Abstract

We analyze various scenarios of the dynamics of a spatial soliton interacting with a sharp potential barrier (SPB) in the complex Ginzburg–Landau model with the cubic-quintic nonlinearity. In optical realizations of the model, the SPB corresponds to a local notch in the refractive-index field. Possible outcomes of the interaction include splitting of the soliton, its lateral drift, formation of tree-like multi-jet patterns, and destruction of the soliton. The results suggest applications to the design of two- and multi-route splitters of light beams.

© 2010 Optical Society of America

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References

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  1. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).
  2. G. I. Stegeman, D. N. Christodoulides, and M. Segev, “Optical spatial solitons: historical perspectives,” IEEE J. Sel. Top. Quantum Electron. 6, 1419–1427 (2000).
    [CrossRef]
  3. N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, 1997).
  4. W. J. Firth, in Self-Organization in Optical Systems and Applications in Information Technology, M.A.Vorontsov and W.B.Miller, eds. (Springer-Verlag, 1995), p. 69.
    [CrossRef]
  5. A. Fratalocchi and G. Assanto, “Governing soliton splitting in one-dimensional lattices,” Phys. Rev. E 73, 046603 (2006).
    [CrossRef]
  6. R. Yang and X. Wu, “Spatial soliton tunneling, compression and splitting,” Opt. Express 16, 17759–17767 (2008).
    [CrossRef] [PubMed]
  7. J. Holmer, J. Marzuola, and M. Zworski, “Soliton splitting by external delta potentials,” J. Nonlinear Sci. 17, 349–367 (2007).
    [CrossRef]
  8. B. A. Malomed, “Complex Ginzburg–Landau equation,” in Encyclopedia of Nonlinear Science, A.Scott, ed. (Routledge, 2005) pp. 157–160; B. A. MalomedChaos 17, 037117 (2007).
    [CrossRef]
  9. D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
    [CrossRef] [PubMed]
  10. D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg–Landau equation,” Phys. Rev. A 77, 033817 (2008).
    [CrossRef]
  11. J. M. Soto-Crespo, N. Akhmediev, C. Mejía-Cortés, and N. Devine, “Dissipative ring solitons with vorticity,” Opt. Express 17, 4236–4250 (2009).
    [CrossRef] [PubMed]
  12. H. Leblond, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the Ginzburg–Landau model of a two-dimensional lasing medium with a transverse grating,” Phys. Rev. A 80, 033835 (2009).
    [CrossRef]
  13. H. Sakaguchi, “Splitting instability of cellular structures in the Ginzburg–Landau model under feedback control,” Phys. Rev. E 80, 017202 (2009).
    [CrossRef]

2009 (3)

J. M. Soto-Crespo, N. Akhmediev, C. Mejía-Cortés, and N. Devine, “Dissipative ring solitons with vorticity,” Opt. Express 17, 4236–4250 (2009).
[CrossRef] [PubMed]

H. Leblond, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the Ginzburg–Landau model of a two-dimensional lasing medium with a transverse grating,” Phys. Rev. A 80, 033835 (2009).
[CrossRef]

H. Sakaguchi, “Splitting instability of cellular structures in the Ginzburg–Landau model under feedback control,” Phys. Rev. E 80, 017202 (2009).
[CrossRef]

2008 (2)

R. Yang and X. Wu, “Spatial soliton tunneling, compression and splitting,” Opt. Express 16, 17759–17767 (2008).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg–Landau equation,” Phys. Rev. A 77, 033817 (2008).
[CrossRef]

2007 (2)

J. Holmer, J. Marzuola, and M. Zworski, “Soliton splitting by external delta potentials,” J. Nonlinear Sci. 17, 349–367 (2007).
[CrossRef]

B. A. Malomed, “Complex Ginzburg–Landau equation,” in Encyclopedia of Nonlinear Science, A.Scott, ed. (Routledge, 2005) pp. 157–160; B. A. MalomedChaos 17, 037117 (2007).
[CrossRef]

B. A. Malomed, “Complex Ginzburg–Landau equation,” in Encyclopedia of Nonlinear Science, A.Scott, ed. (Routledge, 2005) pp. 157–160; B. A. MalomedChaos 17, 037117 (2007).
[CrossRef]

2006 (2)

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

A. Fratalocchi and G. Assanto, “Governing soliton splitting in one-dimensional lattices,” Phys. Rev. E 73, 046603 (2006).
[CrossRef]

2000 (1)

G. I. Stegeman, D. N. Christodoulides, and M. Segev, “Optical spatial solitons: historical perspectives,” IEEE J. Sel. Top. Quantum Electron. 6, 1419–1427 (2000).
[CrossRef]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Akhmediev, N.

Akhmediev, N. N.

N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, 1997).

Ankiewicz, A.

