Abstract

The interaction of optical filaments in bulk self-focusing media is investigated theoretically and numerically. The nature of this interaction is shown to vary with the incident individual powers and relative phases of the beamlets. By means of virial arguments supported by numerical results it is found that three distinct evolution regimes characterize two in-phase interacting filaments: (i) When each filament has a power below Nc/4, where Nc is the critical self-focusing threshold for a single wave, both filaments disperse along their propagation axis. (ii) When their respective powers lie between Nc/4 and Nc, they fuse into a single central lobe that may self-focus until collapse, depending on their initial separation distance. The critical distance below which a central lobe forms and collapses is estimated analytically. (iii) When their incident powers both exceed Nc, initially separated filaments individually self-focus without mutual interaction. In contrast to in-phase beamlets, two light cells with opposite phase are shown to never coalesce. The extension of the self-focusing dynamics to optical filaments in bulk media with anomalous group-velocity dispersion is discussed.

© 1997 Optical Society of America

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1996 (6)

S. Hüller, Ph. Mounaix, and D. Pesme, “Numerical simulation of filamentation and its interplay with SBS inunderdense plasmas,” Phys. Scr. T63, 151 (1996).
[CrossRef]

I. M. Uzunov, V. S. Gerdjikov, M. Gölles, and F. Lederer, “On the description of N-soliton interaction in opticalfibers,” Opt. Commun. 125, 237 (1996).
[CrossRef]

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetrybreaking to spatial turbulence,” Phys. Rev. A 54, 870 (1996).
[CrossRef] [PubMed]

V. Tikhonenko, J. Christov, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in asaturable nonlinear medium,” Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

L. Bergé and J. Juul Rasmussen, “Multi-splitting and collapse of self-focusing anisotropic beams innormal/anomalous dispersive media,” Phys. Plasmas 3, 824 (1996).
[CrossRef]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Break-up of two-dimensional bright spatial solitons due to transversemodulation instability,” Europhys. Lett. 35, 25 (1996).
[CrossRef]

1995 (4)

J. S. Hesthaven, J. P. Lynov, A. H. Nielsen, J. Juul Rasmussen, M. R. Schmidt, E. A. Shapiro, and S. K. Turitsyn, “Dynamics of a nonlinear dipole vortex,” Phys. Fluids 7, 2220 (1995).
[CrossRef]

Yu. B. Gaididei, K. Ø. Rasmussen, and P. L. Christiansen, “Nonlinear excitations in two-dimensional molecular structures withimpurities,” Phys. Rev. E 52, 2951 (1995).
[CrossRef]

T. Okamawari, A. Hasegawa, and Y. Kodama, “Analyses of soliton interactions by means of a perturbed inverse-scatteringtransform,” Phys. Rev. A 51, 3203 (1995).
[CrossRef] [PubMed]

R. McLeod, K. Wagner, and S. Blair, “(3+1)-dimensional optical soliton dragging logic,” Phys. Rev. A 52, 3254 (1995).
[CrossRef] [PubMed]

1993 (2)

N. Akhmediev and J. M. Soto-Crespo, “Generation of a train of three-dimensional optical solitons in a self-focusingmedium,” Phys. Rev. A 47, 1358 (1993).
[CrossRef] [PubMed]

D. E. Edmundson and R. H. Enns, “Fully three-dimensional collisions of bistable light bullets,” Opt. Lett. 18, 1609 (1993).
[CrossRef] [PubMed]

1992 (1)

1991 (1)

M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Stability of isotropic singularities for the nonlinear Schrödingerequation,” Physica D 47, 393 (1991).
[CrossRef]

1990 (1)

1989 (1)

K. Rypdal and J. Juul Rasmussen, “Stability of solitary structures in the nonlinear Schrödingerequation,” Phys. Scr. 40, 192 (1989).
[CrossRef]

1988 (1)

C. J. McKinstrie and D. A. Russel, “Nonlinear focusing of coupled waves,” Phys. Rev. Lett. 61, 2929 (1988).
[CrossRef] [PubMed]

1986 (3)

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, “Soliton stability in plasmas and hydrodynamics,” Phys. Rep. 142, 103 (1986).
[CrossRef]

