Abstract

The dynamical equations that govern the interaction between a weakly modulated plane wave and its second harmonic are derived for materials with asymmetric crystal structure, in which the effects of both the quadratic and the cubic nonlinear susceptibility tensors must be considered. Unlike in the case of pure quadratic nonlinearity, the equations for wave packets that describe temporal and spatial solitary waves do not have the same form unless the crystal structure of the material is nearly centrosymmetric, such that the lowest-order quadratic and cubic nonlinear terms balance. For lossless materials and nonresonant conditions the Hamiltonian structure of the equations is discussed, conserved quantities described, and a stable one-parameter family of bright ground-state solitary wave solutions found numerically for fixed material parameters.

© 1997 Optical Society of America

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  1. A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, 1992), Chap. 2.
  2. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118 (1971) [Sov. Phys. JETP 34, 62 (1972)].
  3. J. J. Rasmussen and K. Rypdal, “Blow-up in nonlinear Schrödinger equations—I, A general review,” Phys. Scr. 33, 481 (1986).
    [Crossref]
  4. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  5. C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2):χ(2) cascading,” J. Opt. Soc. Am. B 11, 2434 (1994).
    [Crossref]
  6. L. A. Ostrovskii, “Self-action of light in crystals,” Pis’ma Zh. Eksp. Teor. Fiz. 5, 331 (1967) [JETP Lett. 5, 272 (1967)].
  7. Yu. N. Karamzin and A. P. Sukhorukov, “Nonlinear interaction of diffracted light beams in a medium with quadratic nonlinearity: mutual focusing of beams and limitation on the efficiency of optical frequency converters,” JETP Lett. 20, 339 (1974).
  8. A. A. Kanashov and A. M. Rubenshik, “On diffraction and dispersion effect on three wave interaction,” Physica D 4, 122 (1981).
    [Crossref]
  9. A. V. Buryak and Y. S. Kivshar, “Spatial optical solitons governed by quadratic nonlinearity,” Opt. Lett. 19, 1612 (1994); ‘Solitons due to second harmonic generation,” Phys. Lett. A 197, 407 (1995).
    [Crossref] [PubMed]
  10. A. V. Buryak, Y. S. Kivshar, and V. V. Steblina, “Self-trapping of light beams and parametric solitons in diffractive quadratic materials,” Phys. Rev. A 52, 1670 (1995).
    [Crossref] [PubMed]
  11. L. Torner, C. R. Menyuk, W. E. Torruellas, and G. I. Stegeman, “Two-dimensional solitons with second-order nonlinearities,” Opt. Lett. 20, 13 (1995).
    [Crossref] [PubMed]
  12. L. Bergé, V. K. Mezentsev, J. J. Rasmussen, and J. Wyller, “Formation of stable solitons in quadratic nonlinear media,” Phys. Rev. A 52, R28 (1995).
    [Crossref] [PubMed]
  13. D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, “Instability of solitons governed by quadratic nonlinearities,” Phys. Rev. Lett. 75, 591 (1995).
    [Crossref] [PubMed]
  14. N. R. Belashenkov, S. V. Gagarskii, and M. V. Inochkin, “Nonlinear refraction of light on second-harmonic generation,” Opt. Spectrosc. (USSR) 66, 806 (1989) [Opt. Spektrosk. 66, 1383 (1989)]
  15. R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28 (1992).
    [Crossref] [PubMed]
  16. M. L. Sundheimer, Ch. Bosshard, E. W. Van Stryland, G. I. Stegeman, and J. D. Bierlein, “Large nonlinear phase modulation in quasi-phase-matched KTP waveguides as a result of cascaded second-order processes,” Opt. Lett. 18, 13 (1993).
    [Crossref]
  17. W. E. 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 (1995).
    [Crossref] [PubMed]
  18. X. Baek, R. Schiek, and G. I. Stegeman, “Nonlinear guided waves and their application,” in Nonlinear Guided Wave Phenomena, Vol. 6 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 24.
  19. L. Bergé, O. Bang, J. J. Rasmussen, and V. K. Mezentsev, “Self-focusing and soliton-like structures in materials with competing quadratic and cubic nonlinearities,” Phys. Rev. E (to be published).
  20. A. G. Kalocsai and J. W. Haus, “Self-modulation effects in quadratically nonlinear materials,” Opt. Commun. 97, 239 (1993).
    [Crossref]
  21. Q. Guo, “Non-linear Schrödinger solitons in media with non-zero second-order nonlinear susceptibility,” Quantum Opt. 5, 133 (1993).
    [Crossref]
  22. L. S. Telegin and A. S. Chirkin, “Interaction in frequency doubling of ultrashort laser pulses,” Kvant. Elektron. (Moscow) 9, 2086 (1982) [Sov. J. Quantum Electron. 12, 1354 (1982)].
    [Crossref]
  23. S. Trillo and S. Wabnitz, “Nonlinear parametric mixing instabilities induced by self-phase and cross-phase modulation,” Opt. Lett. 17, 1572 (1992).
    [Crossref] [PubMed]
  24. S. Trillo, A. V. Buryak, and Y. S. Kivshar, “Modulational instabilities and optical solitons due to competition of χ(2) and χ(3) nonlinearities,” Opt. Commun. 122, 200 (1996).
    [Crossref]
  25. A. V. Buryak, Y. S. Kivshar, and S. Trillo, “Optical solitons supported by competing nonlinearities,” Opt. Lett. 20, 1961 (1995).
    [Crossref] [PubMed]
  26. M. A. Karpierz, “Coupled solitons in waveguides with second- and third-order nonlinearities,” Opt. Lett. 20, 1677 (1995).
    [Crossref] [PubMed]
  27. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).
  28. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1980), Chap. 12.
  29. M. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Radiophys. Quantum Electron. 16, 783 (1975).
    [Crossref]
  30. M. Grillakis, J. Shatah, and W. Strauss, “Stability theory of solitary waves in the presence of symmetry, I,” J. Funct. Anal. 74, 160 (1987).
    [Crossref]
  31. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (fortran) (Cambridge U. Press, Cambridge, 1989).

