Abstract

We present the qualitative dynamics of parametric mixing of two waves in a quadratic nonlinear medium by introducing a reduced phase-plane description. We find the nonlinear frequency-conversion eigenmodes and their bifurcations and instabilities. We discuss applications of this description to the phase dependence of seeded second-harmonic generation, frequency division, and spatial chaos in a medium with periodic mismatch.

© 1992 Optical Society of America

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References

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  1. P. A. Franken, A. E. Hill, C. W. Peters, G. Weinreich, Phys. Lett. 7, 118 (1961).
    [Crossref]
  2. J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, Phys. Rev. 6, 1918 (1962).
    [Crossref]
  3. P. F. Byrd, M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer-Verlag, Berlin, 1971).
  4. S. A. Akhmanov, R. V. Khokhlov, Problems of Nonlinear Optics (Gordon and Breach, New York, 1972).
  5. A. Yariv, Quantum Electronics (Wiley, New York, 1989).
  6. X. M. Zhao, D. J. McGraw, “Parametric mode locking,” IEEE J. Quantum Electron. (to be published).
  7. S. Somekh, A. Yariv, Appl. Phys. Lett. 21, 140 (1972).
    [Crossref]
  8. J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983), p. 184.

1972 (2)

S. A. Akhmanov, R. V. Khokhlov, Problems of Nonlinear Optics (Gordon and Breach, New York, 1972).

S. Somekh, A. Yariv, Appl. Phys. Lett. 21, 140 (1972).
[Crossref]

1962 (1)

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

1961 (1)

P. A. Franken, A. E. Hill, C. W. Peters, G. Weinreich, Phys. Lett. 7, 118 (1961).
[Crossref]

Akhmanov, S. A.

S. A. Akhmanov, R. V. Khokhlov, Problems of Nonlinear Optics (Gordon and Breach, New York, 1972).

Armstrong, J. A.

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]

Byrd, P. F.

P. F. Byrd, M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer-Verlag, Berlin, 1971).

Ducuing, J.

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

Franken, P. A.

P. A. Franken, A. E. Hill, C. W. Peters, G. Weinreich, Phys. Lett. 7, 118 (1961).
[Crossref]

Friedman, M. D.

P. F. Byrd, M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer-Verlag, Berlin, 1971).

Guckenheimer, J.

J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983), p. 184.

Hill, A. E.

P. A. Franken, A. E. Hill, C. W. Peters, G. Weinreich, Phys. Lett. 7, 118 (1961).
[Crossref]

Holmes, P.

J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983), p. 184.

Khokhlov, R. V.

S. A. Akhmanov, R. V. Khokhlov, Problems of Nonlinear Optics (Gordon and Breach, New York, 1972).

McGraw, D. J.

X. M. Zhao, D. J. McGraw, “Parametric mode locking,” IEEE J. Quantum Electron. (to be published).

Pershan, P. S.

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

Peters, C. W.

P. A. Franken, A. E. Hill, C. W. Peters, G. Weinreich, Phys. Lett. 7, 118 (1961).
[Crossref]

Somekh, S.

S. Somekh, A. Yariv, Appl. Phys. Lett. 21, 140 (1972).
[Crossref]

Weinreich, G.

P. A. Franken, A. E. Hill, C. W. Peters, G. Weinreich, Phys. Lett. 7, 118 (1961).
[Crossref]

Yariv, A.

S. Somekh, A. Yariv, Appl. Phys. Lett. 21, 140 (1972).
[Crossref]

A. Yariv, Quantum Electronics (Wiley, New York, 1989).

Zhao, X. M.

X. M. Zhao, D. J. McGraw, “Parametric mode locking,” IEEE J. Quantum Electron. (to be published).

Appl. Phys. Lett. (1)

S. Somekh, A. Yariv, Appl. Phys. Lett. 21, 140 (1972).
[Crossref]

Phys. Lett. (1)

P. A. Franken, A. E. Hill, C. W. Peters, G. Weinreich, Phys. Lett. 7, 118 (1961).
[Crossref]

Phys. Rev. (1)

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

Problems of Nonlinear Optics (1)

S. A. Akhmanov, R. V. Khokhlov, Problems of Nonlinear Optics (Gordon and Breach, New York, 1972).

Other (4)

A. Yariv, Quantum Electronics (Wiley, New York, 1989).

X. M. Zhao, D. J. McGraw, “Parametric mode locking,” IEEE J. Quantum Electron. (to be published).

P. F. Byrd, M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer-Verlag, Berlin, 1971).

J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983), p. 184.

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Figures (4)

Fig. 1
Fig. 1

Bifurcation diagram: fraction of second-harmonic power ηe of the stable (solid curves) and unstable (dashed line) nonlinear eigenmodes versus the normalized mismatch κ.

Fig. 2
Fig. 2

Phase-plane portraits for (a) κ = −5 and (b) κ = 0.

Fig. 3
Fig. 3

Fraction of second-harmonic power versus the propagation distance z/L for different values of the initial relative phase for κ = 0 and (a) η0 = 0.1, γ = 8, (b) η0 = 0.999, γ = 15.

Fig. 4
Fig. 4

Fraction of second-harmonic power versus the propagation distance z/L with = 0.1 and γ = 32 different values of the spatial frequency of the periodic mismatch: Ω = 0.5 (solid curve) and Ω = 1 (dashed curve).

Equations (7)

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- i d A 0 d z = ω 0 R A 2 A 0 * exp ( i Δ k z ) ,
- i d A 2 d z = ω 0 R A 0 2 exp ( - i Δ k z ) ,
H = H ( a 0 , a 2 , a 0 * , a 2 * ) = 1 2 [ a 0 2 a 2 * + ( a 0 * ) 2 a 2 ] + κ a 2 2 .
H = H ( η , ϕ ) = 2 η ( 1 - η ) cos ϕ + κ η .
ξ = η 0 η ( ξ ) [ H ( η , ϕ ) ϕ ] - 1 d η = 1 2 η 0 η ( ξ ) 1 E - V ( η ) d η ,
H = H 0 ( η , ϕ ) + H 1 ( η , ϕ , ξ ) = 2 η ( 1 - η ) cos ϕ + cos ( Ω ξ ) η .
M ( ξ 0 ) = - + ( H 0 η H 1 ϕ - H 1 η H 0 ϕ ) ( ξ + ξ 0 ) d ξ = π Ω 2 sin ( Ω ξ 0 ) sinh ( π Ω / 2 ) ,

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