Abstract

We present a comprehensive theoretical investigation of the formation and propagation of multicolor ringlike soliton clusters in quadratic nonlinear materials under conditions for type I second-harmonic generation. We show the several existing propagation regimes for the soliton clusters, which include fusion, expansion, and quasi-stationary propagation. The physical mechanism that leads to each evolution is elucidated and discussed. In the case of purely expanding clusters, accurate estimates of the expansion rate are obtained with a Hamiltonian approach. The experimental formation of the different soliton clusters by induced breakup techniques is discussed in detail, and the practical limitations of the concept of quasi-stationary clusters are highlighted.

© 2002 Optical Society of America

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  1. G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (2996).
    [CrossRef]
  2. L. Torner and A. P. Sukhorukov, “Quadratic solitons,” Opt. Photon. News 13(2), 42–47 (2002).
    [CrossRef]
  3. A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities,” Phys. Rep. (to be published).
  4. A. Bramati, W. Chinaglia, S. Minardi, and P. Di Trapani, “Reconstruction of blurred images by controlled formation of spatial solitons,” Opt. Lett. 26, 1409–1411 (2001).
    [CrossRef]
  5. M. Soljacic, S. Sears, and M. Segev, “Self-trapping of necklace beams in self-focusing Kerr media,” Phys. Rev. Lett. 81, 4851–4854 (1998).
    [CrossRef]
  6. M. Soljacic and M. Segev, “Self-trapping of necklace-ring beams in self-focusing Kerr media,” Phys. Rev. E 62, 2810–2820 (2000).
    [CrossRef]
  7. M. Soljacic and M. Segev, “Integer and fractional angular momentum borne on self-trapped necklace-ring beams,” Phys. Rev. Lett. 86, 420–423 (2001).
    [CrossRef] [PubMed]
  8. A. S. Desyatnikov and Yu. S. Kivshar, “Necklace-ring vector solitons,” Phys. Rev. Lett. 87, 033901 (2001).
    [CrossRef] [PubMed]
  9. A. S. Desyatnikov and Yu. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. 88, 053901 (2002).
    [CrossRef] [PubMed]
  10. L. Torner, J. P. Torres, D. V. Petrov, and J. M. Soto-Crespo, “From topological charge information to sets of solitons in quadratic non-linear media,” Opt. Quantum Electron. 30, 809–827 (1998).
    [CrossRef]
  11. L. Torner and D. V. Petrov, “Azimuthal instabilities and self-breaking of beams into sets of solitons in bulk second-harmonic generation,” Electron. Lett. 33, 608–609 (1997).
    [CrossRef]
  12. L. Torner and D. V. Petrov, “Splitting of light beams with spiral phase dislocations into solitons in bulk quadratic nonlinear media,” J. Opt. Soc. Am. B 14, 2017–2023 (1997).
    [CrossRef]
  13. W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450–2453 (1997).
    [CrossRef]
  14. D. V. Skryabin and W. J. Firth, “Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media,” Phys. Rev. E 58, 3916–3930 (1998).
    [CrossRef]
  15. J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in quadratic nonlinear media,” J. Opt. Soc. Am. B 15, 625–627 (1998).
    [CrossRef]
  16. D. V. Petrov, L. Torner, J. Martorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, “Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal,” Opt. Lett. 23, 1444–1446 (1998).
    [CrossRef]
  17. S. Minardi, G. Molina-Terriza, P. Di Trapani, J. P. Torres, and L. Torner, “Soliton algebra by vortex-beam splitting,” Opt. Lett. 26, 1004–1006 (2001).
    [CrossRef]
  18. W. E. Torruellas, Z. Wang, D. Hagan, E. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
    [CrossRef] [PubMed]
  19. A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, “Self-trapping of light beams and parametric solitons in diffractive quadratic media,” Phys. Rev. A 52, 1670–1674 (1995).
    [CrossRef] [PubMed]
  20. L. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruellas, and G. I. Stegeman, “Stationary trapping of light beams in bulk second-order nonlinear media,” Opt. Commun. 121, 149–155 (1995).
    [CrossRef]
  21. D.-M. Baboiu, G. I. Stegeman, and L. Torner, “Interaction of one-dimensional bright solitary waves in quadratic media,” Opt. Lett. 20, 2282–2284 (1995).
    [CrossRef] [PubMed]
  22. C. Etrich, U. Peschel, F. Lederer, and B. Malomed, “Collision of solitary waves in media with a second-order nonlinearity,” Phys. Rev. A 52, R3444–3447 (1995).
    [CrossRef] [PubMed]
  23. G. Leo, G. Assanto, and W. E. Torruellas, “Intensity-controlled interactions between vectorial spatial solitary waves in quadratic nonlinear media,” Opt. Lett. 22, 7–9 (1997).
    [CrossRef] [PubMed]
  24. D.-M. Baboiu and G. I. Stegeman, “Solitary-wave interactions in quadratic media near type I phase matching conditions,” J. Opt. Soc. Am. B 14, 3143–3150 (1997).
    [CrossRef]
  25. G. Leo and G. Assanto, “Collisional interaction of vectorial spatial solitary waves in type II frequency-doubling media,” J. Opt. Soc. Am. B 14, 3151–3161 (1997).
    [CrossRef]
  26. R. Schiek, Y. Baek, G. I. Stegeman, and W. Sohler, “Interactions between one-dimensional quadratic soliton-like beams,” Opt. Quantum Electron. 30, 861–879 (1998).
    [CrossRef]
  27. A. Barthelemy, B. Bourliaguet, V. Kermene, B. Constantini, C. De Angelis, D. Modotto, and G. Assanto, “Interaction of type II vectorial spatial solitary waves in materials with quadratic non-linearity,” Opt. Quantum Electron. 30, 923–935 (1998).
    [CrossRef]
  28. V. V. Steblina, Yu. S. Kivshar, and A. V. Buryak, “Scattering and spiraling of solitons in a bulk quadratic medium,” Opt. Lett. 23, 156–158 (1998).
    [CrossRef]
  29. B. Constantini, C. De Angelis, A. Barthelemy, B. Bourliaguet, and V. Kermene, “Collisions between type II two-dimensional quadratic solitons,” Opt. Lett. 23, 424–426 (1998).
    [CrossRef]
  30. A. V. Buryak and V. V. Steblina, “Soliton collisions in bulk quadratic media: comprehensive analytical and numerical study,” J. Opt. Soc. Am. B 16, 245–255 (1999).
    [CrossRef]

