Abstract

The existence of Turing patterns and of spatial solitary waves that arise from a polarization instability of the fundamental field is predicted in type  II second-harmonic generation in a quadratically nonlinear optical cavity. The analogy between these dissipative structures and those studied in absorptive optical bistability is pointed out.

© 1998 Optical Society of America

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References

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  1. A. V. Buryak and Y. S. Kivshar, Opt. Lett. 19, 1612 (1994); Opt. Lett. 20, 834 (1995); L. Torner, C. R. Menyuk, W. E. Torruellas, and G. I. Stegeman, Opt. Lett. 20, 13 (1995); A. V. Buryak, Y. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995).
    [CrossRef] [PubMed]
  2. W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
    [CrossRef] [PubMed]
  3. R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, Phys. Rev. Lett. 78, 2756 (1997).
    [CrossRef]
  4. A. V. Buryak, Y. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
    [CrossRef] [PubMed]
  5. G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994); S. Longhi, Phys. Rev. A 53, 4488 (1996); S. Longhi and A. Geraci, Phys. Rev. A 54, 4581 (1996); G. J. De Valcarcel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
    [CrossRef] [PubMed]
  6. S. Longhi, Opt. Lett. 20, 695 (1995); Opt. Lett. 21, 860 (1996); Phys. Scr. 56, 611 (1997).
    [CrossRef] [PubMed]
  7. S. Trillo and M. Haelterman, Opt. Lett. 21, 1114 (1996).
    [CrossRef] [PubMed]
  8. K. Staliunas and V. J. Sanchez-Morcillo, Opt. Commun. 139, 306 (1997).
    [CrossRef]
  9. S. Trillo, M. Haelterman, and A. Sheppard, Opt. Lett. 22, 970 (1997).
    [CrossRef] [PubMed]
  10. C. Etrich, U. Peschel, and F. Lederer, Phys. Rev. Lett. 79, 2454 (1997).
    [CrossRef]
  11. M. Tlidi, M. Georgiou, and P. Mandel, Phys. Rev. A 48, 4605 (1993).
    [CrossRef] [PubMed]
  12. M. Tlidi, P. Mandel, and R. Lefever, Phys. Rev. Lett. 73, 640 (1994).
    [CrossRef] [PubMed]
  13. J. B. Geddes, J. V. Moloney, E. M. Wright, and W. J. Firth, Opt. Commun. 111, 623 (1994).
    [CrossRef]
  14. Z. Y. Ou, Phys. Rev. A 49, 4902 (1994).
    [CrossRef] [PubMed]
  15. C. B. Clausen, O. Bang, and Y. S. Kivshar, Phys. Rev. Lett. 78, 4749 (1997).
    [CrossRef]
  16. In practice, one can achieve this by varying the polarization angle ? of the driving field around ?/4 [see Fig.??1(b)], keeping constant its amplitude E; in this way, for small angle variations, the effect at leading order is to change E- but not E+.

1997 (5)

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

C. Etrich, U. Peschel, and F. Lederer, Phys. Rev. Lett. 79, 2454 (1997).
[CrossRef]

C. B. Clausen, O. Bang, and Y. S. Kivshar, Phys. Rev. Lett. 78, 4749 (1997).
[CrossRef]

K. Staliunas and V. J. Sanchez-Morcillo, Opt. Commun. 139, 306 (1997).
[CrossRef]

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

1996 (2)

S. Trillo and M. Haelterman, Opt. Lett. 21, 1114 (1996).
[CrossRef] [PubMed]

A. V. Buryak, Y. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[CrossRef] [PubMed]

1995 (2)

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

S. Longhi, Opt. Lett. 20, 695 (1995); Opt. Lett. 21, 860 (1996); Phys. Scr. 56, 611 (1997).
[CrossRef] [PubMed]

1994 (5)

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994); S. Longhi, Phys. Rev. A 53, 4488 (1996); S. Longhi and A. Geraci, Phys. Rev. A 54, 4581 (1996); G. J. De Valcarcel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef] [PubMed]

A. V. Buryak and Y. S. Kivshar, Opt. Lett. 19, 1612 (1994); Opt. Lett. 20, 834 (1995); L. Torner, C. R. Menyuk, W. E. Torruellas, and G. I. Stegeman, Opt. Lett. 20, 13 (1995); A. V. Buryak, Y. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995).
[CrossRef] [PubMed]