1996 (1)

S. Trillo, A. V. Buryak, and Y. S. Kivshar, “Modulational instabilities and optical solitons due to competition of χ(2) and χ(3) nonlinearities,” Opt. Commun. 122, 200 (1996).
[Crossref]

1995 (7)

A. V. Buryak, Y. S. Kivshar, and S. Trillo, “Optical solitons supported by competing nonlinearities,” Opt. Lett. 20, 1961 (1995).
[Crossref] [PubMed]

M. A. Karpierz, “Coupled solitons in waveguides with second- and third-order nonlinearities,” Opt. Lett. 20, 1677 (1995).
[Crossref] [PubMed]

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

L. Torner, C. R. Menyuk, W. E. Torruellas, and G. I. Stegeman, “Two-dimensional solitons with second-order nonlinearities,” Opt. Lett. 20, 13 (1995).
[Crossref] [PubMed]

L. Bergé, V. K. Mezentsev, J. J. Rasmussen, and J. Wyller, “Formation of stable solitons in quadratic nonlinear media,” Phys. Rev. A 52, R28 (1995).
[Crossref] [PubMed]

D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, “Instability of solitons governed by quadratic nonlinearities,” Phys. Rev. Lett. 75, 591 (1995).
[Crossref] [PubMed]

W. E. 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 (1995).
[Crossref] [PubMed]

1994 (2)

1993 (3)

A. G. Kalocsai and J. W. Haus, “Self-modulation effects in quadratically nonlinear materials,” Opt. Commun. 97, 239 (1993).
[Crossref]

Q. Guo, “Non-linear Schrödinger solitons in media with non-zero second-order nonlinear susceptibility,” Quantum Opt. 5, 133 (1993).
[Crossref]

M. L. Sundheimer, Ch. Bosshard, E. W. Van Stryland, G. I. Stegeman, and J. D. Bierlein, “Large nonlinear phase modulation in quasi-phase-matched KTP waveguides as a result of cascaded second-order processes,” Opt. Lett. 18, 13 (1993).
[Crossref]

1992 (2)

1989 (1)

N. R. Belashenkov, S. V. Gagarskii, and M. V. Inochkin, “Nonlinear refraction of light on second-harmonic generation,” Opt. Spectrosc. (USSR) 66, 806 (1989) [Opt. Spektrosk. 66, 1383 (1989)]

1987 (1)

M. Grillakis, J. Shatah, and W. Strauss, “Stability theory of solitary waves in the presence of symmetry, I,” J. Funct. Anal. 74, 160 (1987).
[Crossref]