See, for a review, J. Juul Rasmussen and K. Rypdal, “Blow-up in nonlinear Schrödinger equations I. A general review,” Phys. Scr. 33, 481 (1986).
[CrossRef]

D. Anderson and M. Lisak, “Bandwidth limits due to mutual pulse interaction in optical solitoncommunication systems,” Opt. Lett. 11, 174 (1986).
[CrossRef]

1983 (1)

1981 (2)

V. I. Karpman and V. V. Solov'ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487 (1981).
[CrossRef]

K. A. Gorshkov and L. A. Ostrovsky, “Interactions of solitons in non-integrable systems: direct perturbationmethod and applications,” Physica D 3, 428 (1981).
[CrossRef]

Akhmediev, N.

N. Akhmediev and J. M. Soto-Crespo, “Generation of a train of three-dimensional optical solitons in a self-focusingmedium,” Phys. Rev. A 47, 1358 (1993).
[CrossRef] [PubMed]

Akhmediev, N. N.

Anderson, D.

Anderson, D. Z.

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetrybreaking to spatial turbulence,” Phys. Rev. A 54, 870 (1996).
[CrossRef] [PubMed]

Bergé, L.

L. Bergé and J. Juul Rasmussen, “Multi-splitting and collapse of self-focusing anisotropic beams innormal/anomalous dispersive media,” Phys. Plasmas 3, 824 (1996).
[CrossRef]

Blair, S.

R. McLeod, K. Wagner, and S. Blair, “(3+1)-dimensional optical soliton dragging logic,” Phys. Rev. A 52, 3254 (1995).
[CrossRef] [PubMed]

Christiansen, P. L.

Yu. B. Gaididei, K. Ø. Rasmussen, and P. L. Christiansen, “Nonlinear excitations in two-dimensional molecular structures withimpurities,” Phys. Rev. E 52, 2951 (1995).
[CrossRef]

Christov, J.

V. Tikhonenko, J. Christov, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in asaturable nonlinear medium,” Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

Edmundson, D. E.

Enns, R. H.

Gaididei, Yu. B.

Yu. B. Gaididei, K. Ø. Rasmussen, and P. L. Christiansen, “Nonlinear excitations in two-dimensional molecular structures withimpurities,” Phys. Rev. E 52, 2951 (1995).
[CrossRef]

Gerdjikov, V. S.

I. M. Uzunov, V. S. Gerdjikov, M. Gölles, and F. Lederer, “On the description of N-soliton interaction in opticalfibers,” Opt. Commun. 125, 237 (1996).
[CrossRef]

Gölles, M.

I. M. Uzunov, V. S. Gerdjikov, M. Gölles, and F. Lederer, “On the description of N-soliton interaction in opticalfibers,” Opt. Commun. 125, 237 (1996).
[CrossRef]

Gordon, J. P.

Gorshkov, K. A.

K. A. Gorshkov and L. A. Ostrovsky, “Interactions of solitons in non-integrable systems: direct perturbationmethod and applications,” Physica D 3, 428 (1981).
[CrossRef]

Hasegawa, A.

T. Okamawari, A. Hasegawa, and Y. Kodama, “Analyses of soliton interactions by means of a perturbed inverse-scatteringtransform,” Phys. Rev. A 51, 3203 (1995).
[CrossRef] [PubMed]

Hesthaven, J. S.

J. S. Hesthaven, J. P. Lynov, A. H. Nielsen, J. Juul Rasmussen, M. R. Schmidt, E. A. Shapiro, and S. K. Turitsyn, “Dynamics of a nonlinear dipole vortex,” Phys. Fluids 7, 2220 (1995).
[CrossRef]

Hüller, S.

S. Hüller, Ph. Mounaix, and D. Pesme, “Numerical simulation of filamentation and its interplay with SBS inunderdense plasmas,” Phys. Scr. T63, 151 (1996).
[CrossRef]

Karpman, V. I.

V. I. Karpman and V. V. Solov'ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487 (1981).
[CrossRef]

Kodama, Y.

T. Okamawari, A. Hasegawa, and Y. Kodama, “Analyses of soliton interactions by means of a perturbed inverse-scatteringtransform,” Phys. Rev. A 51, 3203 (1995).
[CrossRef] [PubMed]

Korneev, V. I.

