Abstract

We analyze pattern formation in doubly resonant intracavity second-harmonic generation in the presence of competing nondegenerate parametric downconversion. We show that for positive cavity detuning of the fundamental frequency the threshold for parametric oscillation is lower than that of transverse, pattern forming instabilities. The parametric oscillation strongly modifies the pattern dynamics found previously in a simplified analysis that neglects parametric instability [Phys. Rev. E 56, 4803 (1997)]. Stationary and dynamic patterns in the presence of parametric oscillation are found numerically.

© 2000 Optical Society of America

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References

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  1. G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994).
    [CrossRef] [PubMed]
  2. K. Staliunas, J. Mod. Opt. 42, 1261 (1995); G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
    [CrossRef]
  3. S. Longhi, Phys. Rev. A 53, 4488 (1996).
    [CrossRef] [PubMed]
  4. C. Etrich, U. Peschel, and F. Lederer, Phys. Rev. E 56, 4803 (1997).
    [CrossRef]
  5. S. Longhi, Opt. Lett. 23, 346 (1998).
    [CrossRef]
  6. P. Lodahl and M. Saffman, Phys. Rev. A 60, 3251 (1999).
    [CrossRef]
  7. L. A. Lugiato and A. Gatti, Phys. Rev. Lett. 70, 3868 (1993); A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G.-L. Oppo, and S. M. Barnett, Phys. Rev. A 56, 877 (1997).
    [CrossRef] [PubMed]
  8. M. A. M. Marte, Phys. Rev. Lett. 74, 4815 (1995).
    [CrossRef] [PubMed]
  9. S. Schiller and R. L. Byer, J. Opt. Soc. Am. B 10, 1696 (1993); A. G. White, P. K. Lam, M. S. Taubman, M. A. M. Marte, S. Schiller, D. E. McClelland, and H.-A. Bachor, Phys. Rev. A 55, 4511 (1997).
    [CrossRef]
  10. M. A. M. Marte, Phys. Rev. A 49, R3166 (1994).
    [CrossRef]

1999 (1)

P. Lodahl and M. Saffman, Phys. Rev. A 60, 3251 (1999).
[CrossRef]

1998 (1)

1997 (1)

C. Etrich, U. Peschel, and F. Lederer, Phys. Rev. E 56, 4803 (1997).
[CrossRef]

1996 (1)

S. Longhi, Phys. Rev. A 53, 4488 (1996).
[CrossRef] [PubMed]

1995 (2)

K. Staliunas, J. Mod. Opt. 42, 1261 (1995); G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef]

M. A. M. Marte, Phys. Rev. Lett. 74, 4815 (1995).
[CrossRef] [PubMed]

1994 (2)

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994).
[CrossRef] [PubMed]

M. A. M. Marte, Phys. Rev. A 49, R3166 (1994).
[CrossRef]

1993 (2)

S. Schiller and R. L. Byer, J. Opt. Soc. Am. B 10, 1696 (1993); A. G. White, P. K. Lam, M. S. Taubman, M. A. M. Marte, S. Schiller, D. E. McClelland, and H.-A. Bachor, Phys. Rev. A 55, 4511 (1997).
[CrossRef]

L. A. Lugiato and A. Gatti, Phys. Rev. Lett. 70, 3868 (1993); A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G.-L. Oppo, and S. M. Barnett, Phys. Rev. A 56, 877 (1997).
[CrossRef] [PubMed]

Brambilla, M.

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994).
[CrossRef] [PubMed]

Byer, R. L.

Etrich, C.

C. Etrich, U. Peschel, and F. Lederer, Phys. Rev. E 56, 4803 (1997).
[CrossRef]

Gatti, A.

L. A. Lugiato and A. Gatti, Phys. Rev. Lett. 70, 3868 (1993); A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G.-L. Oppo, and S. M. Barnett, Phys. Rev. A 56, 877 (1997).
[CrossRef] [PubMed]

Lederer, F.

C. Etrich, U. Peschel, and F. Lederer, Phys. Rev. E 56, 4803 (1997).
[CrossRef]

Lodahl, P.

P. Lodahl and M. Saffman, Phys. Rev. A 60, 3251 (1999).
[CrossRef]

Longhi, S.

Lugiato, L. A.

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994).
[CrossRef] [PubMed]

L. A. Lugiato and A. Gatti, Phys. Rev. Lett. 70, 3868 (1993); A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G.-L. Oppo, and S. M. Barnett, Phys. Rev. A 56, 877 (1997).
[CrossRef] [PubMed]

Marte, M. A. M.

M. A. M. Marte, Phys. Rev. Lett. 74, 4815 (1995).
[CrossRef] [PubMed]

M. A. M. Marte, Phys. Rev. A 49, R3166 (1994).
[CrossRef]

Oppo, G.-L.

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994).
[CrossRef] [PubMed]

Peschel, U.

C. Etrich, U. Peschel, and F. Lederer, Phys. Rev. E 56, 4803 (1997).
[CrossRef]

Saffman, M.

P. Lodahl and M. Saffman, Phys. Rev. A 60, 3251 (1999).
[CrossRef]

Schiller, S.

Staliunas, K.

K. Staliunas, J. Mod. Opt. 42, 1261 (1995); G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef]

J. Mod. Opt. (1)

K. Staliunas, J. Mod. Opt. 42, 1261 (1995); G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef]

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

Opt. Lett. (1)

Phys. Rev. A (4)

P. Lodahl and M. Saffman, Phys. Rev. A 60, 3251 (1999).
[CrossRef]

M. A. M. Marte, Phys. Rev. A 49, R3166 (1994).
[CrossRef]

S. Longhi, Phys. Rev. A 53, 4488 (1996).
[CrossRef] [PubMed]

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994).
[CrossRef] [PubMed]

Phys. Rev. E (1)

C. Etrich, U. Peschel, and F. Lederer, Phys. Rev. E 56, 4803 (1997).
[CrossRef]

Phys. Rev. Lett. (2)

L. A. Lugiato and A. Gatti, Phys. Rev. Lett. 70, 3868 (1993); A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G.-L. Oppo, and S. M. Barnett, Phys. Rev. A 56, 877 (1997).
[CrossRef] [PubMed]

M. A. M. Marte, Phys. Rev. Lett. 74, 4815 (1995).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Pump amplitude E for the instability thresholds as a function of second-harmonic detuning Δ2 and with γ=0.6 and Δ1=2. Thin solid curve, parametric process; thick solid curve, stationary transverse instability; thin dashed curve, self-pulsing instability; thick dashed curve, oscillatory transverse instability.

Fig. 2
Fig. 2

Bifurcation diagram for Δ1=-2 and otherwise the same parameters and notation as for Fig. 1.

Fig. 3
Fig. 3

Comparison between analytical (solid curve) and numerical (filled squares) solutions for Δ1=2, Δ2=-2, γ=0.6.

Fig. 4
Fig. 4

Near-field intensity distributions for Δ1=-2, Δ2=-1, γ=0.6, E=3.4. The window size is 36×36.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

A1/t=-1+iΔ1A1+iA1*A2+i2A1+E,
A2/t=-γ+iΔ2A2+iA12+2iA+A-+i22A2,
A±/t=-1+iΔ±A±+iA*A2+i2A±,
λ˜2+2λ˜+1+k˜2-Δ12-A2022=0.
A1,p02=γ2+Δ221/21+k˜2-Δ121/2.

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