M. Tlidi, P. Mandel, and R. Lefever, Phys. Rev. Lett. 73, 640 (1994).
[CrossRef] [PubMed]

J. B. Geddes, J. V. Moloney, E. M. Wright, and W. J. Firth, Opt. Commun. 111, 623 (1994).
[CrossRef]

Z. Y. Ou, Phys. Rev. A 49, 4902 (1994).
[CrossRef] [PubMed]

1993 (1)

M. Tlidi, M. Georgiou, and P. Mandel, Phys. Rev. A 48, 4605 (1993).
[CrossRef] [PubMed]

Baboiu, D.-M.

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

Bang, O.

C. B. Clausen, O. Bang, and Y. S. Kivshar, Phys. Rev. Lett. 78, 4749 (1997).
[CrossRef]

Brambilla, M.

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994); S. Longhi, Phys. Rev. A 53, 4488 (1996); S. Longhi and A. Geraci, Phys. Rev. A 54, 4581 (1996); G. J. De Valcarcel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef] [PubMed]

Buryak, A. V.

Clausen, C. B.

C. B. Clausen, O. Bang, and Y. S. Kivshar, Phys. Rev. Lett. 78, 4749 (1997).
[CrossRef]

Etrich, C.

C. Etrich, U. Peschel, and F. Lederer, Phys. Rev. Lett. 79, 2454 (1997).
[CrossRef]

Firth, W. J.

J. B. Geddes, J. V. Moloney, E. M. Wright, and W. J. Firth, Opt. Commun. 111, 623 (1994).
[CrossRef]

Fuerst, R. A.

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

Geddes, J. B.

J. B. Geddes, J. V. Moloney, E. M. Wright, and W. J. Firth, Opt. Commun. 111, 623 (1994).
[CrossRef]

Georgiou, M.

M. Tlidi, M. Georgiou, and P. Mandel, Phys. Rev. A 48, 4605 (1993).
[CrossRef] [PubMed]

Haelterman, M.

Hagan, D. J.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Kivshar, Y. S.

Lawrence, B.

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

Lederer, F.

C. Etrich, U. Peschel, and F. Lederer, Phys. Rev. Lett. 79, 2454 (1997).
[CrossRef]

Lefever, R.

M. Tlidi, P. Mandel, and R. Lefever, Phys. Rev. Lett. 73, 640 (1994).
[CrossRef] [PubMed]

Longhi, S.

Lugiato, L. A.

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994); S. Longhi, Phys. Rev. A 53, 4488 (1996); S. Longhi and A. Geraci, Phys. Rev. A 54, 4581 (1996); G. J. De Valcarcel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef] [PubMed]

Mandel, P.

M. Tlidi, P. Mandel, and R. Lefever, Phys. Rev. Lett. 73, 640 (1994).
[CrossRef] [PubMed]

M. Tlidi, M. Georgiou, and P. Mandel, Phys. Rev. A 48, 4605 (1993).
[CrossRef] [PubMed]

Menyuk, C. R.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Moloney, J. V.

J. B. Geddes, J. V. Moloney, E. M. Wright, and W. J. Firth, Opt. Commun. 111, 623 (1994).
[CrossRef]

Oppo, G.-L.

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994); S. Longhi, Phys. Rev. A 53, 4488 (1996); S. Longhi and A. Geraci, Phys. Rev. A 54, 4581 (1996); G. J. De Valcarcel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef] [PubMed]

Ou, Z. Y.

Z. Y. Ou, Phys. Rev. A 49, 4902 (1994).
[CrossRef] [PubMed]

Peschel, U.

C. Etrich, U. Peschel, and F. Lederer, Phys. Rev. Lett. 79, 2454 (1997).
[CrossRef]

Sanchez-Morcillo, V. J.

K. Staliunas and V. J. Sanchez-Morcillo, Opt. Commun. 139, 306 (1997).
[CrossRef]

Sheppard, A.

Staliunas, K.

K. Staliunas and V. J. Sanchez-Morcillo, Opt. Commun. 139, 306 (1997).
[CrossRef]

Stegeman, G. I.

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

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Tlidi, M.