Kuznetsov, E. A.

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, “Soliton stability in plasmas and hydrodynamics,” Phys. Rep. 142, 103 (1986).
[CrossRef]

Landman, M. J.

M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Stability of isotropic singularities for the nonlinear Schrödingerequation,” Physica D 47, 393 (1991).
[CrossRef]

Lederer, F.

I. M. Uzunov, V. S. Gerdjikov, M. Gölles, and F. Lederer, “On the description of N-soliton interaction in opticalfibers,” Opt. Commun. 125, 237 (1996).
[CrossRef]

Lisak, M.

Luther-Davies, B.

V. Tikhonenko, J. Christov, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in asaturable nonlinear medium,” Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

Lynov, J. P.

J. S. Hesthaven, J. P. Lynov, A. H. Nielsen, J. Juul Rasmussen, M. R. Schmidt, E. A. Shapiro, and S. K. Turitsyn, “Dynamics of a nonlinear dipole vortex,” Phys. Fluids 7, 2220 (1995).
[CrossRef]

Mamaev, A. V.

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetrybreaking to spatial turbulence,” Phys. Rev. A 54, 870 (1996).
[CrossRef] [PubMed]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Break-up of two-dimensional bright spatial solitons due to transversemodulation instability,” Europhys. Lett. 35, 25 (1996).
[CrossRef]

McKinstrie, C. J.

C. J. McKinstrie and D. A. Russel, “Nonlinear focusing of coupled waves,” Phys. Rev. Lett. 61, 2929 (1988).
[CrossRef] [PubMed]

McLeod, R.

R. McLeod, K. Wagner, and S. Blair, “(3+1)-dimensional optical soliton dragging logic,” Phys. Rev. A 52, 3254 (1995).
[CrossRef] [PubMed]

Mounaix, Ph.

S. Hüller, Ph. Mounaix, and D. Pesme, “Numerical simulation of filamentation and its interplay with SBS inunderdense plasmas,” Phys. Scr. T63, 151 (1996).
[CrossRef]

Nabiev, R. F.

Nielsen, A. H.

J. S. Hesthaven, J. P. Lynov, A. H. Nielsen, J. Juul Rasmussen, M. R. Schmidt, E. A. Shapiro, and S. K. Turitsyn, “Dynamics of a nonlinear dipole vortex,” Phys. Fluids 7, 2220 (1995).
[CrossRef]

Okamawari, T.

T. Okamawari, A. Hasegawa, and Y. Kodama, “Analyses of soliton interactions by means of a perturbed inverse-scatteringtransform,” Phys. Rev. A 51, 3203 (1995).
[CrossRef] [PubMed]

Ostrovsky, L. A.

K. A. Gorshkov and L. A. Ostrovsky, “Interactions of solitons in non-integrable systems: direct perturbationmethod and applications,” Physica D 3, 428 (1981).
[CrossRef]

Papanicolaou, G. C.

M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Stability of isotropic singularities for the nonlinear Schrödingerequation,” Physica D 47, 393 (1991).
[CrossRef]

Pesme, D.

S. Hüller, Ph. Mounaix, and D. Pesme, “Numerical simulation of filamentation and its interplay with SBS inunderdense plasmas,” Phys. Scr. T63, 151 (1996).
[CrossRef]

Rasmussen, J. Juul

L. Bergé and J. Juul Rasmussen, “Multi-splitting and collapse of self-focusing anisotropic beams innormal/anomalous dispersive media,” Phys. Plasmas 3, 824 (1996).
[CrossRef]

J. S. Hesthaven, J. P. Lynov, A. H. Nielsen, J. Juul Rasmussen, M. R. Schmidt, E. A. Shapiro, and S. K. Turitsyn, “Dynamics of a nonlinear dipole vortex,” Phys. Fluids 7, 2220 (1995).
[CrossRef]

K. Rypdal and J. Juul Rasmussen, “Stability of solitary structures in the nonlinear Schrödingerequation,” Phys. Scr. 40, 192 (1989).
[CrossRef]

See, for a review, J. Juul Rasmussen and K. Rypdal, “Blow-up in nonlinear Schrödinger equations I. A general review,” Phys. Scr. 33, 481 (1986).
[CrossRef]

Rasmussen, K. Ø.