2996 (1)

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (2996).
[CrossRef]

2002 (2)

L. Torner and A. P. Sukhorukov, “Quadratic solitons,” Opt. Photon. News 13(2), 42–47 (2002).
[CrossRef]

A. S. Desyatnikov and Yu. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. 88, 053901 (2002).
[CrossRef] [PubMed]

2001 (4)

2000 (1)

M. Soljacic and M. Segev, “Self-trapping of necklace-ring beams in self-focusing Kerr media,” Phys. Rev. E 62, 2810–2820 (2000).
[CrossRef]

1999 (1)

1998 (9)

V. V. Steblina, Yu. S. Kivshar, and A. V. Buryak, “Scattering and spiraling of solitons in a bulk quadratic medium,” Opt. Lett. 23, 156–158 (1998).
[CrossRef]

B. Constantini, C. De Angelis, A. Barthelemy, B. Bourliaguet, and V. Kermene, “Collisions between type II two-dimensional quadratic solitons,” Opt. Lett. 23, 424–426 (1998).
[CrossRef]

D. V. Petrov, L. Torner, J. Martorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, “Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal,” Opt. Lett. 23, 1444–1446 (1998).
[CrossRef]

J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in quadratic nonlinear media,” J. Opt. Soc. Am. B 15, 625–627 (1998).
[CrossRef]

M. Soljacic, S. Sears, and M. Segev, “Self-trapping of necklace beams in self-focusing Kerr media,” Phys. Rev. Lett. 81, 4851–4854 (1998).
[CrossRef]

L. Torner, J. P. Torres, D. V. Petrov, and J. M. Soto-Crespo, “From topological charge information to sets of solitons in quadratic non-linear media,” Opt. Quantum Electron. 30, 809–827 (1998).
[CrossRef]

