Abstract

We investigate the effect of coupling between diffraction and walk-off on secondary instabilities in nondegenerate optical parametric oscillators. We show that traveling waves that propagate in the walk-off direction, which are generated at the onset of absolute instability, experience Eckhaus and zigzag phase instabilities. Each of these secondary instabilities splits into absolute and convective instabilities that modify the Eckhaus and zigzag instability boundaries. As a consequence, the stability domain of modulated traveling waves is enlarged and may coexist with uniform steady states. The predictions are consistent with the numerical solutions of the optical parametric oscillator model.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Vaupel, A. Maître, and C. Fabre, Phys. Rev. Lett. 83, 5278 (1999).
    [CrossRef]
  2. T. Nishikawa and N. Uesugi, Opt. Commun. 124, 512 (1996).
    [CrossRef]
  3. A. V. Smith, W. J. Alford, T. D. Raymond, and M. S. Bowers, J. Opt. Soc. Am. B 12, 473 (1995).
  4. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. E 58, 3843 (1998).
    [CrossRef]
  5. H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, Eur. Phys. J. D 3, 275–288 (1998).
    [CrossRef]
  6. M. Taki, M. San Miguel, and M. Santagiustina, Phys. Rev. E 61, 2133 (2000).
    [CrossRef]
  7. M. Taki, N. Ouarzazi, H. Ward, and P. Glorieux, J. Opt. Soc. Am. B 17, 997 (2000).
    [CrossRef]
  8. G. Izùs, M. Santagiustina, M. San Miguel, and P. Colet, J. Opt. Soc. Am. B 16, 1592 (1999).
    [CrossRef]
  9. H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, Phys. Rev. E 63, 016604-1 (2001).
  10. G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994).
    [CrossRef] [PubMed]
  11. M. Tlidi, P. Mandel, and R. Lefever, Phys. Rev. Lett. 81, 979 (1998).
    [CrossRef]
  12. G. J. de Varcacel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
    [CrossRef]
  13. S. Longhi, J. Mod. Opt. 43, 1089 (1996).
  14. J. N. Kutz, T. Erneux, S. Trillo, and M. Haelterman, J. Opt. Soc. Am. B 16, 1936 (1999).
    [CrossRef]
  15. M. Le Berre, D. Leduc, S. Patrascu, E. Ressayre, and A. Tallet, Chaos Solitons Fractals 10, 627 (1999).
    [CrossRef]
  16. J. M. Chomaz, A. Couairon, and S. Julien, Phys. Fluids 11, 3369 (1999).
    [CrossRef]

2001 (1)

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, Phys. Rev. E 63, 016604-1 (2001).

2000 (2)

M. Taki, M. San Miguel, and M. Santagiustina, Phys. Rev. E 61, 2133 (2000).
[CrossRef]

M. Taki, N. Ouarzazi, H. Ward, and P. Glorieux, J. Opt. Soc. Am. B 17, 997 (2000).
[CrossRef]

1999 (5)

M. Le Berre, D. Leduc, S. Patrascu, E. Ressayre, and A. Tallet, Chaos Solitons Fractals 10, 627 (1999).
[CrossRef]

J. M. Chomaz, A. Couairon, and S. Julien, Phys. Fluids 11, 3369 (1999).
[CrossRef]

G. Izùs, M. Santagiustina, M. San Miguel, and P. Colet, J. Opt. Soc. Am. B 16, 1592 (1999).
[CrossRef]

J. N. Kutz, T. Erneux, S. Trillo, and M. Haelterman, J. Opt. Soc. Am. B 16, 1936 (1999).
[CrossRef]

M. Vaupel, A. Maître, and C. Fabre, Phys. Rev. Lett. 83, 5278 (1999).
[CrossRef]

1998 (3)

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. E 58, 3843 (1998).
[CrossRef]

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, Eur. Phys. J. D 3, 275–288 (1998).
[CrossRef]

M. Tlidi, P. Mandel, and R. Lefever, Phys. Rev. Lett. 81, 979 (1998).
[CrossRef]

1996 (3)

G. J. de Varcacel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef]

S. Longhi, J. Mod. Opt. 43, 1089 (1996).

T. Nishikawa and N. Uesugi, Opt. Commun. 124, 512 (1996).
[CrossRef]

1995 (1)

A. V. Smith, W. J. Alford, T. D. Raymond, and M. S. Bowers, J. Opt. Soc. Am. B 12, 473 (1995).

1994 (1)

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

Alford, W. J.

A. V. Smith, W. J. Alford, T. D. Raymond, and M. S. Bowers, J. Opt. Soc. Am. B 12, 473 (1995).

Bowers, M. S.

