Abstract

We demonstrate, in both two and three dimensions, how a self-guided beam in a non-Kerr medium is split into two beams on weak illumination. We also provide an elegant physical explanation that predicts the universal character of the observed phenomenon. Possible applications of our findings to guiding light with light are also discussed.

© 1998 Optical Society of America

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References

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  1. A. W. Snyder, L. Poladian, and D. J. Mitchell, presented at the Fifteenth Australian Conference on Optical Fiber Technology, Sydney, Australia, December 2–6, 1990; A. W. Synder, D. J. Mitchell, L. Poladian, and F. Ladouceur, Opt. Lett. 16, 21 (1991); L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991); T. Thwaites, New Sci. 129 (1751), 14 (1991).
    [CrossRef]
  2. A. W. Snyder, D. J. Mitchell, L. Poladian, and F. Ladouceur, Opt. Lett. 16, 21 (1991).
    [CrossRef] [PubMed]
  3. Y. Silberberg, in Experiments and Advances in Integrated Optics, S. Martilucci, A. N. Chester, and M. Bertolotti, eds. (Plenum, New York, 1994), pp. 103–111.
  4. A. W. Snyder and A. P. Sheppard, Opt. Lett. 18, 482 (1992). This Letter also presented a qualitative theory of soliton collisions that describes whether solitons fuse or give birth to new solitons.
    [CrossRef]
  5. M. Segev, M. Shih, and G. Salamo, Phys. Rev. Lett. 78, 2551 (1997).
    [CrossRef]
  6. V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996); W. Królikowski and S. A. Holmstrom, Opt. Lett. 22, 369 (1997).
    [CrossRef] [PubMed]
  7. A. V. Buryak and N. N. Akhmediev, Phys. Rev. E 50, 3126 (1994).
    [CrossRef]
  8. A. W. Snyder and D. J. Mitchell, Opt. Lett. 22, 16 (1997).
    [CrossRef] [PubMed]
  9. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1981).
    [CrossRef]
  10. It might be surprising that the weak beam in Fig. 2 does not diffract. We deliberately chose the radius of the weak beam to be that necessary for self-guidance in the log-nonlinearity medium. For this special type of nonlinearity stationary self-guidance occurs at only one radius, independent of beam intensity.8
  11. Y. Silberberg, Opt. Lett. 15, 1282 (1990).
    [CrossRef] [PubMed]
  12. E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, and V. N. Serkin, JETP Lett. 42, 87 (1985); [Pis’ma Zh. Eksp. Teor. Fiz. 42, 74 (1985)].
  13. V. V. Afanasjev, J. S. Aitchison, and Yu. S. Kivshar, Opt. Commun. 116, 331 (1995).
    [CrossRef]
  14. D. Burak, J. Opt. Soc. Am. B 13, 613 (1996); D. Burak and R. Binder, J. Opt. Soc. Am. B 14, 1458 (1997).
    [CrossRef]

1997 (2)

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

A. W. Snyder and D. J. Mitchell, Opt. Lett. 22, 16 (1997).
[CrossRef] [PubMed]

1996 (2)

D. Burak, J. Opt. Soc. Am. B 13, 613 (1996); D. Burak and R. Binder, J. Opt. Soc. Am. B 14, 1458 (1997).
[CrossRef]

V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996); W. Królikowski and S. A. Holmstrom, Opt. Lett. 22, 369 (1997).
[CrossRef] [PubMed]

1995 (1)

V. V. Afanasjev, J. S. Aitchison, and Yu. S. Kivshar, Opt. Commun. 116, 331 (1995).
[CrossRef]

1994 (1)

A. V. Buryak and N. N. Akhmediev, Phys. Rev. E 50, 3126 (1994).
[CrossRef]

1992 (1)

1991 (1)

1990 (1)

1985 (1)

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, and V. N. Serkin, JETP Lett. 42, 87 (1985); [Pis’ma Zh. Eksp. Teor. Fiz. 42, 74 (1985)].

Ablowitz, M. J.

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1981).
[CrossRef]

Afanasjev, V. V.

V. V. Afanasjev, J. S. Aitchison, and Yu. S. Kivshar, Opt. Commun. 116, 331 (1995).
[CrossRef]

Aitchison, J. S.