R. Schiek, Y. Baek, G. I. Stegeman, and W. Sohler, “Interactions between one-dimensional quadratic soliton-like beams,” Opt. Quantum Electron. 30, 861–879 (1998).
[CrossRef]

A. Barthelemy, B. Bourliaguet, V. Kermene, B. Constantini, C. De Angelis, D. Modotto, and G. Assanto, “Interaction of type II vectorial spatial solitary waves in materials with quadratic non-linearity,” Opt. Quantum Electron. 30, 923–935 (1998).
[CrossRef]

D. V. Skryabin and W. J. Firth, “Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media,” Phys. Rev. E 58, 3916–3930 (1998).
[CrossRef]

1997 (6)

1995 (5)

W. E. Torruellas, Z. Wang, D. Hagan, E. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, “Self-trapping of light beams and parametric solitons in diffractive quadratic media,” Phys. Rev. A 52, 1670–1674 (1995).
[CrossRef] [PubMed]

L. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruellas, and G. I. Stegeman, “Stationary trapping of light beams in bulk second-order nonlinear media,” Opt. Commun. 121, 149–155 (1995).
[CrossRef]

D.-M. Baboiu, G. I. Stegeman, and L. Torner, “Interaction of one-dimensional bright solitary waves in quadratic media,” Opt. Lett. 20, 2282–2284 (1995).
[CrossRef] [PubMed]

C. Etrich, U. Peschel, F. Lederer, and B. Malomed, “Collision of solitary waves in media with a second-order nonlinearity,” Phys. Rev. A 52, R3444–3447 (1995).
[CrossRef] [PubMed]

Assanto, G.

Baboiu, D.-M.

Baek, Y.

R. Schiek, Y. Baek, G. I. Stegeman, and W. Sohler, “Interactions between one-dimensional quadratic soliton-like beams,” Opt. Quantum Electron. 30, 861–879 (1998).
[CrossRef]

Barthelemy, A.

A. Barthelemy, B. Bourliaguet, V. Kermene, B. Constantini, C. De Angelis, D. Modotto, and G. Assanto, “Interaction of type II vectorial spatial solitary waves in materials with quadratic non-linearity,” Opt. Quantum Electron. 30, 923–935 (1998).
[CrossRef]

B. Constantini, C. De Angelis, A. Barthelemy, B. Bourliaguet, and V. Kermene, “Collisions between type II two-dimensional quadratic solitons,” Opt. Lett. 23, 424–426 (1998).
[CrossRef]

Bourliaguet, B.

B. Constantini, C. De Angelis, A. Barthelemy, B. Bourliaguet, and V. Kermene, “Collisions between type II two-dimensional quadratic solitons,” Opt. Lett. 23, 424–426 (1998).
[CrossRef]

A. Barthelemy, B. Bourliaguet, V. Kermene, B. Constantini, C. De Angelis, D. Modotto, and G. Assanto, “Interaction of type II vectorial spatial solitary waves in materials with quadratic non-linearity,” Opt. Quantum Electron. 30, 923–935 (1998).
[CrossRef]

Bramati, A.

Buryak, A. V.

Chinaglia, W.

Cojocaru, C.

Constantini, B.

B. Constantini, C. De Angelis, A. Barthelemy, B. Bourliaguet, and V. Kermene, “Collisions between type II two-dimensional quadratic solitons,” Opt. Lett. 23, 424–426 (1998).
[CrossRef]

A. Barthelemy, B. Bourliaguet, V. Kermene, B. Constantini, C. De Angelis, D. Modotto, and G. Assanto, “Interaction of type II vectorial spatial solitary waves in materials with quadratic non-linearity,” Opt. Quantum Electron. 30, 923–935 (1998).
[CrossRef]

De Angelis, C.

