Abstract

We report the first experimental observation, to our knowledge, of phase-controlled three-wave interactions between two spatial solitons in a bulk quadratic nonlinear crystal. We demonstrate that the nature of the interaction can be attractive or repulsive, depending on the phase relationships between the input beams at the fundamental frequency and their power imbalance. Full control of the interaction is achieved by the adjustment of the relative phases and/or the orientation of the input waves, leading to scattering, fusion, spiraling, and power-exchange phenomena. Experimental observations are in good qualitative agreement with the corresponding theoretical and numerical results.

© 2003 Optical Society of America

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  1. M. Toda, Nonlinear Waves and Solitons (Kluwer Academic, Boston, Mass., 1990).
  2. M. Segev and G. I. Stegeman, “Self-trapping of optical beams: spatial solitons,” Phys. Today 51(8), 42–48 (1998).
    [CrossRef]
  3. K. A. Gorshkov and L. A. Ostrovsky, “Interactions of solitons in nonintegrable systems: direct perturbation method and applications,” Physica D 3, 428–438 (1981).
    [CrossRef]
  4. V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487–502 (1981).
    [CrossRef]
  5. 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]
  6. 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]
  7. M. Segev, B. Crosignani, A. Yariv, and B. Fisher, “Spatial soliton on photorepractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
    [CrossRef] [PubMed]
  8. A. V. Buryak, Yu. S. Kivshar, M.-f. Shih, and M. Segev, “Induced coherence and stable soliton spiraling,” Phys. Rev. Lett. 82, 81–84 (1999).
    [CrossRef]
  9. W. Torruellas, Z. Wang, D. J. Hagan, E. W. 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]
  10. M.-F. Shih and M. Segev, “Three dimensional spiraling of interacting spatial solitons,” Phys. Rev. Lett. 78, 13–16 (1997).
    [CrossRef]
  11. B. Costantini, 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]
  12. M. R. Belié, A. Stepken, and F. Kaiser, “Spiraling behavior of photorefractive screening solitons,” Phys. Rev. Lett. 82, 544–547 (1999).
    [CrossRef]
  13. X. Liu, K. Beckwitt, and F. W. Wise, “Noncollinear generation of optical spatio-temporal solitons and application to ultra-fast digital logic,” Phys. Rev. E 61, R4722–R4725 (2000).
    [CrossRef]
  14. V. Couderc, E. Lopez Lago, C. Simos, and A. Barthelemy, “Experiments in quadratic soliton generation and steering in a noncollinear geometry,” Opt. Lett. 26, 905–907 (2001).
    [CrossRef]
  15. A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
    [CrossRef]
  16. Y. Baek, R. Schiek, G. I. Stegeman, I. Baumann, and W. Sohler, “Interactions between one-dimensional quadratic solitons,” Opt. Lett. 22, 1550–1552 (1997).
    [CrossRef]
  17. R. Schiek, Y. Baek, G. I. Stegeman, I. Baumann, and W. Sohler, “Interactions between one-dimensional quadratic soliton-like beams,” Opt. Quantum Electron. 30, 861–879 (1998).
    [CrossRef]
  18. G. Leo and G. Assanto, “Collisional interactions of vectorial spatial solitary waves in type II frequency-doubling crystal,” J. Opt. Soc. Am. B 14, 3151–3161 (1997).
    [CrossRef]
  19. A. V. Buryak, Yu. S. Kivshar, and S. Trillo, “Parametric spatial solitary waves due to type-II second harmonic generation,” J. Opt. Soc. Am. B 14, 3110–3118 (1997).
    [CrossRef]
  20. P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
    [CrossRef]
  21. C. Conti, S. Trillo, P. Di Trapani, J. Kilius, A. Bramati, S. Minardi, W. Chinaglia, and G. Valiulis, “Effective lensing effects in parametric frequency conversion,” J. Opt. Soc. Am. B 19, 852–859 (2002).
    [CrossRef]

2002

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

C. Conti, S. Trillo, P. Di Trapani, J. Kilius, A. Bramati, S. Minardi, W. Chinaglia, and G. Valiulis, “Effective lensing effects in parametric frequency conversion,” J. Opt. Soc. Am. B 19, 852–859 (2002).
[CrossRef]

2001

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

V. Couderc, E. Lopez Lago, C. Simos, and A. Barthelemy, “Experiments in quadratic soliton generation and steering in a noncollinear geometry,” Opt. Lett. 26, 905–907 (2001).
[CrossRef]