M. Tlidi, P. Mandel, and R. Lefever, Phys. Rev. Lett. 73, 640 (1994).
[CrossRef] [PubMed]

M. Tlidi, M. Georgiou, and P. Mandel, Phys. Rev. A 48, 4605 (1993).
[CrossRef] [PubMed]

Torner, L.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Torruellas, W. E.

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

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Trillo, S.

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

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

S. Trillo and M. Haelterman, Opt. Lett. 21, 1114 (1996).
[CrossRef] [PubMed]

A. V. Buryak, Y. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[CrossRef] [PubMed]

Van Stryland, E. W.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Wabnitz, S.

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

Wang, Z.

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

Wright, E. M.

J. B. Geddes, J. V. Moloney, E. M. Wright, and W. J. Firth, Opt. Commun. 111, 623 (1994).
[CrossRef]

Opt. Commun. (2)

J. B. Geddes, J. V. Moloney, E. M. Wright, and W. J. Firth, Opt. Commun. 111, 623 (1994).
[CrossRef]

K. Staliunas and V. J. Sanchez-Morcillo, Opt. Commun. 139, 306 (1997).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (3)

M. Tlidi, M. Georgiou, and P. Mandel, Phys. Rev. A 48, 4605 (1993).
[CrossRef] [PubMed]

Z. Y. Ou, Phys. Rev. A 49, 4902 (1994).
[CrossRef] [PubMed]

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, Phys. Rev. A 49, 2028 (1994); S. Longhi, Phys. Rev. A 53, 4488 (1996); S. Longhi and A. Geraci, Phys. Rev. A 54, 4581 (1996); G. J. De Valcarcel, K. Staliunas, E. Roldan, and V. J. Sanchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef] [PubMed]

Phys. Rev. Lett. (6)

C. Etrich, U. Peschel, and F. Lederer, Phys. Rev. Lett. 79, 2454 (1997).
[CrossRef]

W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, Phys. Rev. Lett. 74, 5036 (1995).
[CrossRef] [PubMed]

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

A. V. Buryak, Y. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996).
[CrossRef] [PubMed]

C. B. Clausen, O. Bang, and Y. S. Kivshar, Phys. Rev. Lett. 78, 4749 (1997).
[CrossRef]

M. Tlidi, P. Mandel, and R. Lefever, Phys. Rev. Lett. 73, 640 (1994).
[CrossRef] [PubMed]

Other (1)

In practice, one can achieve this by varying the polarization angle ? of the driving field around ?/4 [see Fig.??1(b)], keeping constant its amplitude E; in this way, for small angle variations, the effect at leading order is to change E- but not E+.

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Figures (3)

Fig. 1
Fig. 1

Schematic of type  II SHG in an optical cavity. (a) The plane–plane cavity with a χ2 medium externally driven by a linearly polarized fundamental frequency field, (b) the polarization pumping geometry (ϑ=π/4 for symmetric pumping).

Fig. 2
Fig. 2

Neutral stability curves for I1 (dashed curves) and I2 (solid curves) instabilities for (a) Δ+=Δ0=0, (b) Δ+=Δ0=3. We have assumed that γ=γ0 and have scaled the transverse wave number k to a.

Fig. 3
Fig. 3

Domains of prevalence for instabilities I1 and I2 in the plane Δ0, Δ+ for γ=γ0.

Equations (5)

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

tA0=γ0-1+iΔ0A0+ia02A0-γ0A1A2, tA1=γ-1+iΔ1A1+ia2A1+γA2*A0+γE cos ϑ, tA2=γ-1+iΔ2A2+ia2A2+γA1*A0+γE sin ϑ,
tA0=γ0-1+iΔ0A0+ia02A0-γ02A+2-A-2, tA±=γ-1+iΔ+A±+ia2A±±γA±*A0-iγΔ-A+γE±,
XX+2-2Δ0Δ+2+4Δ0+Δ+2=41+Δ02E2.
1γtψ=-a224ψ+aΔ+2ψ+μψ-28ψ3+S,
A0=-1+iΔ++2ψ2-22E+-22+Δ0-Δ+2+4Δ+2/8, A+=2+iΔ0-Δ+2+22ψ2+4E+-22+2Δ02-3Δ+2+6Δ0Δ+16 A-=ψ-i2Δ-/2+ia2-2Δ+ψ/2.

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