A. Barthelemy, B. Bourliaguet, V. Kermene, B. Constantini, C. De Angelis, D. Modotto, and G. Assanto, “Interaction of type II vectorial spatial solitary waves in materials with quadratic non-linearity,” Opt. Quantum Electron. 30, 923–935 (1998).
[CrossRef]

B. Constantini, C. De Angelis, A. Barthelemy, B. Bourliaguet, and V. Kermene, “Collisions between type II two-dimensional quadratic solitons,” Opt. Lett. 23, 424–426 (1998).
[CrossRef]

Desyatnikov, A. S.

A. S. Desyatnikov and Yu. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. 88, 053901 (2002).
[CrossRef] [PubMed]

A. S. Desyatnikov and Yu. S. Kivshar, “Necklace-ring vector solitons,” Phys. Rev. Lett. 87, 033901 (2001).
[CrossRef] [PubMed]

Di Trapani, P.

Etrich, C.

C. Etrich, U. Peschel, F. Lederer, and B. Malomed, “Collision of solitary waves in media with a second-order nonlinearity,” Phys. Rev. A 52, R3444–3447 (1995).
[CrossRef] [PubMed]

Firth, W. J.

D. V. Skryabin and W. J. Firth, “Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media,” Phys. Rev. E 58, 3916–3930 (1998).
[CrossRef]

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450–2453 (1997).
[CrossRef]

Hagan, D.

W. E. Torruellas, Z. Wang, D. Hagan, E. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Hagan, D. J.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (2996).
[CrossRef]

Kermene, V.

A. Barthelemy, B. Bourliaguet, V. Kermene, B. Constantini, C. De Angelis, D. Modotto, and G. Assanto, “Interaction of type II vectorial spatial solitary waves in materials with quadratic non-linearity,” Opt. Quantum Electron. 30, 923–935 (1998).
[CrossRef]

B. Constantini, C. De Angelis, A. Barthelemy, B. Bourliaguet, and V. Kermene, “Collisions between type II two-dimensional quadratic solitons,” Opt. Lett. 23, 424–426 (1998).
[CrossRef]

Kivshar, Yu. S.

A. S. Desyatnikov and Yu. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. 88, 053901 (2002).
[CrossRef] [PubMed]

A. S. Desyatnikov and Yu. S. Kivshar, “Necklace-ring vector solitons,” Phys. Rev. Lett. 87, 033901 (2001).
[CrossRef] [PubMed]

V. V. Steblina, Yu. S. Kivshar, and A. V. Buryak, “Scattering and spiraling of solitons in a bulk quadratic medium,” Opt. Lett. 23, 156–158 (1998).
[CrossRef]

A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, “Self-trapping of light beams and parametric solitons in diffractive quadratic media,” Phys. Rev. A 52, 1670–1674 (1995).
[CrossRef] [PubMed]

Lederer, F.

C. Etrich, U. Peschel, F. Lederer, and B. Malomed, “Collision of solitary waves in media with a second-order nonlinearity,” Phys. Rev. A 52, R3444–3447 (1995).
[CrossRef] [PubMed]

Leo, G.

Malomed, B.

C. Etrich, U. Peschel, F. Lederer, and B. Malomed, “Collision of solitary waves in media with a second-order nonlinearity,” Phys. Rev. A 52, R3444–3447 (1995).
[CrossRef] [PubMed]

Martorell, J.

Mazilu, D.

L. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruellas, and G. I. Stegeman, “Stationary trapping of light beams in bulk second-order nonlinear media,” Opt. Commun. 121, 149–155 (1995).
[CrossRef]

Menyuk, C. R.

W. E. Torruellas, Z. Wang, D. Hagan, E. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Mihalache, D.

L. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruellas, and G. I. Stegeman, “Stationary trapping of light beams in bulk second-order nonlinear media,” Opt. Commun. 121, 149–155 (1995).
[CrossRef]

Minardi, S.

Modotto, D.

A. Barthelemy, B. Bourliaguet, V. Kermene, B. Constantini, C. De Angelis, D. Modotto, and G. Assanto, “Interaction of type II vectorial spatial solitary waves in materials with quadratic non-linearity,” Opt. Quantum Electron. 30, 923–935 (1998).
[CrossRef]

Molina-Terriza, G.

Peschel, U.

