Abstract

We find families of vortex solitary waves in bulk quadratic nonlinear media under conditions for second-harmonic generation. We show that the vortex solitary waves are azimuthally unstable and that they decay into sets of stable spatial solitons. We calculate the growth rates of the azimuthal perturbations and show how those affect the pattern of output light.

© 1998 Optical Society of America

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References

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  1. J. F. Nye and M. V. Berry, Proc. R. Soc. London, Ser. A 336, 165 (1974).
    [CrossRef]
  2. G. Indebetouw, J. Mod. Opt. 40, 73 (1993); F. S. Roux, J. Opt. Soc. Am. B 12, 1215 (1995).
    [CrossRef]
  3. V. I. Kruglov and R. A. Vlasov, Phys. Lett. A 111, 401 (1985); I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 422 (1993); N. N. Rosanov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1996), Vol. 35.
    [CrossRef]
  4. G. A. Swartzlander and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992); V. Tikhonenko, J. Christou, and B. Luther-Davies, J. Opt. Soc. Am. B 12, 2046 (1995).
    [CrossRef] [PubMed]
  5. A. V. Mamaev, M. Saffman, and A. A. Zozulya, Phys. Rev. Lett. 78, 2108 (1997); Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, Phys. Rev. Lett. 78, 2948 (1997).
    [CrossRef]
  6. L. Torner and D. V. Petrov, Electron. Lett. 33, 608 (1997).
    [CrossRef]
  7. C. R. Menyuk, R. Schiek, and L. Torner, J. Opt. Soc. Am. B 11, 2434 (1994).
    [CrossRef]
  8. A. V. Buryak, Y. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995); L. Torner, D. Mihalache, D. Mazilu, E. W. Wright, W. E. Torruellas, and G. I. Stegeman, Opt. Commun. 121, 149 (1995).
    [CrossRef] [PubMed]
  9. 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]
  10. A. A. Kanashov and A. M. Rubenchik, Physica D 4, 122 (1981); S. Trillo and P. Ferro, Opt. Lett. 20, 438 (1995).
    [CrossRef] [PubMed]
  11. N. N. Akhmediev, V. I. Korneev, and Yu. V. Kuz’menko, Sov. Phys. JETP 61, 62 (1985); J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, and N. N. Akhmediev, Phys. Rev. A 44, 636 (1991).
    [CrossRef] [PubMed]

1997

L. Torner and D. V. Petrov, Electron. Lett. 33, 608 (1997).
[CrossRef]

1995

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]

1994

1974

J. F. Nye and M. V. Berry, Proc. R. Soc. London, Ser. A 336, 165 (1974).
[CrossRef]

Berry, M. V.

J. F. Nye and M. V. Berry, Proc. R. Soc. London, Ser. A 336, 165 (1974).
[CrossRef]

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]

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]

C. R. Menyuk, R. Schiek, and L. Torner, J. Opt. Soc. Am. B 11, 2434 (1994).
[CrossRef]

Nye, J. F.

J. F. Nye and M. V. Berry, Proc. R. Soc. London, Ser. A 336, 165 (1974).
[CrossRef]

Petrov, D. V.

L. Torner and D. V. Petrov, Electron. Lett. 33, 608 (1997).
[CrossRef]

Schiek, R.

Stegeman, G. I.

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]

Torner, L.

L. Torner and D. V. Petrov, Electron. Lett. 33, 608 (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]

C. R. Menyuk, R. Schiek, and L. Torner, J. Opt. Soc. Am. B 11, 2434 (1994).
[CrossRef]

Torruellas, W. E.

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]

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]

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]

Electron. Lett.

L. Torner and D. V. Petrov, Electron. Lett. 33, 608 (1997).
[CrossRef]

J. Opt. Soc. Am. B

Phys. Rev. Lett.

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]

Proc. R. Soc. London, Ser. A

J. F. Nye and M. V. Berry, Proc. R. Soc. London, Ser. A 336, 165 (1974).
[CrossRef]

Other

G. Indebetouw, J. Mod. Opt. 40, 73 (1993); F. S. Roux, J. Opt. Soc. Am. B 12, 1215 (1995).
[CrossRef]

V. I. Kruglov and R. A. Vlasov, Phys. Lett. A 111, 401 (1985); I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 422 (1993); N. N. Rosanov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1996), Vol. 35.
[CrossRef]

G. A. Swartzlander and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992); V. Tikhonenko, J. Christou, and B. Luther-Davies, J. Opt. Soc. Am. B 12, 2046 (1995).
[CrossRef] [PubMed]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, Phys. Rev. Lett. 78, 2108 (1997); Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, Phys. Rev. Lett. 78, 2948 (1997).
[CrossRef]

A. A. Kanashov and A. M. Rubenchik, Physica D 4, 122 (1981); S. Trillo and P. Ferro, Opt. Lett. 20, 438 (1995).
[CrossRef] [PubMed]

N. N. Akhmediev, V. I. Korneev, and Yu. V. Kuz’menko, Sov. Phys. JETP 61, 62 (1985); J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, and N. N. Akhmediev, Phys. Rev. A 44, 636 (1991).
[CrossRef] [PubMed]

A. V. Buryak, Y. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995); L. Torner, D. Mihalache, D. Mazilu, E. W. Wright, W. E. Torruellas, and G. I. Stegeman, Opt. Commun. 121, 149 (1995).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Families of solitary wave solutions represented in an energy-flow–Hamiltonian diagram at β=±3. Dashed curves, families of lowest-order, stable solitons; dotted curves, higher-order vorticityless solutions; solid curves, vortex solutions. (a) β=-3; (b) β=3.

Fig. 2
Fig. 2

Shape of typical vortex solutions with two different charges. Solid curves, fundamental beam; dashed curves, second-harmonic beam. Conditions: β=3, κ1=3. (a) m1=1; (b) m2=2.

Fig. 3
Fig. 3

Growth rate of perturbations with different azimuthal indices as a function of κ1 for two different values of the wave-vector mismatch and two different charges. (a) β=-3, m1=1; (b) β=3, m1=1; (c) β=-3, m1=2; (d) β=3, m1=2.

Fig. 4
Fig. 4

Typical decay of two vortex solitary waves; the plots show the fundamental beams at ξ=10. The second harmonic exhibits identical features. The inputs are the vortices of Fig. 2.

Equations (8)

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i a1ξ-r2 2a1+a1*a2 exp(-iβξ)=0,
i a2ξ-α2 2a2-iδa2+a12 exp(iβξ)=0,
H=-12 r|A1|2+α2 |A2|2-β|A2|2+(A1*2A2+A12A2*)dr,
I=(|A1|2+|A2|2)dr
r2 1ρ ddρ ρ dU1dρ+κ1-rm122ρ2U1-U1U2=0,
α2 1ρ ddρ ρ dU2dρ+κ2-αm222ρ2U2-U12=0.
aj=[Uj(ρ)+fj(ρ, ξ)exp(inφ)+gj(ρ, ξ)exp(-inφ)]exp(imjφ)exp(iκjξ),
j=1, 2.

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