2000

X. Liu, K. Beckwitt, and F. W. Wise, “Noncollinear generation of optical spatio-temporal solitons and application to ultra-fast digital logic,” Phys. Rev. E 61, R4722–R4725 (2000).
[CrossRef]

1999

M. R. Belié, A. Stepken, and F. Kaiser, “Spiraling behavior of photorefractive screening solitons,” Phys. Rev. Lett. 82, 544–547 (1999).
[CrossRef]

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]

A. V. Buryak, Yu. S. Kivshar, M.-f. Shih, and M. Segev, “Induced coherence and stable soliton spiraling,” Phys. Rev. Lett. 82, 81–84 (1999).
[CrossRef]

1998

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]

M. Segev and G. I. Stegeman, “Self-trapping of optical beams: spatial solitons,” Phys. Today 51(8), 42–48 (1998).
[CrossRef]

B. Costantini, 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]

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

1997

1995

W. Torruellas, Z. Wang, D. J. Hagan, E. W. 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]

1992

M. Segev, B. Crosignani, A. Yariv, and B. Fisher, “Spatial soliton on photorepractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
[CrossRef] [PubMed]

1981

K. A. Gorshkov and L. A. Ostrovsky, “Interactions of solitons in nonintegrable systems: direct perturbation method and applications,” Physica D 3, 428–438 (1981).
[CrossRef]

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487–502 (1981).
[CrossRef]

Assanto, G.

Baek, Y.

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

Y. Baek, R. Schiek, G. I. Stegeman, I. Baumann, and W. Sohler, “Interactions between one-dimensional quadratic solitons,” Opt. Lett. 22, 1550–1552 (1997).
[CrossRef]

Barthelemy, A.

Baumann, I.

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

Y. Baek, R. Schiek, G. I. Stegeman, I. Baumann, and W. Sohler, “Interactions between one-dimensional quadratic solitons,” Opt. Lett. 22, 1550–1552 (1997).
[CrossRef]

Beckwitt, K.

X. Liu, K. Beckwitt, and F. W. Wise, “Noncollinear generation of optical spatio-temporal solitons and application to ultra-fast digital logic,” Phys. Rev. E 61, R4722–R4725 (2000).
[CrossRef]

Belié, M. R.

M. R. Belié, A. Stepken, and F. Kaiser, “Spiraling behavior of photorefractive screening solitons,” Phys. Rev. Lett. 82, 544–547 (1999).
[CrossRef]

Bourliaguet, B.

Bramati, A.

C. Conti, S. Trillo, P. Di Trapani, J. Kilius, A. Bramati, S. Minardi, W. Chinaglia, and G. Valiulis, “Effective lensing effects in parametric frequency conversion,” J. Opt. Soc. Am. B 19, 852–859 (2002).
[CrossRef]

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Buryak, A. V.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

A. V. Buryak, Yu. S. Kivshar, M.-f. Shih, and M. Segev, “Induced coherence and stable soliton spiraling,” Phys. Rev. Lett. 82, 81–84 (1999).
[CrossRef]

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 S. Trillo, “Parametric spatial solitary waves due to type-II second harmonic generation,” J. Opt. Soc. Am. B 14, 3110–3118 (1997).
[CrossRef]

Buryak, V.

Chinaglia, W.

C. Conti, S. Trillo, P. Di Trapani, J. Kilius, A. Bramati, S. Minardi, W. Chinaglia, and G. Valiulis, “Effective lensing effects in parametric frequency conversion,” J. Opt. Soc. Am. B 19, 852–859 (2002).
[CrossRef]

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Conti, C.

C. Conti, S. Trillo, P. Di Trapani, J. Kilius, A. Bramati, S. Minardi, W. Chinaglia, and G. Valiulis, “Effective lensing effects in parametric frequency conversion,” J. Opt. Soc. Am. B 19, 852–859 (2002).
[CrossRef]

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Costantini, B.

Couderc, V.

Crosignani, B.

M. Segev, B. Crosignani, A. Yariv, and B. Fisher, “Spatial soliton on photorepractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
[CrossRef] [PubMed]

De Angelis, C.

Di Trapani, P.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

C. Conti, S. Trillo, P. Di Trapani, J. Kilius, A. Bramati, S. Minardi, W. Chinaglia, and G. Valiulis, “Effective lensing effects in parametric frequency conversion,” J. Opt. Soc. Am. B 19, 852–859 (2002).
[CrossRef]

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Fisher, B.