C. Etrich, U. Peschel, F. Lederer, and B. Malomed, “Collision of solitary waves in media with a second-order nonlinearity,” Phys. Rev. A 52, R3444–3447 (1995).
[CrossRef] [PubMed]

Petrov, D. V.

Schiek, R.

R. Schiek, Y. Baek, G. I. Stegeman, and W. Sohler, “Interactions between one-dimensional quadratic soliton-like beams,” Opt. Quantum Electron. 30, 861–879 (1998).
[CrossRef]

Sears, S.

M. Soljacic, S. Sears, and M. Segev, “Self-trapping of necklace beams in self-focusing Kerr media,” Phys. Rev. Lett. 81, 4851–4854 (1998).
[CrossRef]

Segev, M.

M. Soljacic and M. Segev, “Integer and fractional angular momentum borne on self-trapped necklace-ring beams,” Phys. Rev. Lett. 86, 420–423 (2001).
[CrossRef] [PubMed]

M. Soljacic and M. Segev, “Self-trapping of necklace-ring beams in self-focusing Kerr media,” Phys. Rev. E 62, 2810–2820 (2000).
[CrossRef]

M. Soljacic, S. Sears, and M. Segev, “Self-trapping of necklace beams in self-focusing Kerr media,” Phys. Rev. Lett. 81, 4851–4854 (1998).
[CrossRef]

Skryabin, D. V.

D. V. Skryabin and W. J. Firth, “Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media,” Phys. Rev. E 58, 3916–3930 (1998).
[CrossRef]

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450–2453 (1997).
[CrossRef]

Sohler, W.

R. Schiek, Y. Baek, G. I. Stegeman, and W. Sohler, “Interactions between one-dimensional quadratic soliton-like beams,” Opt. Quantum Electron. 30, 861–879 (1998).
[CrossRef]

Soljacic, M.

M. Soljacic and M. Segev, “Integer and fractional angular momentum borne on self-trapped necklace-ring beams,” Phys. Rev. Lett. 86, 420–423 (2001).
[CrossRef] [PubMed]

M. Soljacic and M. Segev, “Self-trapping of necklace-ring beams in self-focusing Kerr media,” Phys. Rev. E 62, 2810–2820 (2000).
[CrossRef]

M. Soljacic, S. Sears, and M. Segev, “Self-trapping of necklace beams in self-focusing Kerr media,” Phys. Rev. Lett. 81, 4851–4854 (1998).
[CrossRef]

Soto-Crespo, J. M.

L. Torner, J. P. Torres, D. V. Petrov, and J. M. Soto-Crespo, “From topological charge information to sets of solitons in quadratic non-linear media,” Opt. Quantum Electron. 30, 809–827 (1998).
[CrossRef]

J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in quadratic nonlinear media,” J. Opt. Soc. Am. B 15, 625–627 (1998).
[CrossRef]

Steblina, V. V.

Stegeman, G. I.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (2996).
[CrossRef]

R. Schiek, Y. Baek, G. I. Stegeman, and W. Sohler, “Interactions between one-dimensional quadratic soliton-like beams,” Opt. Quantum Electron. 30, 861–879 (1998).
[CrossRef]

D.-M. Baboiu and G. I. Stegeman, “Solitary-wave interactions in quadratic media near type I phase matching conditions,” J. Opt. Soc. Am. B 14, 3143–3150 (1997).
[CrossRef]

D.-M. Baboiu, G. I. Stegeman, and L. Torner, “Interaction of one-dimensional bright solitary waves in quadratic media,” Opt. Lett. 20, 2282–2284 (1995).
[CrossRef] [PubMed]

L. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruellas, and G. I. Stegeman, “Stationary trapping of light beams in bulk second-order nonlinear media,” Opt. Commun. 121, 149–155 (1995).
[CrossRef]

W. E. Torruellas, Z. Wang, D. Hagan, E. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Sukhorukov, A. P.

L. Torner and A. P. Sukhorukov, “Quadratic solitons,” Opt. Photon. News 13(2), 42–47 (2002).
[CrossRef]

Torner, L.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (2996).
[CrossRef]