V. V. Afanasjev, J. S. Aitchison, and Yu. S. Kivshar, Opt. Commun. 116, 331 (1995).
[CrossRef]

Akhmediev, N. N.

A. V. Buryak and N. N. Akhmediev, Phys. Rev. E 50, 3126 (1994).
[CrossRef]

Burak, D.

Buryak, A. V.

A. V. Buryak and N. N. Akhmediev, Phys. Rev. E 50, 3126 (1994).
[CrossRef]

Christou, J.

V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996); W. Królikowski and S. A. Holmstrom, Opt. Lett. 22, 369 (1997).
[CrossRef] [PubMed]

Dianov, E. M.

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, and V. N. Serkin, JETP Lett. 42, 87 (1985); [Pis’ma Zh. Eksp. Teor. Fiz. 42, 74 (1985)].

Golovchenko, E. A.

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, and V. N. Serkin, JETP Lett. 42, 87 (1985); [Pis’ma Zh. Eksp. Teor. Fiz. 42, 74 (1985)].

Kivshar, Yu. S.

V. V. Afanasjev, J. S. Aitchison, and Yu. S. Kivshar, Opt. Commun. 116, 331 (1995).
[CrossRef]

Ladouceur, F.

Luther-Davies, B.

V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996); W. Królikowski and S. A. Holmstrom, Opt. Lett. 22, 369 (1997).
[CrossRef] [PubMed]

Mitchell, D. J.

A. W. Snyder and D. J. Mitchell, Opt. Lett. 22, 16 (1997).
[CrossRef] [PubMed]

A. W. Snyder, D. J. Mitchell, L. Poladian, and F. Ladouceur, Opt. Lett. 16, 21 (1991).
[CrossRef] [PubMed]

A. W. Snyder, L. Poladian, and D. J. Mitchell, presented at the Fifteenth Australian Conference on Optical Fiber Technology, Sydney, Australia, December 2–6, 1990; A. W. Synder, D. J. Mitchell, L. Poladian, and F. Ladouceur, Opt. Lett. 16, 21 (1991); L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991); T. Thwaites, New Sci. 129 (1751), 14 (1991).
[CrossRef]

Poladian, L.

A. W. Snyder, D. J. Mitchell, L. Poladian, and F. Ladouceur, Opt. Lett. 16, 21 (1991).
[CrossRef] [PubMed]

A. W. Snyder, L. Poladian, and D. J. Mitchell, presented at the Fifteenth Australian Conference on Optical Fiber Technology, Sydney, Australia, December 2–6, 1990; A. W. Synder, D. J. Mitchell, L. Poladian, and F. Ladouceur, Opt. Lett. 16, 21 (1991); L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991); T. Thwaites, New Sci. 129 (1751), 14 (1991).
[CrossRef]

Prokhorov, A. M.

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, and V. N. Serkin, JETP Lett. 42, 87 (1985); [Pis’ma Zh. Eksp. Teor. Fiz. 42, 74 (1985)].

Salamo, G.

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

Segev, M.

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

Segur, H.

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1981).
[CrossRef]

Serkin, V. N.

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, and V. N. Serkin, JETP Lett. 42, 87 (1985); [Pis’ma Zh. Eksp. Teor. Fiz. 42, 74 (1985)].

Sheppard, A. P.

Shih, M.

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

Silberberg, Y.

Y. Silberberg, Opt. Lett. 15, 1282 (1990).
[CrossRef] [PubMed]

Y. Silberberg, in Experiments and Advances in Integrated Optics, S. Martilucci, A. N. Chester, and M. Bertolotti, eds. (Plenum, New York, 1994), pp. 103–111.

Snyder, A. W.

A. W. Snyder and D. J. Mitchell, Opt. Lett. 22, 16 (1997).
[CrossRef] [PubMed]

A. W. Snyder and A. P. Sheppard, Opt. Lett. 18, 482 (1992). This Letter also presented a qualitative theory of soliton collisions that describes whether solitons fuse or give birth to new solitons.
[CrossRef]

A. W. Snyder, D. J. Mitchell, L. Poladian, and F. Ladouceur, Opt. Lett. 16, 21 (1991).
[CrossRef] [PubMed]

A. W. Snyder, L. Poladian, and D. J. Mitchell, presented at the Fifteenth Australian Conference on Optical Fiber Technology, Sydney, Australia, December 2–6, 1990; A. W. Synder, D. J. Mitchell, L. Poladian, and F. Ladouceur, Opt. Lett. 16, 21 (1991); L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991); T. Thwaites, New Sci. 129 (1751), 14 (1991).
[CrossRef]

Tikhonenko, V.

