Abstract

We analyze the interactions of 2+1-dimensional parametric solitons and demonstrate nonplanar beam switching in a bulk χ2 medium, which includes controllable soliton repulsion, spiraling, and fusion. An analytical model predicting the results of the soliton scattering is derived and verified by direct numerics.

© 1998 Optical Society of America

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References

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  1. See, e.g., F. Reynaud and A. Barthelemy, in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer, Dordrecht, The Netherlands, 1992), p. 319.
    [CrossRef]
  2. G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Share, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
    [CrossRef]
  3. 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]
  4. V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
    [CrossRef] [PubMed]
  5. M. Shih, M. Segev, and G. Salamo, Phys. Rev. Lett. 78, 2551 (1997).
    [CrossRef]
  6. See, e.g., L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991).
    [CrossRef]
  7. G. I. Stegeman, D. J. Hagan, and L. Torner, J. Opt. Quantum Electron. 28, 1691 (1996).
    [CrossRef]
  8. A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995); V. V. Steblina, Yu. S. Kivshar, M. Lisak, and B. A. Malomed, Opt. Commun. 118, 345 (1995).
    [CrossRef] [PubMed]
  9. See also the stability analysis for the parametric three-wave mixing by A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996); A. V. Buryak and Yu. S. Kivshar, Phys. Rev. Lett. 78, 3286 (1997).
    [CrossRef] [PubMed]
  10. M. J. Werner and P. D. Drummond, J. Opt. Soc. Am. B 10, 2390 (1993); D. M. Baboiu, G. I. Stegeman, and L. Torner, Opt. Lett. 20, 2282 (1995); C. Etrich, U. Peschel, F. Lederer, and B. Malomed, Phys. Rev. A 52, R3444 (1995).
    [CrossRef]
  11. Y. Baek, R. Schiek, G. I. Stegeman, I. Baumann, and W. Sohler, Opt. Lett. 22, 1550 (1997).
    [CrossRef]
  12. G. Leo, G. Assanto, and W. E. Torruellas, Opt. Lett. 22, 7 (1997).
    [CrossRef] [PubMed]
  13. C. Clausen, O. Bang, and Yu. Kivshar, Phys. Rev. Lett. 78, 4749 (1997).
    [CrossRef]
  14. K. A. Gorshkov and L. A. Ostrovsky, Physica D 3, 428 (1981).
    [CrossRef]
  15. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1980), Chap. 3.

1997

M. Shih, M. Segev, and G. Salamo, Phys. Rev. Lett. 78, 2551 (1997).
[CrossRef]

Y. Baek, R. Schiek, G. I. Stegeman, I. Baumann, and W. Sohler, Opt. Lett. 22, 1550 (1997).
[CrossRef]

G. Leo, G. Assanto, and W. E. Torruellas, Opt. Lett. 22, 7 (1997).
[CrossRef] [PubMed]

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

1996

V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

G. I. Stegeman, D. J. Hagan, and L. Torner, J. Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

See also the stability analysis for the parametric three-wave mixing by A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996); A. V. Buryak and Yu. S. Kivshar, Phys. Rev. Lett. 78, 3286 (1997).
[CrossRef] [PubMed]

1995

A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995); V. V. Steblina, Yu. S. Kivshar, M. Lisak, and B. A. Malomed, Opt. Commun. 118, 345 (1995).
[CrossRef] [PubMed]

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]

1993

1991

See, e.g., L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991).
[CrossRef]

1981

K. A. Gorshkov and L. A. Ostrovsky, Physica D 3, 428 (1981).
[CrossRef]

Assanto, G.

Baek, Y.

Bang, O.

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

Barthelemy, A.

See, e.g., F. Reynaud and A. Barthelemy, in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer, Dordrecht, The Netherlands, 1992), p. 319.
[CrossRef]

Baumann, I.

Buryak, A. V.

See also the stability analysis for the parametric three-wave mixing by A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996); A. V. Buryak and Yu. S. Kivshar, Phys. Rev. Lett. 78, 3286 (1997).
[CrossRef] [PubMed]

A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995); V. V. Steblina, Yu. S. Kivshar, M. Lisak, and B. A. Malomed, Opt. Commun. 118, 345 (1995).
[CrossRef] [PubMed]

Christou, J.

V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

Clausen, C.

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

Crosignani, B.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Share, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
[CrossRef]

Di Porto, P.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Share, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
[CrossRef]

Drummond, P. D.

Duree, G. C.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Share, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
[CrossRef]

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1980), Chap. 3.

Gorshkov, K. A.

K. A. Gorshkov and L. A. Ostrovsky, Physica D 3, 428 (1981).
[CrossRef]

Hagan, D. J.

G. I. Stegeman, D. J. Hagan, and L. Torner, J. Opt. Quantum Electron. 28, 1691 (1996).
[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]

Kivshar, Yu.

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

Kivshar, Yu. S.

See also the stability analysis for the parametric three-wave mixing by A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996); A. V. Buryak and Yu. S. Kivshar, Phys. Rev. Lett. 78, 3286 (1997).
[CrossRef] [PubMed]

A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995); V. V. Steblina, Yu. S. Kivshar, M. Lisak, and B. A. Malomed, Opt. Commun. 118, 345 (1995).
[CrossRef] [PubMed]

Leo, G.