Yu. B. Gaididei, K. Ø. Rasmussen, and P. L. Christiansen, “Nonlinear excitations in two-dimensional molecular structures withimpurities,” Phys. Rev. E 52, 2951 (1995).
[CrossRef]

Rubenchik, A. M.

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, “Soliton stability in plasmas and hydrodynamics,” Phys. Rep. 142, 103 (1986).
[CrossRef]

Russel, D. A.

C. J. McKinstrie and D. A. Russel, “Nonlinear focusing of coupled waves,” Phys. Rev. Lett. 61, 2929 (1988).
[CrossRef] [PubMed]

Rypdal, K.

K. Rypdal and J. Juul Rasmussen, “Stability of solitary structures in the nonlinear Schrödingerequation,” Phys. Scr. 40, 192 (1989).
[CrossRef]

See, for a review, J. Juul Rasmussen and K. Rypdal, “Blow-up in nonlinear Schrödinger equations I. A general review,” Phys. Scr. 33, 481 (1986).
[CrossRef]

Saffman, M.

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetrybreaking to spatial turbulence,” Phys. Rev. A 54, 870 (1996).
[CrossRef] [PubMed]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Break-up of two-dimensional bright spatial solitons due to transversemodulation instability,” Europhys. Lett. 35, 25 (1996).
[CrossRef]

Schmidt, M. R.

J. S. Hesthaven, J. P. Lynov, A. H. Nielsen, J. Juul Rasmussen, M. R. Schmidt, E. A. Shapiro, and S. K. Turitsyn, “Dynamics of a nonlinear dipole vortex,” Phys. Fluids 7, 2220 (1995).
[CrossRef]

Shapiro, E. A.

J. S. Hesthaven, J. P. Lynov, A. H. Nielsen, J. Juul Rasmussen, M. R. Schmidt, E. A. Shapiro, and S. K. Turitsyn, “Dynamics of a nonlinear dipole vortex,” Phys. Fluids 7, 2220 (1995).
[CrossRef]

Silberberg, Y.

Solov'ev, V. V.

V. I. Karpman and V. V. Solov'ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487 (1981).
[CrossRef]

Soto-Crespo, J. M.

N. Akhmediev and J. M. Soto-Crespo, “Generation of a train of three-dimensional optical solitons in a self-focusingmedium,” Phys. Rev. A 47, 1358 (1993).
[CrossRef] [PubMed]

Sulem, C.

M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Stability of isotropic singularities for the nonlinear Schrödingerequation,” Physica D 47, 393 (1991).
[CrossRef]

Sulem, P. L.

M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Stability of isotropic singularities for the nonlinear Schrödingerequation,” Physica D 47, 393 (1991).
[CrossRef]

Tikhonenko, V.

V. Tikhonenko, J. Christov, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in asaturable nonlinear medium,” Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

Turitsyn, S. K.

J. S. Hesthaven, J. P. Lynov, A. H. Nielsen, J. Juul Rasmussen, M. R. Schmidt, E. A. Shapiro, and S. K. Turitsyn, “Dynamics of a nonlinear dipole vortex,” Phys. Fluids 7, 2220 (1995).
[CrossRef]

Uzunov, I. M.

I. M. Uzunov, V. S. Gerdjikov, M. Gölles, and F. Lederer, “On the description of N-soliton interaction in opticalfibers,” Opt. Commun. 125, 237 (1996).
[CrossRef]

Wagner, K.

R. McLeod, K. Wagner, and S. Blair, “(3+1)-dimensional optical soliton dragging logic,” Phys. Rev. A 52, 3254 (1995).
[CrossRef] [PubMed]

Wang, X. P.

M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Stability of isotropic singularities for the nonlinear Schrödingerequation,” Physica D 47, 393 (1991).
[CrossRef]

Zakharov, V. E.