N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, 1997).

Assanto, G.

A. Fratalocchi and G. Assanto, “Governing soliton splitting in one-dimensional lattices,” Phys. Rev. E 73, 046603 (2006).
[CrossRef]

Christodoulides, D. N.

G. I. Stegeman, D. N. Christodoulides, and M. Segev, “Optical spatial solitons: historical perspectives,” IEEE J. Sel. Top. Quantum Electron. 6, 1419–1427 (2000).
[CrossRef]

Crasovan, L. -C.

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

Devine, N.

Firth, W. J.

W. J. Firth, in Self-Organization in Optical Systems and Applications in Information Technology, M.A.Vorontsov and W.B.Miller, eds. (Springer-Verlag, 1995), p. 69.
[CrossRef]

Fratalocchi, A.

A. Fratalocchi and G. Assanto, “Governing soliton splitting in one-dimensional lattices,” Phys. Rev. E 73, 046603 (2006).
[CrossRef]

Holmer, J.

J. Holmer, J. Marzuola, and M. Zworski, “Soliton splitting by external delta potentials,” J. Nonlinear Sci. 17, 349–367 (2007).
[CrossRef]

Kartashov, Y. V.

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

Kivshar, Y. S.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Leblond, H.

H. Leblond, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the Ginzburg–Landau model of a two-dimensional lasing medium with a transverse grating,” Phys. Rev. A 80, 033835 (2009).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg–Landau equation,” Phys. Rev. A 77, 033817 (2008).
[CrossRef]

Lederer, F.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg–Landau equation,” Phys. Rev. A 77, 033817 (2008).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

Malomed, B. A.

H. Leblond, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the Ginzburg–Landau model of a two-dimensional lasing medium with a transverse grating,” Phys. Rev. A 80, 033835 (2009).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg–Landau equation,” Phys. Rev. A 77, 033817 (2008).
[CrossRef]

B. A. Malomed, “Complex Ginzburg–Landau equation,” in Encyclopedia of Nonlinear Science, A.Scott, ed. (Routledge, 2005) pp. 157–160; B. A. MalomedChaos 17, 037117 (2007).
[CrossRef]

B. A. Malomed, “Complex Ginzburg–Landau equation,” in Encyclopedia of Nonlinear Science, A.Scott, ed. (Routledge, 2005) pp. 157–160; B. A. MalomedChaos 17, 037117 (2007).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

Marzuola, J.

J. Holmer, J. Marzuola, and M. Zworski, “Soliton splitting by external delta potentials,” J. Nonlinear Sci. 17, 349–367 (2007).
[CrossRef]

Mazilu, D.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg–Landau equation,” Phys. Rev. A 77, 033817 (2008).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

Mejía-Cortés, C.

Mihalache, D.

H. Leblond, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the Ginzburg–Landau model of a two-dimensional lasing medium with a transverse grating,” Phys. Rev. A 80, 033835 (2009).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg–Landau equation,” Phys. Rev. A 77, 033817 (2008).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

Sakaguchi, H.

H. Sakaguchi, “Splitting instability of cellular structures in the Ginzburg–Landau model under feedback control,” Phys. Rev. E 80, 017202 (2009).
[CrossRef]

Segev, M.

G. I. Stegeman, D. N. Christodoulides, and M. Segev, “Optical spatial solitons: historical perspectives,” IEEE J. Sel. Top. Quantum Electron. 6, 1419–1427 (2000).
[CrossRef]

Soto-Crespo, J. M.

Stegeman, G. I.

G. I. Stegeman, D. N. Christodoulides, and M. Segev, “Optical spatial solitons: historical perspectives,” IEEE J. Sel. Top. Quantum Electron. 6, 1419–1427 (2000).
[CrossRef]

Torner, L.

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

Wu, X.

Yang, R.

Zworski, M.

J. Holmer, J. Marzuola, and M. Zworski, “Soliton splitting by external delta potentials,” J. Nonlinear Sci. 17, 349–367 (2007).
[CrossRef]

Chaos (1)

B. A. Malomed, “Complex Ginzburg–Landau equation,” in Encyclopedia of Nonlinear Science, A.Scott, ed. (Routledge, 2005) pp. 157–160; B. A. MalomedChaos 17, 037117 (2007).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

G. I. Stegeman, D. N. Christodoulides, and M. Segev, “Optical spatial solitons: historical perspectives,” IEEE J. Sel. Top. Quantum Electron. 6, 1419–1427 (2000).
[CrossRef]

J. Nonlinear Sci. (1)

J. Holmer, J. Marzuola, and M. Zworski, “Soliton splitting by external delta potentials,” J. Nonlinear Sci. 17, 349–367 (2007).
[CrossRef]

Opt. Express (2)

Phys. Rev. A (2)

