Abstract

We show that the eigenmodes of the two-wave interaction in a dispersive quadratic medium are unstable with respect to periodic perturbations. This mechanism represents a novel kind of modulational instability that involves the spontaneous generation of sideband pairs around both the fundamental and the second-harmonic frequencies.

© 1995 Optical Society of America

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References

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  1. J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 6, 1918 (1962).
    [CrossRef]
  2. R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. I. Stegeman, E. W. Van Stryland, H. Vanherzeele, Opt. Lett. 17, 28 (1992).
    [CrossRef] [PubMed]
  3. G. I. Stegeman, M. Sheik-Bahae, E. W. Van Stryland, G. Assanto, Opt. Lett. 18, 13 (1993).
    [CrossRef] [PubMed]
  4. S. Trillo, S. Wabnitz, R. Chisari, G. Cappellini, Opt. Lett. 17, 637 (1992).
    [CrossRef] [PubMed]
  5. A. E. Kaplan, Opt. Lett. 18, 1223 (1993).
    [CrossRef] [PubMed]
  6. T. B. Benjamin, J. E. Feir, J. Fluid Mech. 27, 417 (1967).
    [CrossRef]
  7. G. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
    [CrossRef] [PubMed]
  8. S. Wabnitz, Phys. Rev. A 38, 2018 (1988).
    [CrossRef] [PubMed]
  9. S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
    [CrossRef]
  10. P. Pliszka, P. P. Banerjee, J. Opt. Soc. Am. B 10, 1810 (1993).
    [CrossRef]
  11. Y. N. Karamzin, A. P. Sukhorukov, JETP Lett. 20, 339 (1974); Sov. Phys. JETP 41, 414 (1976).
  12. M. J. Werner, P. D. Drummond, J. Opt. Soc. Am. B 10, 2390 (1993); Opt. Lett. 19, 813 (1994).
    [CrossRef]
  13. M. A. Karpierz, M. Sypek, Opt. Commun. 110, 75 (1994); K. Hayata, M. Koshiba, Phys. Rev. A 50, 675 (1994).
    [CrossRef]

1994 (1)

M. A. Karpierz, M. Sypek, Opt. Commun. 110, 75 (1994); K. Hayata, M. Koshiba, Phys. Rev. A 50, 675 (1994).
[CrossRef]

1993 (4)

1992 (2)

1988 (1)

S. Wabnitz, Phys. Rev. A 38, 2018 (1988).
[CrossRef] [PubMed]

1987 (1)

G. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[CrossRef] [PubMed]

1974 (1)

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

1968 (1)

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

1967 (1)

T. B. Benjamin, J. E. Feir, J. Fluid Mech. 27, 417 (1967).
[CrossRef]

1962 (1)

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

Agrawal, G.

G. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[CrossRef] [PubMed]

Akhmanov, S. A.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

Armstrong, J. A.

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

Assanto, G.

Banerjee, P. P.

Benjamin, T. B.

T. B. Benjamin, J. E. Feir, J. Fluid Mech. 27, 417 (1967).
[CrossRef]

Bloembergen, N.

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

Cappellini, G.

Chirkin, A. S.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

Chisari, R.

DeSalvo, R.

Drabovich, K. N.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

Drummond, P. D.

Ducuing, J.

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

Feir, J. E.

T. B. Benjamin, J. E. Feir, J. Fluid Mech. 27, 417 (1967).
[CrossRef]

Hagan, D. J.

Kaplan, A. E.

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); K. Hayata, M. Koshiba, Phys. Rev. A 50, 675 (1994).
[CrossRef]

Khokhlov, R. V.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

Kovrigin, A. I.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

Pershan, P. S.

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

Pliszka, P.

Sheik-Bahae, M.

Stegeman, G. I.

Sukhorukov, A. P.

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

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

Sypek, M.

M. A. Karpierz, M. Sypek, Opt. Commun. 110, 75 (1994); K. Hayata, M. Koshiba, Phys. Rev. A 50, 675 (1994).
[CrossRef]

Trillo, S.

Van Stryland, E. W.

Vanherzeele, H.

Wabnitz, S.

Werner, M. J.

IEEE J. Quantum Electron. (1)

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, A. P. Sukhorukov, IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

J. Fluid Mech. (1)

T. B. Benjamin, J. E. Feir, J. Fluid Mech. 27, 417 (1967).
[CrossRef]

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

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); K. Hayata, M. Koshiba, Phys. Rev. A 50, 675 (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. A (1)

S. Wabnitz, Phys. Rev. A 38, 2018 (1988).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

G. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[CrossRef] [PubMed]

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Figures (2)

Fig. 1
Fig. 1

Modulation instability gain versus frequency detuning Ω, and the normalized wave-vector mismatch δk, for v = 0 and (a) β1 = β2 = −1 (i.e., in the normal dispersion regime), (b) β1 = β2 = 1 (i.e., in the anomalous dispersion regime).

Fig. 2
Fig. 2

Propagation of the dark solitary wave in the normal dispersion regime (β1 = −1, β2 = −2), for δk = 0.

Equations (8)

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i u 1 ξ + β 1 2 2 u 1 τ 2 + u 2 u 1 * exp ( i δ k ξ ) = 0 , i u 2 ξ + i v u 2 τ + β 2 2 + 2 u 2 τ 2 u 1 2 2 exp ( - i δ k ξ ) = 0 ,
u ¯ 1 = ± [ 2 μ ( 2 μ - δ k ) ] 1 / 2 ,             u ¯ 2 = μ ,
u 1 ( ξ , τ ) = [ u ¯ 1 + a 1 ( ξ , τ ) ] exp ( i μ ξ ) , u 2 ( ξ , τ ) = [ u ¯ 2 + a 2 ( ξ , τ ) ] exp i ( 2 μ - δ k ) ξ ] ,
i a 1 ξ + β 1 a 1 τ τ / 2 - μ a 1 + u ¯ 2 a 1 * + u ¯ 1 a 2 = 0 , i a 2 ξ + i v a 2 τ + β 2 a 2 τ τ / 2 - μ 2 a 2 + u ¯ 1 a 1 = 0 ,
a j ( ξ , τ ) = j s ( ξ ) exp ( i Ω τ ) + j a ( ξ ) exp ( - i Ω τ )             ( j = 1 , 2 ) ,
Q = i [ - Ω ¯ 1 2 u ¯ 2 u ¯ 1 0 - u ¯ 2 Ω ¯ 1 2 0 - u ¯ 1 u ¯ 1 0 - Ω ¯ 2 2 - v Ω 0 0 - u ¯ 1 0 Ω ¯ 2 2 - v Ω ] ,
2 λ ± = { f 0 ( Ω ) ± [ f 0 ( Ω ) 2 - f 1 ( Ω ) ] 1 / 2 } 1 / 2 ,
u 1 = u ¯ 1 f ( τ ) exp ( i μ ξ ) , u 2 = u ¯ 2 f ( τ ) exp [ i ( 2 μ - δ k ) ξ ] ,

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