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, “Soliton stability in plasmas and hydrodynamics,” Phys. Rep. 142, 103 (1986).
[CrossRef]

Zozulya, A. A.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Break-up of two-dimensional bright spatial solitons due to transversemodulation instability,” Europhys. Lett. 35, 25 (1996).
[CrossRef]

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetrybreaking to spatial turbulence,” Phys. Rev. A 54, 870 (1996).
[CrossRef] [PubMed]

Europhys. Lett. (1)

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Break-up of two-dimensional bright spatial solitons due to transversemodulation instability,” Europhys. Lett. 35, 25 (1996).
[CrossRef]

Opt. Commun. (1)

I. M. Uzunov, V. S. Gerdjikov, M. Gölles, and F. Lederer, “On the description of N-soliton interaction in opticalfibers,” Opt. Commun. 125, 237 (1996).
[CrossRef]

Opt. Lett. (5)

Phys. Fluids (1)

J. S. Hesthaven, J. P. Lynov, A. H. Nielsen, J. Juul Rasmussen, M. R. Schmidt, E. A. Shapiro, and S. K. Turitsyn, “Dynamics of a nonlinear dipole vortex,” Phys. Fluids 7, 2220 (1995).
[CrossRef]

Phys. Plasmas (1)

L. Bergé and J. Juul Rasmussen, “Multi-splitting and collapse of self-focusing anisotropic beams innormal/anomalous dispersive media,” Phys. Plasmas 3, 824 (1996).
[CrossRef]

Phys. Rep. (1)

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, “Soliton stability in plasmas and hydrodynamics,” Phys. Rep. 142, 103 (1986).
[CrossRef]

Phys. Rev. A (4)

R. McLeod, K. Wagner, and S. Blair, “(3+1)-dimensional optical soliton dragging logic,” Phys. Rev. A 52, 3254 (1995).
[CrossRef] [PubMed]

N. Akhmediev and J. M. Soto-Crespo, “Generation of a train of three-dimensional optical solitons in a self-focusingmedium,” Phys. Rev. A 47, 1358 (1993).
[CrossRef] [PubMed]

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetrybreaking to spatial turbulence,” Phys. Rev. A 54, 870 (1996).
[CrossRef] [PubMed]

T. Okamawari, A. Hasegawa, and Y. Kodama, “Analyses of soliton interactions by means of a perturbed inverse-scatteringtransform,” Phys. Rev. A 51, 3203 (1995).
[CrossRef] [PubMed]

Phys. Rev. E (1)

Yu. B. Gaididei, K. Ø. Rasmussen, and P. L. Christiansen, “Nonlinear excitations in two-dimensional molecular structures withimpurities,” Phys. Rev. E 52, 2951 (1995).
[CrossRef]

Phys. Rev. Lett. (2)

V. Tikhonenko, J. Christov, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in asaturable nonlinear medium,” Phys. Rev. Lett. 76, 2698 (1996).
[CrossRef] [PubMed]

C. J. McKinstrie and D. A. Russel, “Nonlinear focusing of coupled waves,” Phys. Rev. Lett. 61, 2929 (1988).
[CrossRef] [PubMed]

Phys. Scr. (3)

K. Rypdal and J. Juul Rasmussen, “Stability of solitary structures in the nonlinear Schrödingerequation,” Phys. Scr. 40, 192 (1989).
[CrossRef]

See, for a review, J. Juul Rasmussen and K. Rypdal, “Blow-up in nonlinear Schrödinger equations I. A general review,” Phys. Scr. 33, 481 (1986).
[CrossRef]

S. Hüller, Ph. Mounaix, and D. Pesme, “Numerical simulation of filamentation and its interplay with SBS inunderdense plasmas,” Phys. Scr. T63, 151 (1996).
[CrossRef]

Physica D (3)

M. J. Landman, G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Stability of isotropic singularities for the nonlinear Schrödingerequation,” Physica D 47, 393 (1991).
[CrossRef]

V. I. Karpman and V. V. Solov'ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487 (1981).
[CrossRef]

K. A. Gorshkov and L. A. Ostrovsky, “Interactions of solitons in non-integrable systems: direct perturbationmethod and applications,” Physica D 3, 428 (1981).
[CrossRef]

Other (4)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479 (1964); P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005 (1965); V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” Zh. Eksp. Teor. Fiz. Pis'ma Red. 3, 471 (1966) [JETP Lett. JTPLA2 3, 307 (1966)].
[CrossRef]