L. Torner and A. P. Sukhorukov, “Quadratic solitons,” Opt. Photon. News 13(2), 42–47 (2002).
[CrossRef]

S. Minardi, G. Molina-Terriza, P. Di Trapani, J. P. Torres, and L. Torner, “Soliton algebra by vortex-beam splitting,” Opt. Lett. 26, 1004–1006 (2001).
[CrossRef]

D. V. Petrov, L. Torner, J. Martorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, “Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal,” Opt. Lett. 23, 1444–1446 (1998).
[CrossRef]

J. P. Torres, J. M. Soto-Crespo, L. Torner, and D. V. Petrov, “Solitary-wave vortices in quadratic nonlinear media,” J. Opt. Soc. Am. B 15, 625–627 (1998).
[CrossRef]

L. Torner, J. P. Torres, D. V. Petrov, and J. M. Soto-Crespo, “From topological charge information to sets of solitons in quadratic non-linear media,” Opt. Quantum Electron. 30, 809–827 (1998).
[CrossRef]

L. Torner and D. V. Petrov, “Azimuthal instabilities and self-breaking of beams into sets of solitons in bulk second-harmonic generation,” Electron. Lett. 33, 608–609 (1997).
[CrossRef]

L. Torner and D. V. Petrov, “Splitting of light beams with spiral phase dislocations into solitons in bulk quadratic nonlinear media,” J. Opt. Soc. Am. B 14, 2017–2023 (1997).
[CrossRef]

D.-M. Baboiu, G. I. Stegeman, and L. Torner, “Interaction of one-dimensional bright solitary waves in quadratic media,” Opt. Lett. 20, 2282–2284 (1995).
[CrossRef] [PubMed]

W. E. Torruellas, Z. Wang, D. Hagan, E. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

L. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruellas, and G. I. Stegeman, “Stationary trapping of light beams in bulk second-order nonlinear media,” Opt. Commun. 121, 149–155 (1995).
[CrossRef]

Torres, J. P.

Torruellas, W. E.

G. Leo, G. Assanto, and W. E. Torruellas, “Intensity-controlled interactions between vectorial spatial solitary waves in quadratic nonlinear media,” Opt. Lett. 22, 7–9 (1997).
[CrossRef] [PubMed]

L. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruellas, and G. I. Stegeman, “Stationary trapping of light beams in bulk second-order nonlinear media,” Opt. Commun. 121, 149–155 (1995).
[CrossRef]

W. E. Torruellas, Z. Wang, D. Hagan, E. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Van Stryland, E.

W. E. Torruellas, Z. Wang, D. Hagan, E. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Vilaseca, R.

Wang, Z.

W. E. Torruellas, Z. Wang, D. Hagan, E. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Wright, E. M.

L. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruellas, and G. I. Stegeman, “Stationary trapping of light beams in bulk second-order nonlinear media,” Opt. Commun. 121, 149–155 (1995).
[CrossRef]

Electron. Lett. (1)

L. Torner and D. V. Petrov, “Azimuthal instabilities and self-breaking of beams into sets of solitons in bulk second-harmonic generation,” Electron. Lett. 33, 608–609 (1997).
[CrossRef]

J. Opt. Soc. Am. B (5)

Opt. Commun. (1)

L. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. E. Torruellas, and G. I. Stegeman, “Stationary trapping of light beams in bulk second-order nonlinear media,” Opt. Commun. 121, 149–155 (1995).
[CrossRef]

Opt. Lett. (7)

Opt. Photon. News (1)

L. Torner and A. P. Sukhorukov, “Quadratic solitons,” Opt. Photon. News 13(2), 42–47 (2002).
[CrossRef]

Opt. Quantum Electron. (4)

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (2996).
[CrossRef]

L. Torner, J. P. Torres, D. V. Petrov, and J. M. Soto-Crespo, “From topological charge information to sets of solitons in quadratic non-linear media,” Opt. Quantum Electron. 30, 809–827 (1998).
[CrossRef]

R. Schiek, Y. Baek, G. I. Stegeman, and W. Sohler, “Interactions between one-dimensional quadratic soliton-like beams,” Opt. Quantum Electron. 30, 861–879 (1998).
[CrossRef]