M. Segev, B. Crosignani, A. Yariv, and B. Fisher, “Spatial soliton on photorepractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
[CrossRef] [PubMed]

Gorshkov, K. A.

K. A. Gorshkov and L. A. Ostrovsky, “Interactions of solitons in nonintegrable systems: direct perturbation method and applications,” Physica D 3, 428–438 (1981).
[CrossRef]

Hagan, D. J.

W. Torruellas, Z. Wang, D. J. Hagan, E. W. 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]

Kaiser, F.

M. R. Belié, A. Stepken, and F. Kaiser, “Spiraling behavior of photorefractive screening solitons,” Phys. Rev. Lett. 82, 544–547 (1999).
[CrossRef]

Karpman, V. I.

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487–502 (1981).
[CrossRef]

Kermene, V.

Kilius, J.

C. Conti, S. Trillo, P. Di Trapani, J. Kilius, A. Bramati, S. Minardi, W. Chinaglia, and G. Valiulis, “Effective lensing effects in parametric frequency conversion,” J. Opt. Soc. Am. B 19, 852–859 (2002).
[CrossRef]

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Kivshar, Yu. S.

Leo, G.

Liu, X.

X. Liu, K. Beckwitt, and F. W. Wise, “Noncollinear generation of optical spatio-temporal solitons and application to ultra-fast digital logic,” Phys. Rev. E 61, R4722–R4725 (2000).
[CrossRef]

Lopez Lago, E.

Menyuk, C. R.

W. Torruellas, Z. Wang, D. J. Hagan, E. W. 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]

Minardi, S.

C. Conti, S. Trillo, P. Di Trapani, J. Kilius, A. Bramati, S. Minardi, W. Chinaglia, and G. Valiulis, “Effective lensing effects in parametric frequency conversion,” J. Opt. Soc. Am. B 19, 852–859 (2002).
[CrossRef]

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Ostrovsky, L. A.

K. A. Gorshkov and L. A. Ostrovsky, “Interactions of solitons in nonintegrable systems: direct perturbation method and applications,” Physica D 3, 428–438 (1981).
[CrossRef]

Schiek, R.

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

Y. Baek, R. Schiek, G. I. Stegeman, I. Baumann, and W. Sohler, “Interactions between one-dimensional quadratic solitons,” Opt. Lett. 22, 1550–1552 (1997).
[CrossRef]

Segev, M.

A. V. Buryak, Yu. S. Kivshar, M.-f. Shih, and M. Segev, “Induced coherence and stable soliton spiraling,” Phys. Rev. Lett. 82, 81–84 (1999).
[CrossRef]

M. Segev and G. I. Stegeman, “Self-trapping of optical beams: spatial solitons,” Phys. Today 51(8), 42–48 (1998).
[CrossRef]

M.-F. Shih and M. Segev, “Three dimensional spiraling of interacting spatial solitons,” Phys. Rev. Lett. 78, 13–16 (1997).
[CrossRef]

M. Segev, B. Crosignani, A. Yariv, and B. Fisher, “Spatial soliton on photorepractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
[CrossRef] [PubMed]

Shih, M.-f.

A. V. Buryak, Yu. S. Kivshar, M.-f. Shih, and M. Segev, “Induced coherence and stable soliton spiraling,” Phys. Rev. Lett. 82, 81–84 (1999).
[CrossRef]

M.-F. Shih and M. Segev, “Three dimensional spiraling of interacting spatial solitons,” Phys. Rev. Lett. 78, 13–16 (1997).
[CrossRef]

Simos, C.

Skryabin, D. V.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

Sohler, W.

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

Y. Baek, R. Schiek, G. I. Stegeman, I. Baumann, and W. Sohler, “Interactions between one-dimensional quadratic solitons,” Opt. Lett. 22, 1550–1552 (1997).
[CrossRef]

Solov’ev, V. V.

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487–502 (1981).
[CrossRef]

Steblina, V. V.

Stegeman, G. I.

M. Segev and G. I. Stegeman, “Self-trapping of optical beams: spatial solitons,” Phys. Today 51(8), 42–48 (1998).
[CrossRef]

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

Y. Baek, R. Schiek, G. I. Stegeman, I. Baumann, and W. Sohler, “Interactions between one-dimensional quadratic solitons,” Opt. Lett. 22, 1550–1552 (1997).
[CrossRef]

W. Torruellas, Z. Wang, D. J. Hagan, E. W. 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]

Stepken, A.

