Soc. B (1)

C. R. Menyuk, R. Schiek, L. Torner, J. Opt. Soc. B 11, 2434 (1994 ); L. Torner, C. R. Menyuk, G. Stegeman, Opt. Lett. 19, 1615 (1994); L. Torner, Opt. Commun. 114, 136 (1995).
[CrossRef] [PubMed]

JETP Lett. (1)

Y. N. Karamzin, A. P. Sukhorukov, JETP Lett. 20, 339 (1974 ); Sov. Phys. JETP 41, 414 (1976).

Opt. Commun. (1)

M. A. Karpierz, M. Sypek, Opt. Commun. 110, 75 (1994).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 6, 1918 (1962).
[CrossRef]

Phys. Rev. Lett. (2)

W. E. Torruelas, A. Wang, C. R. Manyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef]

K. Hayata, M. Koshiba, Phys. Rev. Lett. 20, 3275 (1993 ); Phys. Rev. A 50, 675 (1994).
[CrossRef] [PubMed]

Stud. Appl. Math. (1)

D. J. Kaup, Stud. Appl. Math. 59, 25 (1978).

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Figures (3)

Fig. 1
Fig. 1

Phase-plane picture of separatrix trajectories of Eqs. (4) and (5) for different values of θ. For θ → −θ, separatrices are mirror images with respect to the θ = 0 trajectory.

Fig. 2
Fig. 2

Subharmonic (I1, solid curve) and harmonic (I2, dashed curve) intensities and phase ψ (dotted curve, in units of π) versus time τ for θ = 0.5 (here I set δ1 = 1).

Fig. 3
Fig. 3

Instability gain versus modulation frequency Ω, and θ, for δk = 0, δ1 = 1, δ2 = 2, and β1 = 0.1.

Equations (14)

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i ( u 1 ξ + δ 1 u 1 τ ) β 1 2 2 u 1 τ 2 + u 2 u 1 * exp ( i δ k ξ ) = 0 ,
i ( u 2 ξ + δ 2 u 2 τ ) β 2 2 2 u 2 τ 2 + u 1 2 2 exp ( i δ k ξ ) = 0 ,
u 1 = ( δ 2 / δ 1 ) x 1 ( τ ) exp ( i u ξ ) , u 2 = x 2 ( τ ) exp [ i ( 2 μ δ k ) ξ ] ,
x ˙ 1 = i ( x 2 x 1 * μ x 1 ) = i H 1 x 1 * ,
x ˙ 2 = i ( x 1 2 2 μ 2 x 2 ) = i H 1 x 2 * ,
ϕ ˙ = H r η , η ˙ = H r ϕ ,
H r = H r ( η , ϕ ) = θ η + 2 η ( 1 η ) cos ϕ .
θ 2 μ ( 1 δ 1 δ 2 ) + δ 1 δ 2 δ k
ϕ ˙ 1 = H r 2 μ + ( 2 θ ) η 2 ( 1 η ) ,
ϕ ˙ 2 = ( 4 μ θ ) η 2 + ( θ 4 μ H r ) η + H r 2 η ( 1 η ) .
u 1 = ( 2 a δ 2 / δ 1 ) 1 / 2 sech ( α τ ) exp ( i b τ + i μ ξ ) ,
u 2 = [ 1 a sec h 2 ( α τ ) ] 1 / 2 exp [ i 2 b τ + i ( 2 μ δ k ) ξ + i ψ ] ,
ψ = tan 1 { [ a / ( 1 a ) ] 1 / 2 tanh ( α τ ) } + ϕ 0 ,
λ ± = i b β 1 Ω ± 1 d ( Ω ) 2 ,

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