V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996); W. Królikowski and S. A. Holmstrom, Opt. Lett. 22, 369 (1997).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B (1)

JETP Lett. (1)

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, and V. N. Serkin, JETP Lett. 42, 87 (1985); [Pis’ma Zh. Eksp. Teor. Fiz. 42, 74 (1985)].

Opt. Commun. (1)

V. V. Afanasjev, J. S. Aitchison, and Yu. S. Kivshar, Opt. Commun. 116, 331 (1995).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. E (1)

A. V. Buryak and N. N. Akhmediev, Phys. Rev. E 50, 3126 (1994).
[CrossRef]

Phys. Rev. Lett. (2)

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

V. Tikhonenko, J. Christou, and B. Luther-Davies, Phys. Rev. Lett. 76, 2698 (1996); W. Królikowski and S. A. Holmstrom, Opt. Lett. 22, 369 (1997).
[CrossRef] [PubMed]

Other (4)

A. W. Snyder, L. Poladian, and D. J. Mitchell, presented at the Fifteenth Australian Conference on Optical Fiber Technology, Sydney, Australia, December 2–6, 1990; A. W. Synder, D. J. Mitchell, L. Poladian, and F. Ladouceur, Opt. Lett. 16, 21 (1991); L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991); T. Thwaites, New Sci. 129 (1751), 14 (1991).
[CrossRef]

Y. Silberberg, in Experiments and Advances in Integrated Optics, S. Martilucci, A. N. Chester, and M. Bertolotti, eds. (Plenum, New York, 1994), pp. 103–111.

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1981).
[CrossRef]

It might be surprising that the weak beam in Fig. 2 does not diffract. We deliberately chose the radius of the weak beam to be that necessary for self-guidance in the log-nonlinearity medium. For this special type of nonlinearity stationary self-guidance occurs at only one radius, independent of beam intensity.8

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Figures (4)

Fig. 1
Fig. 1

Propagation of quasi-periodic self-guiding beams in a near-Kerr medium. For both examples =0.003, and the form of the initial soliton is given by U=22/coshx. (a) Beam propagation without weak illumination, (b) beam propagation under the influence of a weak probe beam. Initially the probe beam has 0.2/cosh0.4x amplitude and a velocity of 6.32.

Fig. 2
Fig. 2

Same as Fig.  1(b) but for a highly saturating medium (log-nonlinearity model of Ref.  8). For this example the amplitude of the initial large beam is given by U=10 exp-x2/2ρ2, where ρ=3.3. The weak beam has an initial amplitude of 0.1 exp-x2/2 and an initial velocity of 0.5.

Fig. 3
Fig. 3

Splitting of self-guided beams of circular symmetry (log-nonlinearity model of Ref.  8). For this example the initial amplitude of the periodic beam is given by U=10 exp-x2/2-y2/2/ρ2, where ρ=3.3. The weak beam has an initial amplitude exp-x2/2-y2/2 and an initial velocity of 0.7.

Fig. 4
Fig. 4

Collisions and steering of self-guided beams in a highly saturating medium (log-nonlinearity model of Ref.  8). (a) Collisions of two symmetric beams under the influence of a weak probe beam in two dimensions. For this example the amplitudes of the initial large beams are given by U=10 exp-x2/2, and their velocities are ±0.05. The weak beam is 10 times weaker than the large beams and has an initial velocity of 0.5. (b) Same as (a) but for three dimensions. The collision occurs in the xz plane. For this example the amplitudes of the initial large beams are given by U=10 exp-x2/2-y2/2, and their initial velocities are ±0.15. The weak beam is 20 times weaker than the large beams and has an initial velocity of 0.45. Note that for both examples splitting does not occur without weak beam influence for the propagation distances shown.

Equations (1)

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iUz+2U+FU2U=0.

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