1986 (1)

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

1982 (1)

L. S. Telegin and A. S. Chirkin, “Interaction in frequency doubling of ultrashort laser pulses,” Kvant. Elektron. (Moscow) 9, 2086 (1982) [Sov. J. Quantum Electron. 12, 1354 (1982)].
[Crossref]

1981 (1)

A. A. Kanashov and A. M. Rubenshik, “On diffraction and dispersion effect on three wave interaction,” Physica D 4, 122 (1981).
[Crossref]

1975 (1)

M. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Radiophys. Quantum Electron. 16, 783 (1975).
[Crossref]

1974 (1)

Yu. N. Karamzin and A. P. Sukhorukov, “Nonlinear interaction of diffracted light beams in a medium with quadratic nonlinearity: mutual focusing of beams and limitation on the efficiency of optical frequency converters,” JETP Lett. 20, 339 (1974).

1971 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118 (1971) [Sov. Phys. JETP 34, 62 (1972)].

1967 (1)

L. A. Ostrovskii, “Self-action of light in crystals,” Pis’ma Zh. Eksp. Teor. Fiz. 5, 331 (1967) [JETP Lett. 5, 272 (1967)].

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).

Baek, X.

X. Baek, R. Schiek, and G. I. Stegeman, “Nonlinear guided waves and their application,” in Nonlinear Guided Wave Phenomena, Vol. 6 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 24.

Bang, O.

L. Bergé, O. Bang, J. J. Rasmussen, and V. K. Mezentsev, “Self-focusing and soliton-like structures in materials with competing quadratic and cubic nonlinearities,” Phys. Rev. E (to be published).

Belashenkov, N. R.

N. R. Belashenkov, S. V. Gagarskii, and M. V. Inochkin, “Nonlinear refraction of light on second-harmonic generation,” Opt. Spectrosc. (USSR) 66, 806 (1989) [Opt. Spektrosk. 66, 1383 (1989)]

Bergé, L.

L. Bergé, V. K. Mezentsev, J. J. Rasmussen, and J. Wyller, “Formation of stable solitons in quadratic nonlinear media,” Phys. Rev. A 52, R28 (1995).
[Crossref] [PubMed]

L. Bergé, O. Bang, J. J. Rasmussen, and V. K. Mezentsev, “Self-focusing and soliton-like structures in materials with competing quadratic and cubic nonlinearities,” Phys. Rev. E (to be published).

Bierlein, J. D.

Bosshard, Ch.

Buryak, A. V.

S. Trillo, A. V. Buryak, and Y. S. Kivshar, “Modulational instabilities and optical solitons due to competition of χ(2) and χ(3) nonlinearities,” Opt. Commun. 122, 200 (1996).
[Crossref]

D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, “Instability of solitons governed by quadratic nonlinearities,” Phys. Rev. Lett. 75, 591 (1995).
[Crossref] [PubMed]

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

A. V. Buryak, Y. S. Kivshar, and S. Trillo, “Optical solitons supported by competing nonlinearities,” Opt. Lett. 20, 1961 (1995).
[Crossref] [PubMed]

A. V. Buryak and Y. S. Kivshar, “Spatial optical solitons governed by quadratic nonlinearity,” Opt. Lett. 19, 1612 (1994); ‘Solitons due to second harmonic generation,” Phys. Lett. A 197, 407 (1995).
[Crossref] [PubMed]

Chirkin, A. S.

L. S. Telegin and A. S. Chirkin, “Interaction in frequency doubling of ultrashort laser pulses,” Kvant. Elektron. (Moscow) 9, 2086 (1982) [Sov. J. Quantum Electron. 12, 1354 (1982)].
[Crossref]

DeSalvo, R.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (fortran) (Cambridge U. Press, Cambridge, 1989).

Gagarskii, S. V.

N. R. Belashenkov, S. V. Gagarskii, and M. V. Inochkin, “Nonlinear refraction of light on second-harmonic generation,” Opt. Spectrosc. (USSR) 66, 806 (1989) [Opt. Spektrosk. 66, 1383 (1989)]

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1980), Chap. 12.

Grillakis, M.

M. Grillakis, J. Shatah, and W. Strauss, “Stability theory of solitary waves in the presence of symmetry, I,” J. Funct. Anal. 74, 160 (1987).
[Crossref]

Guo, Q.