A. I. D'yachenko, V. E. Zakharov, A. N. Pushkarev, V. F. Shvets, and V. V. Yan'kov, “Soliton turbulence in non-integrable wave systems,” Zh. Eksp. Teor. Fiz. 96, 2026 (1989) [Sov. Phys. JETP 69, 1144 (1989)]; V. E. Zakharov, A. N. Pushkarev, V. F. Shvets, and V. V. Yan'kov, “Soliton turbulence,” Pis'ma Zh. Eksp. Teor. Fiz. 48, 79 (1988) [JETP Lett. 48, 83 (1988)]; for a review on optical turbulence, see A. I. D'yachenko, A. C. Newell, A. N. Pushkarev, and V. E. Zakharov, “Optical turbulence: weak turbulence, condensates and collapsing filamentsin the nonlinear Schrödinger equation,” Physica D PDNPDT 57, 96 (1992).
[CrossRef]

A. J. Campillo, S. L. Shapiro, and B. R. Suydam, “Periodic breakup of optical beams due to self-focusing,” Appl. Phys. Lett. 23, 628 (1973); “Relationship of self-focusing to spatial instability modes,” Appl. Phys. Lett. 24, 178 (1974); B. R. Suydam, “Effect of refractive-index nonlinearity on the optical quality of high-powerlaser beams,” IEEE J. Quantum Electron. IEJQA7 QE-11, 225 (1975).
[CrossRef]

P. G. Saffman, Vortex Dynamics (CambridgeU. Press, Cambridge, 1992).

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Figures (10)

Fig. 1
Fig. 1

Numerical plot of curve C [Eq. (14)] versus X=δc2/4ρ2 for several values of powers N1=N2. From the top to the bottom, the levels of N1 increase as follows: N1=π/2, π, 2π, 3π, 4π, 6π. Only the curves that correspond to N1 chosen inside the bounded range πN14π intersect the X axis.

Fig. 2
Fig. 2

Isocontour plots of the amplitude field |ψ| describing the spreading stages of two in-phase light beamlets. The initial conditions are N1,2=π/2, H=4.52, and δ=2.1. The size of the computational domain is defined by lx=12.0 and ly=10.0, with resolution m=256 and n=128, i.e., Δx0.047 and Δy0.078, for Δz=5×10-5. The isocurves are equidistant, with a spacing of 0.1, and the lowest curve is at 0.05.

Fig. 3
Fig. 3

Isocontour plots of the amplitude field |ψ| describing the merging and ultimate spreading of two in-phase interacting cells. The initial conditions are N1,2=2π, H=6.93, and δ=2.1. The size of the computational domain is lx=ly=10.0, and m=n=256 for Δz=2.5×10-5. The isocurves are equidistant, with a spacing of 0.4, and the lowest curve is at 0.2.

Fig. 4
Fig. 4

Isocontour plots of the amplitude field |ψ| describing the merging and collapse (amalgamation) of two in-phase filaments. The initial conditions are N1,2=2π, H=-1.73, and δ=1.5. The size of the computational domain is lx=ly=10.0, and m=n=256 for Δz=8×10-5. Only a centered window of size 5.0 by 5.0 is shown. The isocurves are equidistant, with a spacing of 0.3, and the lowest curve is at 0.15.

Fig. 5
Fig. 5

Nαδ plane for the interaction of in-phase symmetric beamlets with πN1,24π (in units of π). Filled squares: beamlet amalgamation. Open diamonds: spreading cells. The solid curve shows the values of δc obtained from the zero crossings of curve C plotted in Fig. 1.

Fig. 6
Fig. 6

Isocontour plots of the amplitude field |ψ| describing the individual collapses of two high-power in-phase cells. The initial conditions are N1,2=6π, H=-155, and δ=1.52. The size of the computational domain is lx=ly=12.0, and m=n=512 for Δz=2.5×10-5. Only a centered window of size 6.0 by 6.0 is shown. The isocurves are equidistant, with a spacing of 0.4, and the lowest curve is at 0.2.