H. Leblond, B. A. Malomed, and D. Mihalache, “Stable vortex solitons in the Ginzburg–Landau model of a two-dimensional lasing medium with a transverse grating,” Phys. Rev. A 80, 033835 (2009).
[CrossRef]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg–Landau equation,” Phys. Rev. A 77, 033817 (2008).
[CrossRef]

Phys. Rev. E (2)

H. Sakaguchi, “Splitting instability of cellular structures in the Ginzburg–Landau model under feedback control,” Phys. Rev. E 80, 017202 (2009).
[CrossRef]

A. Fratalocchi and G. Assanto, “Governing soliton splitting in one-dimensional lattices,” Phys. Rev. E 73, 046603 (2006).
[CrossRef]

Phys. Rev. Lett. (1)

D. Mihalache, D. Mazilu, F. Lederer, Y. V. Kartashov, L.-C. Crasovan, L. Torner, and B. A. Malomed, “Stable vortex tori in the three-dimensional cubic-quintic Ginzburg–Landau equation,” Phys. Rev. Lett. 97, 073904 (2006).
[CrossRef] [PubMed]

Other (3)

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, 1997).

W. J. Firth, in Self-Organization in Optical Systems and Applications in Information Technology, M.A.Vorontsov and W.B.Miller, eds. (Springer-Verlag, 1995), p. 69.
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Dynamics of solitons on top of the SPB with n = 1 . (a) The initial soliton’s profile (solid curve) and the shape of the potential (dashed curve). (b) The soliton rolling off from the top of potential, at α = 0.18 . (c) The soliton splitting into a cluster of parallel solitary filaments at α = 0.2 . (d) The soliton splitting into a cluster of solitary filaments, which, however, quickly decay at α = 0.45 . (e) Annihilation of the soliton at α = 0.8 . (f) The transverse profile the of output beam corresponding to (c).

Fig. 2
Fig. 2

Dynamics of solitons on top of the SPB with n = 0.2 . (a) The initial soliton profile and the SPB shape (solid and dashed curves). (b) The soliton rolling down from the SPB with α = 0.3 . (c) The formation of a periodic multi-eyelet pattern at α = 1.4 . (d) Splitting of the soliton into two solitary filaments at α = 2 . (e) Destruction of the soliton at α = 5 .

Fig. 3
Fig. 3

Various outcomes of the evolution of the soliton in the plane of ( α , n ) : soliton rolling off from the top of the SPB (region A); splitting into arrays of solitary filaments, which is quickly followed by the decay of fusion as shown in Figs. 1d, 2c (regions E and B, respectively); splitting into two filaments (region C); splitting into a cluster of solitary filaments (region D); destruction of the soliton (region F).

Fig. 4
Fig. 4

(a) The maximum initial displacement Δ x , beyond which the soliton rolls down from the potential slope without splitting [as shown in (c) for Δ x = 1 and α = 2.5 ] versus the SPB’s strength for n = 0.2 . (b) At Δ x < Δ max , the soliton splits into two filaments. In this example, Δ x = 0.6 and α = 2.5 .

Fig. 5
Fig. 5

(a) and (c) The same as in Figs. 4a, 4c, but for n = 1 , with Δ x = 0.7 and α = 0.26 in (c). (b) displays an example of an asymmetric (single-side) splitting into a cluster of solitary filaments, at Δ x = 0.6 and α = 0.26 .

Fig. 6
Fig. 6

(a) The single-side splitting of the shifted soliton placed on top of the SPB with n = 1 occurs in the gray region in the plane of ( n , α ) . (b) The largest displacement Δ x of the initial soliton, which admits the “single-side splitting,” versus index n of the potential. (c) An example of the single-side splitting, for Δ x = 0.06 .

Fig. 7
Fig. 7

The dynamics of the soliton starting at the top of the SPB with n = 1 , in the case of β = 0 (no viscosity). (a) An example of the soliton rolling off from the top of potential, at α = 0.1 . (b) The spread of the soliton, at α = 0.4 . (c) Splitting of the soliton into two broad fragments, at α = 0.5 . (d) Splitting of the soliton into two solitary fragments, at α = 5 . Other parameters are the same as in Fig. 1.

Fig. 8
Fig. 8

The chart summarizing outcomes of the evolution of the soliton starting at the top of the SPB, in the case of β = 0 . In the black region, the soliton rolls down from the SPB, without splitting. In the gray region, the soliton spreads out (decays). In the white region, the soliton splits into two fragments: broad or soliton-like ones.

Equations (2)

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i u z + ( 1 / 2 ) u x x + | u | 2 u + ν | u | 4 u = i R [ u ] + V ( x ) u .
R [ u ] = δ u + β u x x + ε | u | 2 u + μ | u | 4 u ,

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