M. R. Belié, A. Stepken, and F. Kaiser, “Spiraling behavior of photorefractive screening solitons,” Phys. Rev. Lett. 82, 544–547 (1999).
[CrossRef]

Torner, L.

W. Torruellas, Z. Wang, D. J. Hagan, E. W. 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]

Torruellas, W.

W. Torruellas, Z. Wang, D. J. Hagan, E. W. 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]

Trillo, S.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

C. Conti, S. Trillo, P. Di Trapani, J. Kilius, A. Bramati, S. Minardi, W. Chinaglia, and G. Valiulis, “Effective lensing effects in parametric frequency conversion,” J. Opt. Soc. Am. B 19, 852–859 (2002).
[CrossRef]

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

A. V. Buryak, Yu. S. Kivshar, and S. Trillo, “Parametric spatial solitary waves due to type-II second harmonic generation,” J. Opt. Soc. Am. B 14, 3110–3118 (1997).
[CrossRef]

Valiulis, G.

C. Conti, S. Trillo, P. Di Trapani, J. Kilius, A. Bramati, S. Minardi, W. Chinaglia, and G. Valiulis, “Effective lensing effects in parametric frequency conversion,” J. Opt. Soc. Am. B 19, 852–859 (2002).
[CrossRef]

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

Van Stryland, E. W.

W. Torruellas, Z. Wang, D. J. Hagan, E. W. 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]

Wang, Z.

W. Torruellas, Z. Wang, D. J. Hagan, E. W. 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]

Wise, F. W.

X. Liu, K. Beckwitt, and F. W. Wise, “Noncollinear generation of optical spatio-temporal solitons and application to ultra-fast digital logic,” Phys. Rev. E 61, R4722–R4725 (2000).
[CrossRef]

Yariv, A.

M. Segev, B. Crosignani, A. Yariv, and B. Fisher, “Spatial soliton on photorepractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B

Opt. Lett.

Opt. Quantum Electron.

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

Phys. Rep.

A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, “Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications,” Phys. Rep. 370, 63–235 (2002).
[CrossRef]

Phys. Rev. E

X. Liu, K. Beckwitt, and F. W. Wise, “Noncollinear generation of optical spatio-temporal solitons and application to ultra-fast digital logic,” Phys. Rev. E 61, R4722–R4725 (2000).
[CrossRef]

Phys. Rev. Lett.

M. R. Belié, A. Stepken, and F. Kaiser, “Spiraling behavior of photorefractive screening solitons,” Phys. Rev. Lett. 82, 544–547 (1999).
[CrossRef]

P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001).
[CrossRef]

M. Segev, B. Crosignani, A. Yariv, and B. Fisher, “Spatial soliton on photorepractive media,” Phys. Rev. Lett. 68, 923–926 (1992).
[CrossRef] [PubMed]

A. V. Buryak, Yu. S. Kivshar, M.-f. Shih, and M. Segev, “Induced coherence and stable soliton spiraling,” Phys. Rev. Lett. 82, 81–84 (1999).
[CrossRef]

W. Torruellas, Z. Wang, D. J. Hagan, E. W. 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]

M.-F. Shih and M. Segev, “Three dimensional spiraling of interacting spatial solitons,” Phys. Rev. Lett. 78, 13–16 (1997).
[CrossRef]

Phys. Today

M. Segev and G. I. Stegeman, “Self-trapping of optical beams: spatial solitons,” Phys. Today 51(8), 42–48 (1998).
[CrossRef]

Physica D

K. A. Gorshkov and L. A. Ostrovsky, “Interactions of solitons in nonintegrable systems: direct perturbation method and applications,” Physica D 3, 428–438 (1981).
[CrossRef]

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487–502 (1981).
[CrossRef]

Other

M. Toda, Nonlinear Waves and Solitons (Kluwer Academic, Boston, Mass., 1990).

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Figures (8)

Fig. 1
Fig. 1

Sequences of soliton positions in the (x, y) plane at equally spaced propagation distances in z, which demonstrate phase sensitivity of three-wave soliton interactions: (a) Interaction of two identical imbalanced solitons (|v(1)| = |v(2)|, |u(1)| = |u(2)|, and |w(1)|=|w(2)|) at φ=π, ϑ=π; (b) interaction of two mirror imbalanced solitons (|v(1)| = |u(2)|, |u(1)| = |v(2)|, and |w(1)| = |w(2)|) at φ=π, ϑ=π; (c) the same as (a), but with φ=0, ϑ=π; (d) the same as (b), but with φ=0, ϑ=π. For all four interactions, other soliton parameters are as follows: βv=0.3, βu=1.0, Δ=-1.0, c0=0.1, and s=11.0. Only the second harmonics’ contours are shown.