Fig. 7
Fig. 7

Isocontour plots of the amplitude field |ψ| describing the collapse of two high-power peaks with N1=5π for the left-hand wave and with N2=7π for the right-hand one. The initial conditions satisfy H=-71.1 with δ=2.1. The size of the computational domain is lx=12.0 and ly=10.0, and the resolution is m=256 and n=128 for Δz=3×10-5. Only a centered window of size 6.0 by 5.0 is shown. The isocurves are equidistant, with a spacing of Δ=0.3, and the lowest curve is at |ψ|min=0.2 for z=0.00; Δ=0.6 at |ψ|min=0.6 for z=0.15, and Δ=3.0 at |ψ|min=3.0 for z=0.30.

Fig. 8
Fig. 8

Isocontour plots of the amplitude field |ψ| describing the interaction of one weakly dispersing wave (N1=3π) with a self-focusing neighbor (N2=6π). The initial conditions satisfy H=-14 with δ=4.0. The size of the computational domain is lx=20.0 and ly=10.0, with m=256 and n=128 for Δz=4×10-5. Only a centered window of size 9.6 by 6.3 is shown. The isocurves are equidistant, with a spacing of Δ=0.3, and the lowest curve is at |ψ|min=0.2 for z=0.00; Δ=0.6 at |ψ|min=0.6 for z=0.16, and Δ=1.5 at |ψ|min=1.5 for z=0.28.

Fig. 9
Fig. 9

Isocontour plots of the amplitude field |ψ| describing uncorrelated collapse events of two high-power beamlets with a phase difference θ=π, forming a dipolar structure. The initial conditions are N1,2=5π, H=1.28, and δ=2.1. The size of the computational domain is lx=12.0 and ly=10.0, with m=256 and n=128 for Δz=4×10-5. Only a centered window of size 6.2 by 5.5 is shown. The isocurves are equidistant, with a spacing of 0.25, and the lowest curve is at 0.125.

Fig. 10
Fig. 10

Isocontour plots of the amplitude field |ψ| describing an individual dispersion of two low-power filaments with a phase difference θ=π. The initial conditions are N1,2=3π, H=17.5, and δ=2.1. The size of the computational domain is lx=12.0 and ly=10.0, with m=256 and n=128 for Δz=4×10-5. Only a centered window of size 9.0 by 7.5 is shown. The isocurves are equidistant, with a spacing of 0.2, and the lowest curve is at 0.1.

Equations (17)

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iψz+2ψ+σψττ+f(|ψ|2)ψ=0,
iψz+2ψ+|ψ|2ψ=0.
uα(r-rα, 0)=Nαπρα21/2 exp-(r-rα)22ρα2+iϕα.
N=N1+N2+4N1N2 cos θ ρ1ρ2ρ12+ρ22exp-δ22(ρ12+ρ22),
N0Nδ=0=N1+N2+4N1N2 cos θ ρ1ρ2ρ12+ρ22.
r(z)=r(0)=r2+δN1ρ12N(ρ12+ρ22)+1-N2N ρ22ρ12+ρ22.
H=α=1,2 Nαρα21-Nα4π+8 cos θN1N2 ρ1ρ2(ρ12+ρ22)2×exp-δ22(ρ12+ρ22) 1-δ22(ρ12+ρ22)-N1N2π(ρ12+ρ22)(1+2 cos2 θ)×exp-δ2ρ12+ρ22-4N1N2πρ1ρ2×cos θN1ρ223ρ22+ρ12×exp-3δ22(3ρ22+ρ12)+N2ρ123ρ12+ρ22exp-3δ22(3ρ12+ρ22).
H=H1+H2+Hint(δ),
Hint(δ)1ρ22N1N2 cos θ exp-δ24ρ21-δ24ρ2-N1N22π(1+2 cos2 θ)exp-δ22ρ2-N1+N2πN1N2 cos θ exp-3δ28ρ2
H=H0+ΔH(δ),
z2I(z)z2(r-r)2|ψ|2dr=8H.
F(δc2/4ρ2)=N1N14π-1+N2N24π-1,
F(X)=2 cos θN1N2e-X(1-X)-N1N22π(1+2 cos2 θ)e-2X-N1+N2πN1N2 cos θ e-3X/2.
C1N1+N2F(X)+α=1,2Nα1-Nα4π
γ(k)=k2A-k2,Au fu|u=|Ψ0|2
z2I(z)=42H-12|ψ|4drdτ,
I(z)[(r-r)2+(τ-τ)2]|ψ|2drdτ.

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