A. V. Smith, W. J. Alford, T. D. Raymond, and M. S. Bowers, J. Opt. Soc. Am. B 12, 473 (1995).

Brambilla, M.

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

Chomaz, J. M.

J. M. Chomaz, A. Couairon, and S. Julien, Phys. Fluids 11, 3369 (1999).
[CrossRef]

Colet, P.

G. Izùs, M. Santagiustina, M. San Miguel, and P. Colet, J. Opt. Soc. Am. B 16, 1592 (1999).
[CrossRef]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. E 58, 3843 (1998).
[CrossRef]

Couairon, A.

J. M. Chomaz, A. Couairon, and S. Julien, Phys. Fluids 11, 3369 (1999).
[CrossRef]

de Varcacel, G. J.

G. J. de Varcacel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef]

Erneux, T.

Fabre, C.

M. Vaupel, A. Maître, and C. Fabre, Phys. Rev. Lett. 83, 5278 (1999).
[CrossRef]

Glorieux, P.

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, Phys. Rev. E 63, 016604-1 (2001).

M. Taki, N. Ouarzazi, H. Ward, and P. Glorieux, J. Opt. Soc. Am. B 17, 997 (2000).
[CrossRef]

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, Eur. Phys. J. D 3, 275–288 (1998).
[CrossRef]

Haelterman, M.

Izùs, G.

Julien, S.

J. M. Chomaz, A. Couairon, and S. Julien, Phys. Fluids 11, 3369 (1999).
[CrossRef]

Kutz, J. N.

Le Berre, M.

M. Le Berre, D. Leduc, S. Patrascu, E. Ressayre, and A. Tallet, Chaos Solitons Fractals 10, 627 (1999).
[CrossRef]

Leduc, D.

M. Le Berre, D. Leduc, S. Patrascu, E. Ressayre, and A. Tallet, Chaos Solitons Fractals 10, 627 (1999).
[CrossRef]

Lefever, R.

M. Tlidi, P. Mandel, and R. Lefever, Phys. Rev. Lett. 81, 979 (1998).
[CrossRef]

Longhi, S.

S. Longhi, J. Mod. Opt. 43, 1089 (1996).

Lugiato, L. A.

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

Maître, A.

M. Vaupel, A. Maître, and C. Fabre, Phys. Rev. Lett. 83, 5278 (1999).
[CrossRef]

Mandel, P.

M. Tlidi, P. Mandel, and R. Lefever, Phys. Rev. Lett. 81, 979 (1998).
[CrossRef]

Nishikawa, T.

T. Nishikawa and N. Uesugi, Opt. Commun. 124, 512 (1996).
[CrossRef]

Oppo, G.-L.

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

Ouarzazi, M. N.

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, Phys. Rev. E 63, 016604-1 (2001).

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, Eur. Phys. J. D 3, 275–288 (1998).
[CrossRef]

Ouarzazi, N.

Patrascu, S.

M. Le Berre, D. Leduc, S. Patrascu, E. Ressayre, and A. Tallet, Chaos Solitons Fractals 10, 627 (1999).
[CrossRef]

Raymond, T. D.

A. V. Smith, W. J. Alford, T. D. Raymond, and M. S. Bowers, J. Opt. Soc. Am. B 12, 473 (1995).

Ressayre, E.

M. Le Berre, D. Leduc, S. Patrascu, E. Ressayre, and A. Tallet, Chaos Solitons Fractals 10, 627 (1999).
[CrossRef]

Roldan, E.

G. J. de Varcacel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef]

San Miguel, M.

M. Taki, M. San Miguel, and M. Santagiustina, Phys. Rev. E 61, 2133 (2000).
[CrossRef]

G. Izùs, M. Santagiustina, M. San Miguel, and P. Colet, J. Opt. Soc. Am. B 16, 1592 (1999).
[CrossRef]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. E 58, 3843 (1998).
[CrossRef]

Sanchez-Morcillo, V. J.

G. J. de Varcacel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef]

Santagiustina, M.

M. Taki, M. San Miguel, and M. Santagiustina, Phys. Rev. E 61, 2133 (2000).
[CrossRef]

G. Izùs, M. Santagiustina, M. San Miguel, and P. Colet, J. Opt. Soc. Am. B 16, 1592 (1999).
[CrossRef]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. E 58, 3843 (1998).
[CrossRef]

Smith, A. V.

A. V. Smith, W. J. Alford, T. D. Raymond, and M. S. Bowers, J. Opt. Soc. Am. B 12, 473 (1995).

Staliunas, K.

G. J. de Varcacel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef]

Taki, M.