A. Barthelemy, B. Bourliaguet, V. Kermene, B. Constantini, C. De Angelis, D. Modotto, and G. Assanto, “Interaction of type II vectorial spatial solitary waves in materials with quadratic non-linearity,” Opt. Quantum Electron. 30, 923–935 (1998).
[CrossRef]

Phys. Rev. A (2)

C. Etrich, U. Peschel, F. Lederer, and B. Malomed, “Collision of solitary waves in media with a second-order nonlinearity,” Phys. Rev. A 52, R3444–3447 (1995).
[CrossRef] [PubMed]

A. V. Buryak, Yu. S. Kivshar, and V. V. Steblina, “Self-trapping of light beams and parametric solitons in diffractive quadratic media,” Phys. Rev. A 52, 1670–1674 (1995).
[CrossRef] [PubMed]

Phys. Rev. E (2)

D. V. Skryabin and W. J. Firth, “Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media,” Phys. Rev. E 58, 3916–3930 (1998).
[CrossRef]

M. Soljacic and M. Segev, “Self-trapping of necklace-ring beams in self-focusing Kerr media,” Phys. Rev. E 62, 2810–2820 (2000).
[CrossRef]

Phys. Rev. Lett. (6)

M. Soljacic and M. Segev, “Integer and fractional angular momentum borne on self-trapped necklace-ring beams,” Phys. Rev. Lett. 86, 420–423 (2001).
[CrossRef] [PubMed]

A. S. Desyatnikov and Yu. S. Kivshar, “Necklace-ring vector solitons,” Phys. Rev. Lett. 87, 033901 (2001).
[CrossRef] [PubMed]

A. S. Desyatnikov and Yu. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. 88, 053901 (2002).
[CrossRef] [PubMed]

M. Soljacic, S. Sears, and M. Segev, “Self-trapping of necklace beams in self-focusing Kerr media,” Phys. Rev. Lett. 81, 4851–4854 (1998).
[CrossRef]

W. E. Torruellas, Z. Wang, D. Hagan, E. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. 79, 2450–2453 (1997).
[CrossRef]

Other (1)

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities,” Phys. Rep. (to be published).

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Figures (10)

Fig. 1
Fig. 1

Hamiltonian-energy diagram for the lowest-order solitons at three representative values of the phase mismatch β. Insets show profiles of the fundamental frequency w1 and second harmonic w2 waves forming the solitons. The profiles shown correspond to the points marked by the circles in the Hamiltonian-energy diagram.

Fig. 2
Fig. 2

Sketches that display the phase relation between neighboring solitons in clusters with different topological charges M and fixed number of solitons N=6 and that illustrate the different dynamical evolutions of the clusters that are possible.

Fig. 3
Fig. 3

Different propagation regimes of multicolor soliton clusters. The first line shows the evolution of the cluster radius upon propagation. Contour plots show the intensity distribution in the fundamental wave at various propagation distances. (a) N=6, M=0, R0=3; (b) N=6, M=3, R0=2.8; (c) N=20, M=6, R0=6.5; (d) N=6, M=1, R0=3.5; (e) N=6, M=1, R0=0.8. Clusters (a), (b), (d), and (e) are constructed from solitons with energy flow U=44.45, at β=3. Cluster (c) is made of solitons with U=49.34, at β=0.

Fig. 4
Fig. 4

Dependencies of the expansion rate on the energy flow of the solitons forming the cluster, for three representative values of phase mismatch β. Cluster parameters N=8, M=4, and R0=3.5.

Fig. 5
Fig. 5

Dependencies of the expansion rate on the energy flow of solitons forming the cluster (a) for different N and fixed radius R0=3.5 and (b) for different radiuses R0 and fixed N=8. Phase mismatch β=3. All plots correspond to the cluster charges M=N/2. Circles present analytical estimates of the expansion rate.

Fig. 6
Fig. 6

Expansion rate versus charge of the cluster. Insets show dependencies of the Hamiltonian H and ξ projection of the angular momentum L on the cluster charge. Parameters N=20, R0=6.5, U=49.34, and β=0.