Luther-Davies, B.

V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[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]

Mitchell, D. J.

See, e.g., L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991).
[CrossRef]

Neurgaonkar, R. R.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Share, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
[CrossRef]

Ostrovsky, L. A.

K. A. Gorshkov and L. A. Ostrovsky, Physica D 3, 428 (1981).
[CrossRef]

Poladian, L.

See, e.g., L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991).
[CrossRef]

Reynaud, F.

See, e.g., F. Reynaud and A. Barthelemy, in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer, Dordrecht, The Netherlands, 1992), p. 319.
[CrossRef]

Salamo, G.

M. Shih, M. Segev, and G. Salamo, Phys. Rev. Lett. 78, 2551 (1997).
[CrossRef]

Salamo, G. J.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Share, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
[CrossRef]

Schiek, R.

Segev, M.

M. Shih, M. Segev, and G. Salamo, Phys. Rev. Lett. 78, 2551 (1997).
[CrossRef]

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Share, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
[CrossRef]

Share, E. J.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Share, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
[CrossRef]

Shih, M.

M. Shih, M. Segev, and G. Salamo, Phys. Rev. Lett. 78, 2551 (1997).
[CrossRef]

Shultz, J. L.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Share, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
[CrossRef]

Snyder, A. W.

See, e.g., L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991).
[CrossRef]

Sohler, W.

Steblina, V. V.

A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995); V. V. Steblina, Yu. S. Kivshar, M. Lisak, and B. A. Malomed, Opt. Commun. 118, 345 (1995).
[CrossRef] [PubMed]

Stegeman, G. I.

Y. Baek, R. Schiek, G. I. Stegeman, I. Baumann, and W. Sohler, Opt. Lett. 22, 1550 (1997).
[CrossRef]

G. I. Stegeman, D. J. Hagan, and L. Torner, J. Opt. Quantum Electron. 28, 1691 (1996).
[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]

Tikhonenko, V.

V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

Torner, L.

G. I. Stegeman, D. J. Hagan, and L. Torner, J. Opt. Quantum Electron. 28, 1691 (1996).
[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]

Torruellas, W. E.

G. Leo, G. Assanto, and W. E. Torruellas, Opt. Lett. 22, 7 (1997).
[CrossRef] [PubMed]

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.

See also the stability analysis for the parametric three-wave mixing by A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996); A. V. Buryak and Yu. S. Kivshar, Phys. Rev. Lett. 78, 3286 (1997).
[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]

Werner, M. J.

Yariv, A.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Share, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
[CrossRef]

J. Opt. Quantum Electron.

G. I. Stegeman, D. J. Hagan, and L. Torner, J. Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

See, e.g., L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991).
[CrossRef]

Opt. Lett.

Phys. Rev. A

A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, Phys. Rev. A 52, 1670 (1995); V. V. Steblina, Yu. S. Kivshar, M. Lisak, and B. A. Malomed, Opt. Commun. 118, 345 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett.

See also the stability analysis for the parametric three-wave mixing by A. V. Buryak, Yu. S. Kivshar, and S. Trillo, Phys. Rev. Lett. 77, 5210 (1996); A. V. Buryak and Yu. S. Kivshar, Phys. Rev. Lett. 78, 3286 (1997).
[CrossRef] [PubMed]

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Share, and R. R. Neurgaonkar, Phys. Rev. Lett. 71, 553 (1993).
[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]

V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

M. Shih, M. Segev, and G. Salamo, Phys. Rev. Lett. 78, 2551 (1997).
[CrossRef]

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

Physica D

K. A. Gorshkov and L. A. Ostrovsky, Physica D 3, 428 (1981).
[CrossRef]

Other

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1980), Chap. 3.

See, e.g., F. Reynaud and A. Barthelemy, in Guided Wave Nonlinear Optics, D. B. Ostrowsky and R. Reinisch, eds. (Kluwer, Dordrecht, The Netherlands, 1992), p. 319.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Qualitative sketch of the effective interaction potential UR, ψ for (a) out-of-phase interaction ψ=π and (b) in-phase interaction ψ=0.

Fig. 2
Fig. 2

Snapshots of the soliton positions in the X,Y plane at various propagation distances for (a) soliton scattering ψ=π, s=3.6, (b) soliton reflection by spiraling ψ=0, s=11.6, and (c) soliton fusion ψ=0, s=11.0. All results were obtained for β=0.5, Δ=1.0, and C=0.2.

Fig. 3
Fig. 3

Soliton scattering angle θ versus impact parameter s [see Fig.  2(a)] for out-of-phase collisions. Solid curve, analytical results; filled diamonds, results of numerical simulations. Results are for β=0.5, Δ=1.0, C=0.2, and ψ=π.

Equations (5)

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

2ik1E1z+2E1+χ1E1*E2 exp-iδkz=0, 2ik2E2z+2E2+χ2E12 expiδkz=0,
ivz+2v-βv+wv*=0, iσwz+2w-σ2β+Δw+v2/2=0,
L=½MRR·2+½Mψψ·2-UR, ψ,
UR, ψ=MRs2C22R2+U1Rcosψ+U2Rcos2ψ,
U1R=-Aexp-βR/R, U2R=-Bexp-σΔ+2β1/2R/R,

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