Fig. 2
Fig. 2

Experimental setup. Inset: relative orientation of the physical and crystallographic axes.

Fig. 3
Fig. 3

Illustration (based on experimental recordings) of the nonlinear interaction between two quadratic solitons in the KTP crystal in the near and far fields: (a) input conditions without interaction; (b) repulsive interaction (scattering), and (c) attractive interaction (fusion). Images correspond to the combination of three recordings: one with “M” alone, one with “P” alone, and one with “M” and “P” simultaneously launched into the crystal. The transition from (b) to (c) is effected by a simple control of the relative phase ξ of the input beams. White spots on far-field recordings represent the initial conditions without interaction.

Fig. 4
Fig. 4

Illustration (based on experimental recordings) of the nonlinear interaction between two quadratic solitons in the KTP crystal in the near and far fields: (a) input conditions without interaction, (b) repulsive interaction (scattering), and (c) attractive interaction (spiraling). Images correspond to the combination of three recordings: one with “M” alone, one with “P” alone, and one with “M” and “P” simultaneously launched into the crystal. The transition from (b) to (c) is effected by a simple control of the relative phase ξ of the input beams. White spots on far-field recordings represent the initial conditions without interaction.

Fig. 5
Fig. 5

Input and output positions of the beams with and without interaction in the case of a repulsive interaction. Initial conditions: IP=9.8 GW/cm2, IP/IM=1.3, ΔkL0, and imbalance δ=0.53.

Fig. 6
Fig. 6

Evolution of the relative distance between the output beams versus total input intensity for the interaction illustrated in Fig. 5. Increasing the total intensity reduces the amplitude of the repulsion.

Fig. 7
Fig. 7

Influence of the polarization imbalance in each input beam on the soliton collision of Fig. 5: (a) The nature and the strength of the collision change with respect to the imbalance. (b) The influence of the imbalance is hidden by the lateral shift due to the walk-off.

Fig. 8
Fig. 8

Influence of the phase mismatch on the interaction sketched in Fig. 5. Deflection angles decrease for large positive values of the phase mismatch.

Equations (24)

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i vz+2vx2+2vy2+wu*=0,
i uz+2ux2+2uy2+wv*=0,
2i wz+2wx2+2wy2-2Δw+uv=0,
i vz+2vx2+2vy2+wv*=0,
2i wz+2wx2+2wy2-2Δw+v22=0,
Qv=(|v|2+2|w|2)dxdy,
Qu=(|u|2+2|w|2)dxdy.
φ¨(1)Qv(1)βv(1)+ϑ¨(1)Qu(1)βv(1)+x¨(1)Px(1)βv(1)+y¨(1)Py(1)βv(1)
-12Uφ(1)=0,
φ¨(1)Qv(1)βu(1)+ϑ¨(1)Qu(1)βu(1)+x¨(1)Px(1)βu(1)+y¨(1)Py(1)βu(1)
-12Uϑ(1)=0,
φ¨(1)Qv(1)Cx(1)+ϑ¨(1)Qu(1)Cx(1)+x¨(1)Px(1)Cx(1)+y¨(1)Py(1)Cx(1)
-12Ux(1)=0,
φ¨(1)Qv(1)Cy(1)+ϑ¨(1)Qu(1)Cy(1)+x¨(1)Px(1)Cy(1)+y¨(1)Py(1)Cy(1)
-12Uy(1)=0.
-φ¨ Qvβv-ϑ¨ Quβv+Ueffφ=0,
-φ¨ Qvβu-ϑ¨ Quβu+Ueffϑ=0,
R¨ (Qv+Qu)2+UeffR=0,
UeffU+s2c02(Qv+Qu)4R2
Mψψ¨+Ueffψ=0,
R¨ (Qv+Qu)2+UeffR=0,
Mψ=QvβvQuβu-QuβvQvβu/
Qvβv+Quβu-Quβv-Qvβu
UeffUwcos(ψ)+s2c02(Qv+Qu)4R2,

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