Q. Guo, “Non-linear Schrödinger solitons in media with non-zero second-order nonlinear susceptibility,” Quantum Opt. 5, 133 (1993).
[Crossref]

Hagan, D. J.

W. E. 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 (1995).
[Crossref] [PubMed]

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28 (1992).
[Crossref] [PubMed]

Haus, J. W.

A. G. Kalocsai and J. W. Haus, “Self-modulation effects in quadratically nonlinear materials,” Opt. Commun. 97, 239 (1993).
[Crossref]

Inochkin, M. V.

N. R. Belashenkov, S. V. Gagarskii, and M. V. Inochkin, “Nonlinear refraction of light on second-harmonic generation,” Opt. Spectrosc. (USSR) 66, 806 (1989) [Opt. Spektrosk. 66, 1383 (1989)]

Kalocsai, A. G.

A. G. Kalocsai and J. W. Haus, “Self-modulation effects in quadratically nonlinear materials,” Opt. Commun. 97, 239 (1993).
[Crossref]

Kanashov, A. A.

A. A. Kanashov and A. M. Rubenshik, “On diffraction and dispersion effect on three wave interaction,” Physica D 4, 122 (1981).
[Crossref]

Karamzin, Yu. N.

Yu. N. Karamzin and A. P. Sukhorukov, “Nonlinear interaction of diffracted light beams in a medium with quadratic nonlinearity: mutual focusing of beams and limitation on the efficiency of optical frequency converters,” JETP Lett. 20, 339 (1974).

Karpierz, M. A.

Kivshar, Y. S.

S. Trillo, A. V. Buryak, and Y. S. Kivshar, “Modulational instabilities and optical solitons due to competition of χ(2) and χ(3) nonlinearities,” Opt. Commun. 122, 200 (1996).
[Crossref]

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

D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, “Instability of solitons governed by quadratic nonlinearities,” Phys. Rev. Lett. 75, 591 (1995).
[Crossref] [PubMed]

A. V. Buryak, Y. S. Kivshar, and S. Trillo, “Optical solitons supported by competing nonlinearities,” Opt. Lett. 20, 1961 (1995).
[Crossref] [PubMed]

A. V. Buryak and Y. S. Kivshar, “Spatial optical solitons governed by quadratic nonlinearity,” Opt. Lett. 19, 1612 (1994); ‘Solitons due to second harmonic generation,” Phys. Lett. A 197, 407 (1995).
[Crossref] [PubMed]

Kolokolov, A. A.

M. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Radiophys. Quantum Electron. 16, 783 (1975).
[Crossref]

Menyuk, C. R.

W. E. 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 (1995).
[Crossref] [PubMed]

L. Torner, C. R. Menyuk, W. E. Torruellas, and G. I. Stegeman, “Two-dimensional solitons with second-order nonlinearities,” Opt. Lett. 20, 13 (1995).
[Crossref] [PubMed]

C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2):χ(2) cascading,” J. Opt. Soc. Am. B 11, 2434 (1994).
[Crossref]

Mezentsev, V. K.

L. Bergé, V. K. Mezentsev, J. J. Rasmussen, and J. Wyller, “Formation of stable solitons in quadratic nonlinear media,” Phys. Rev. A 52, R28 (1995).
[Crossref] [PubMed]

L. Bergé, O. Bang, J. J. Rasmussen, and V. K. Mezentsev, “Self-focusing and soliton-like structures in materials with competing quadratic and cubic nonlinearities,” Phys. Rev. E (to be published).

Moloney, J. V.

A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, 1992), Chap. 2.

Newell, A. C.

A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, 1992), Chap. 2.

Ostrovskii, L. A.

L. A. Ostrovskii, “Self-action of light in crystals,” Pis’ma Zh. Eksp. Teor. Fiz. 5, 331 (1967) [JETP Lett. 5, 272 (1967)].

Pelinovsky, D. E.

D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, “Instability of solitons governed by quadratic nonlinearities,” Phys. Rev. Lett. 75, 591 (1995).
[Crossref] [PubMed]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (fortran) (Cambridge U. Press, Cambridge, 1989).

Rasmussen, J. J.

L. Bergé, V. K. Mezentsev, J. J. Rasmussen, and J. Wyller, “Formation of stable solitons in quadratic nonlinear media,” Phys. Rev. A 52, R28 (1995).
[Crossref] [PubMed]

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

L. