Abstract

The effect of walk-off in pattern selection in optical parametric oscillators is theoretically examined. We show that a dynamic mechanism also allows us to observe the formation of structures for positive signal detunings. In this regime the pattern that is generated is a periodic array of kinks that separate regions in which one of two stable steady states is alternately selected. This structure can be regarded as a train of dark soliton stripes because the two solutions have opposite signs. The wavelength of the selected pattern is theoretically predicted, and the prediction agrees with the results of the numerical solutions of the equations governing the device.

© 1998 Optical Society of America

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  1. K. Staliunas, Opt. Commun. 91, 82 (1992); J. Mod. Opt. 42, 1261 (1995); G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994); G.-L. Oppo, M. Brambilla, D. Camesasca, A. Gatti, and L. A. Lugiato, J. Mod. Opt. 41, 1151 (1994); S. Longhi, J. Mod. Opt. 43, 1089 (1996); J. Mod. Opt. 43, 1569 (1996); Phys. Rev. A 53, 4488 (1996); S. Longhi and A. Geraci, Phys. Rev. A 54, 4581 (1996); G. J. de Varcarcel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1069 (1996); Phys. Rev. A 56, 3237 (1997).
    [CrossRef] [PubMed]
  2. R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruelas, G. I. Stegeman, and S. Trillo, Phys. Rev. Lett. 78, 2756 (1997).
    [CrossRef]
  3. H. J. Kimble, in Fundamental Systems in Quantum Optics, J. Dalibard, J. M. Raimond, and J. Zinn-Justine, eds. (Elsevier, Amsterdam, 1992), p. 545.
  4. L. A. Lugiato, A. Gatti, and H. Wiedemann, in Quantum Fluctuations, S. Reynaud, E. Giacobino, and J. Zinn-Justine, eds. (Elsevier, Amsterdam, 1997), p. 431.
  5. K. Staliunas and V. J. Sanchez-Morcillo, Opt. Commun. 139, 306 (1997); Phys. Rev. A 57, 1454; M. Tlidi, P. Mandel, and M. Haelterman, Phys. Rev. E 56, 6524 (1998).
    [CrossRef]
  6. S. Trillo, M. Haelterman, and A. Sheppard, Opt. Lett. 22, 970 (1997).
    [CrossRef] [PubMed]
  7. W. E. Torruelas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995); A. V. Smith, W. J. Alford, T. R. Raymond, and M. S. Bowers, J. Opt. Soc. Am. B 12, 2253 (1995).
    [CrossRef]
  8. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise sustained by structures in optical parametric oscillators,” Phys. Rev. E (to be published).
  9. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. Lett. 79, 3633 (1997).
    [CrossRef]
  10. G. T. Dee and W. Van Sarloos, Phys. Rev. Lett. 60, 2641 (1988).
    [CrossRef] [PubMed]
  11. R. Montagne, A. Amengual, E. Hernandez-Garcia, and M. San Miguel, Phys. Rev. E 50, 377 (1994); M. San Miguel, R. Montagne, A. Amengual, and E. Hernandez-Garcia, in Instabilities in Nonequilibrium Structures V, E. Tirapegui and W. Zeller, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996), p. 85.
    [CrossRef]
  12. P. D. Miller, N. N. Akhmediev, and A. Ankiewicz, Opt. Lett. 21, 1132 (1996).
    [CrossRef] [PubMed]

1997 (4)

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruelas, G. I. Stegeman, and S. Trillo, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

K. Staliunas and V. J. Sanchez-Morcillo, Opt. Commun. 139, 306 (1997); Phys. Rev. A 57, 1454; M. Tlidi, P. Mandel, and M. Haelterman, Phys. Rev. E 56, 6524 (1998).
[CrossRef]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. Lett. 79, 3633 (1997).
[CrossRef]

S. Trillo, M. Haelterman, and A. Sheppard, Opt. Lett. 22, 970 (1997).
[CrossRef] [PubMed]

1996 (1)

1995 (1)

W. E. Torruelas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995); A. V. Smith, W. J. Alford, T. R. Raymond, and M. S. Bowers, J. Opt. Soc. Am. B 12, 2253 (1995).
[CrossRef]

1994 (1)

R. Montagne, A. Amengual, E. Hernandez-Garcia, and M. San Miguel, Phys. Rev. E 50, 377 (1994); M. San Miguel, R. Montagne, A. Amengual, and E. Hernandez-Garcia, in Instabilities in Nonequilibrium Structures V, E. Tirapegui and W. Zeller, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996), p. 85.
[CrossRef]

1992 (1)

K. Staliunas, Opt. Commun. 91, 82 (1992); J. Mod. Opt. 42, 1261 (1995); G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994); G.-L. Oppo, M. Brambilla, D. Camesasca, A. Gatti, and L. A. Lugiato, J. Mod. Opt. 41, 1151 (1994); S. Longhi, J. Mod. Opt. 43, 1089 (1996); J. Mod. Opt. 43, 1569 (1996); Phys. Rev. A 53, 4488 (1996); S. Longhi and A. Geraci, Phys. Rev. A 54, 4581 (1996); G. J. de Varcarcel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1069 (1996); Phys. Rev. A 56, 3237 (1997).
[CrossRef] [PubMed]

1988 (1)

G. T. Dee and W. Van Sarloos, Phys. Rev. Lett. 60, 2641 (1988).
[CrossRef] [PubMed]

Akhmediev, N. N.

Amengual, A.

R. Montagne, A. Amengual, E. Hernandez-Garcia, and M. San Miguel, Phys. Rev. E 50, 377 (1994); M. San Miguel, R. Montagne, A. Amengual, and E. Hernandez-Garcia, in Instabilities in Nonequilibrium Structures V, E. Tirapegui and W. Zeller, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996), p. 85.
[CrossRef]

Ankiewicz, A.

Baboiu, D.-M.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruelas, G. I. Stegeman, and S. Trillo, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

Colet, P.

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. Lett. 79, 3633 (1997).
[CrossRef]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise sustained by structures in optical parametric oscillators,” Phys. Rev. E (to be published).

Dee, G. T.

G. T. Dee and W. Van Sarloos, Phys. Rev. Lett. 60, 2641 (1988).
[CrossRef] [PubMed]

Fuerst, R. A.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruelas, G. I. Stegeman, and S. Trillo, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

Gatti, A.

L. A. Lugiato, A. Gatti, and H. Wiedemann, in Quantum Fluctuations, S. Reynaud, E. Giacobino, and J. Zinn-Justine, eds. (Elsevier, Amsterdam, 1997), p. 431.

Haelterman, M.

Hagan, D. J.

W. E. Torruelas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995); A. V. Smith, W. J. Alford, T. R. Raymond, and M. S. Bowers, J. Opt. Soc. Am. B 12, 2253 (1995).
[CrossRef]

Hernandez-Garcia, E.

R. Montagne, A. Amengual, E. Hernandez-Garcia, and M. San Miguel, Phys. Rev. E 50, 377 (1994); M. San Miguel, R. Montagne, A. Amengual, and E. Hernandez-Garcia, in Instabilities in Nonequilibrium Structures V, E. Tirapegui and W. Zeller, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996), p. 85.
[CrossRef]

Kimble, H. J.

H. J. Kimble, in Fundamental Systems in Quantum Optics, J. Dalibard, J. M. Raimond, and J. Zinn-Justine, eds. (Elsevier, Amsterdam, 1992), p. 545.

Lawrence, B.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruelas, G. I. Stegeman, and S. Trillo, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

Lugiato, L. A.

L. A. Lugiato, A. Gatti, and H. Wiedemann, in Quantum Fluctuations, S. Reynaud, E. Giacobino, and J. Zinn-Justine, eds. (Elsevier, Amsterdam, 1997), p. 431.

Menyuk, C. R.

W. E. Torruelas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995); A. V. Smith, W. J. Alford, T. R. Raymond, and M. S. Bowers, J. Opt. Soc. Am. B 12, 2253 (1995).
[CrossRef]

Miller, P. D.

Montagne, R.

R. Montagne, A. Amengual, E. Hernandez-Garcia, and M. San Miguel, Phys. Rev. E 50, 377 (1994); M. San Miguel, R. Montagne, A. Amengual, and E. Hernandez-Garcia, in Instabilities in Nonequilibrium Structures V, E. Tirapegui and W. Zeller, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996), p. 85.
[CrossRef]

San Miguel, M.

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. Lett. 79, 3633 (1997).
[CrossRef]

R. Montagne, A. Amengual, E. Hernandez-Garcia, and M. San Miguel, Phys. Rev. E 50, 377 (1994); M. San Miguel, R. Montagne, A. Amengual, and E. Hernandez-Garcia, in Instabilities in Nonequilibrium Structures V, E. Tirapegui and W. Zeller, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996), p. 85.
[CrossRef]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise sustained by structures in optical parametric oscillators,” Phys. Rev. E (to be published).

Sanchez-Morcillo, V. J.

K. Staliunas and V. J. Sanchez-Morcillo, Opt. Commun. 139, 306 (1997); Phys. Rev. A 57, 1454; M. Tlidi, P. Mandel, and M. Haelterman, Phys. Rev. E 56, 6524 (1998).
[CrossRef]

Santagiustina, M.

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. Lett. 79, 3633 (1997).
[CrossRef]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise sustained by structures in optical parametric oscillators,” Phys. Rev. E (to be published).

Sheppard, A.

Staliunas, K.

K. Staliunas and V. J. Sanchez-Morcillo, Opt. Commun. 139, 306 (1997); Phys. Rev. A 57, 1454; M. Tlidi, P. Mandel, and M. Haelterman, Phys. Rev. E 56, 6524 (1998).
[CrossRef]

K. Staliunas, Opt. Commun. 91, 82 (1992); J. Mod. Opt. 42, 1261 (1995); G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994); G.-L. Oppo, M. Brambilla, D. Camesasca, A. Gatti, and L. A. Lugiato, J. Mod. Opt. 41, 1151 (1994); S. Longhi, J. Mod. Opt. 43, 1089 (1996); J. Mod. Opt. 43, 1569 (1996); Phys. Rev. A 53, 4488 (1996); S. Longhi and A. Geraci, Phys. Rev. A 54, 4581 (1996); G. J. de Varcarcel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1069 (1996); Phys. Rev. A 56, 3237 (1997).
[CrossRef] [PubMed]

Stegeman, G. I.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruelas, G. I. Stegeman, and S. Trillo, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

W. E. Torruelas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995); A. V. Smith, W. J. Alford, T. R. Raymond, and M. S. Bowers, J. Opt. Soc. Am. B 12, 2253 (1995).
[CrossRef]

Torner, L.

W. E. Torruelas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995); A. V. Smith, W. J. Alford, T. R. Raymond, and M. S. Bowers, J. Opt. Soc. Am. B 12, 2253 (1995).
[CrossRef]

Torruelas, W. E.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruelas, G. I. Stegeman, and S. Trillo, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

W. E. Torruelas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995); A. V. Smith, W. J. Alford, T. R. Raymond, and M. S. Bowers, J. Opt. Soc. Am. B 12, 2253 (1995).
[CrossRef]

Trillo, S.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruelas, G. I. Stegeman, and S. Trillo, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

S. Trillo, M. Haelterman, and A. Sheppard, Opt. Lett. 22, 970 (1997).
[CrossRef] [PubMed]

Van Sarloos, W.

G. T. Dee and W. Van Sarloos, Phys. Rev. Lett. 60, 2641 (1988).
[CrossRef] [PubMed]

Van Stryland, E. W.

W. E. Torruelas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995); A. V. Smith, W. J. Alford, T. R. Raymond, and M. S. Bowers, J. Opt. Soc. Am. B 12, 2253 (1995).
[CrossRef]

Walgraef, D.

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. Lett. 79, 3633 (1997).
[CrossRef]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise sustained by structures in optical parametric oscillators,” Phys. Rev. E (to be published).

Wang, Z.

W. E. Torruelas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995); A. V. Smith, W. J. Alford, T. R. Raymond, and M. S. Bowers, J. Opt. Soc. Am. B 12, 2253 (1995).
[CrossRef]

Wiedemann, H.

L. A. Lugiato, A. Gatti, and H. Wiedemann, in Quantum Fluctuations, S. Reynaud, E. Giacobino, and J. Zinn-Justine, eds. (Elsevier, Amsterdam, 1997), p. 431.

Opt. Commun. (2)

K. Staliunas, Opt. Commun. 91, 82 (1992); J. Mod. Opt. 42, 1261 (1995); G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994); G.-L. Oppo, M. Brambilla, D. Camesasca, A. Gatti, and L. A. Lugiato, J. Mod. Opt. 41, 1151 (1994); S. Longhi, J. Mod. Opt. 43, 1089 (1996); J. Mod. Opt. 43, 1569 (1996); Phys. Rev. A 53, 4488 (1996); S. Longhi and A. Geraci, Phys. Rev. A 54, 4581 (1996); G. J. de Varcarcel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1069 (1996); Phys. Rev. A 56, 3237 (1997).
[CrossRef] [PubMed]

K. Staliunas and V. J. Sanchez-Morcillo, Opt. Commun. 139, 306 (1997); Phys. Rev. A 57, 1454; M. Tlidi, P. Mandel, and M. Haelterman, Phys. Rev. E 56, 6524 (1998).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. E (1)

R. Montagne, A. Amengual, E. Hernandez-Garcia, and M. San Miguel, Phys. Rev. E 50, 377 (1994); M. San Miguel, R. Montagne, A. Amengual, and E. Hernandez-Garcia, in Instabilities in Nonequilibrium Structures V, E. Tirapegui and W. Zeller, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996), p. 85.
[CrossRef]

Phys. Rev. Lett. (4)

W. E. Torruelas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995); A. V. Smith, W. J. Alford, T. R. Raymond, and M. S. Bowers, J. Opt. Soc. Am. B 12, 2253 (1995).
[CrossRef]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. Lett. 79, 3633 (1997).
[CrossRef]

G. T. Dee and W. Van Sarloos, Phys. Rev. Lett. 60, 2641 (1988).
[CrossRef] [PubMed]

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruelas, G. I. Stegeman, and S. Trillo, Phys. Rev. Lett. 78, 2756 (1997).
[CrossRef]

Other (3)

H. J. Kimble, in Fundamental Systems in Quantum Optics, J. Dalibard, J. M. Raimond, and J. Zinn-Justine, eds. (Elsevier, Amsterdam, 1992), p. 545.

L. A. Lugiato, A. Gatti, and H. Wiedemann, in Quantum Fluctuations, S. Reynaud, E. Giacobino, and J. Zinn-Justine, eds. (Elsevier, Amsterdam, 1997), p. 431.

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise sustained by structures in optical parametric oscillators,” Phys. Rev. E (to be published).

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Figures (3)

Fig. 1
Fig. 1

Solid curve, predicted wave vector of the pattern for ρ1=0.15. Asterisks, wave vector selected in the numerical solutions; dashed curve, predicted wave vector for ρ1=0. Numerical solutions are calculated for E0 with a value that is always 1% above the absolute instability threshold (see text) and with Δ0=0. We also set γ0,1=1, a0=a1/2=0.125, and K0=1.

Fig. 2
Fig. 2

Transverse intensity pattern at t=2000. The parameters used are Δ0=0,Δ1=0.3,ρ1=0.15,E0=1.12736,γ0,1=1,a0=a1/2=0.125, and K0=1.

Fig. 3
Fig. 3

Real (solid curve) and imaginary (dashed curve) parts of the electric field envelope as a function of the space variable y for x=0. The values of the uniform regions agree with the solutions predicted by Eq.  (4), i.e., A1±=±0.166+i0.126. Other parameters are as in Fig.  2.

Equations (7)

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

tA0=γ0-1+iΔ0A0+E0+ia02A0+2iK0A12,
tA1=γ1-1+iΔ1A1+ρ1yA1+ia12A1+iK0A1*A0,
A1±=±I21/21+Δ0+Δ1K0E01/2+i1-Δ0+Δ1K0E01/2,
I=12K02Δ0Δ1-1+K02E02-Δ0+Δ121/2.
kλk|ks=0, Re[k2λk|ks]0,
λk=-1+ρ1ky+F2-a12qc2+kx2+ky221/2,
Qy=Imλksρ1.

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