Fig. 7
Fig. 7

Propagation dynamics of ideal quasi-stationary multicolor soliton clusters. (a) N=5, M=1, R0=1.64; (b) N=6, M=1, R0=1.18; (c) N=12, M=1, R0=0.56; (d) N=12, M=2, R0=2.18. All clusters are constructed from solitons with energy flow U=44.45 and propagate under conditions with β=3.

Fig. 8
Fig. 8

Generation of multicolor soliton clusters with the aid of the IBU technique. (a) m1=4, m2=4, and input energy flow of the fundamental wave U1=128π; (b) m1=4, m2=2, U1=128π; (c) m1=1, m2=-3, U1=256π; (d) m1=1, m2=-5, U1=500π. Input energy flow of the second-harmonic wave for all cases is U2=2π, radius R0=2, phase mismatch β=3.

Fig. 9
Fig. 9

Same as in Fig. 8 but in the presence of random Gaussian noise with zero mean and dispersion σ1,22=0.01.

Fig. 10
Fig. 10

Dependencies of radii R of clusters generated with the aid of the IBU technique on propagation distance ξ. (1) m1=1, m2=6; (2) m1=1, m2=-2; (3) m1=3, m2=10; (4) m1=3, m2=2. Contour plots show energy-flow distributions in the fundamental wave at the distance ξ=8. The distributions for equal charges m1 are superimposed onto each other for easy comparison of the corresponding cluster radii. In all cases U1=256π, U2=32π, N=|2m1-m2|=4, R0=2, and β=3.

Equations (22)

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i q1ξ=d12 2η2+2ζ2q1-q1*q2 exp(-iβξ),
i q2ξ+iδη qη+δζ qζ=d22 2η2+2ζ2q2-q12 exp(iβξ),
H=-12 -dη-dζd1|q1|2+d22|q2|2-β|q2|2+q1*2q2 exp(-iβξ)+q12q2* exp(iβξ) ,
U=- dη- dζ(|q1|2+|q2|2),
P=14i - dη- dζ(2q1*q1-2q1q1*+q2*q2-q2q2*),
Lξ=eξL=eξ 14i - dη- dζ[r×(2q1*q1-2q1q1*+q2*q2-q2q2*)].
d2dr2+1r ddrw1=2b1w1-2w1w2,
d2dr2+1r ddrw2=4(β+2b1)w2-4w12.
q1(η, ζ, ξ=0)=n=1N exp(2iπnM/N)×w1n({[η-R0 cos(2πn/N)]2+[ζ-R0 sin(2πn/N)]2}1/2),
q2(η, ζ, ξ=0)=n=1N exp(4iπnM/N)×w2n({[n-R0 cos(2πn/N)]2+[ζ-R0 sin(2πn/N)]2}1/2),
R(ξ)=1U -dη-dζ(η2+ζ2)1/2(|q1|2+|q2|2),
v=limξ+dRdξ.
q1(η, ζ, ξ)=n=1Nw1n(|r-vnξ|)exp(ivnr)×exp[i(b1n-vn2/2)ξ]exp(iϕn),
q2(η, ζ, ξ)=n=1Nw2n(|r-vnξ|)exp(2ivnr)×exp[i(2b1n+β-vn2)ξ]exp(2iϕn),
H=n=1NHn+12 n=1Nvn2Un,
Un=-dη-dζ(w1n2+w2n2),
Hn=-12 -dη-dζ-(w1n)2-14(w2n)2-βw2n2+2w1n2w2n.
v2=2H-NHnNUn=2HU-HnUn.
q1,2(r, φ, ξ=0)=w1,2r|m1,2| exp(-r2/R02)exp(im1,2φ).
U=U1+U2=π2 n=12wn2|mn|!2-|mn|R02|mn|+2,
L=L1+L2=eξ π2 n=12wn2 mn|mn|!n2-|mn|R02|mn|+2,
q1,2(r, φ, ξ=0)=[1+b1,2(r, φ)]w1,2r|m1,2| exp(-r2/R02)exp(im1,2φ).

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