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, Phys. Rev. E 63, 016604-1 (2001).

M. Taki, M. San Miguel, and M. Santagiustina, Phys. Rev. E 61, 2133 (2000).
[CrossRef]

M. Taki, N. Ouarzazi, H. Ward, and P. Glorieux, J. Opt. Soc. Am. B 17, 997 (2000).
[CrossRef]

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, Eur. Phys. J. D 3, 275–288 (1998).
[CrossRef]

Tallet, A.

M. Le Berre, D. Leduc, S. Patrascu, E. Ressayre, and A. Tallet, Chaos Solitons Fractals 10, 627 (1999).
[CrossRef]

Tlidi, M.

M. Tlidi, P. Mandel, and R. Lefever, Phys. Rev. Lett. 81, 979 (1998).
[CrossRef]

Trillo, S.

Uesugi, N.

T. Nishikawa and N. Uesugi, Opt. Commun. 124, 512 (1996).
[CrossRef]

Vaupel, M.

M. Vaupel, A. Maître, and C. Fabre, Phys. Rev. Lett. 83, 5278 (1999).
[CrossRef]

Walgraef, D.

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. E 58, 3843 (1998).
[CrossRef]

Ward, H.

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, Phys. Rev. E 63, 016604-1 (2001).

M. Taki, N. Ouarzazi, H. Ward, and P. Glorieux, J. Opt. Soc. Am. B 17, 997 (2000).
[CrossRef]

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, Eur. Phys. J. D 3, 275–288 (1998).
[CrossRef]

Chaos Solitons Fractals (1)

M. Le Berre, D. Leduc, S. Patrascu, E. Ressayre, and A. Tallet, Chaos Solitons Fractals 10, 627 (1999).
[CrossRef]

Eur. Phys. J. D (1)

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, Eur. Phys. J. D 3, 275–288 (1998).
[CrossRef]

J. Mod. Opt. (1)

S. Longhi, J. Mod. Opt. 43, 1089 (1996).

J. Opt. Soc. Am. B (4)

Opt. Commun. (1)

T. Nishikawa and N. Uesugi, Opt. Commun. 124, 512 (1996).
[CrossRef]

Phys. Fluids (1)

J. M. Chomaz, A. Couairon, and S. Julien, Phys. Fluids 11, 3369 (1999).
[CrossRef]

Phys. Rev. A (2)

G. J. de Varcacel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef]

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

Phys. Rev. E (3)

M. Taki, M. San Miguel, and M. Santagiustina, Phys. Rev. E 61, 2133 (2000).
[CrossRef]

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, Phys. Rev. E 63, 016604-1 (2001).

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. E 58, 3843 (1998).
[CrossRef]

Phys. Rev. Lett. (2)

M. Vaupel, A. Maître, and C. Fabre, Phys. Rev. Lett. 83, 5278 (1999).
[CrossRef]

M. Tlidi, P. Mandel, and R. Lefever, Phys. Rev. Lett. 81, 979 (1998).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

(a) OPO with vanishing group velocity Vg=0, i.e., no walk-off αs=αi=0 and identical diffraction coefficients as=ai=0.5. The Eckhaus and zigzag thresholds are absolute at onset: There is no convective region for the secondary instability. (b) OPO with nonzero group velocity as=0.5,ai=0.42,αs=0.27,αi=0. ES, EC, and EA (ZS, ZC, ZA) indicate stable, convectively unstable, and absolutely unstable regions with respect to Eckhaus (zigzag) perturbations. Other parameters are γs=γi=1 and Δs=Δi=-0.14.

Fig. 2
Fig. 2

Time evolution (in units of γs) of the signal transverse profile in the zigzag absolutely unstable regime obtained by integration of Eqs. (1). The parameters are γp=γs=γi=1, Δp=2, Δs=Δi=-0.14, ap=0.23, as=0.5, ai=0.42, αs=0.27, and αi=0. The external pump is a super-Gaussian beam, E=E0 exp-r2/r02n, with E0=3.5 (50% above threshold), n=5, and r0=112. Top row, near field; middle row, far field; bottom row, real part near field.

Equations (3)

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

tAp=γp-1+iΔpAp+Ex,y-AsAi+iap2Ap, tAs=γs-1+iΔsAs+ApAi*+ias2As-αsxAs, tAi=γi-1+iΔiAi+ApAs*+iai2Ai-αixAi,
T+VgXA=A+ib2A+X-1/2iKcY22A-A2A.
λq=--k2-iVg+2kbqx-qx2-qy22Kc2k+qy22Kc+-k22+2qxk+qy22Kc+ibqx2+qy221/2.

Metrics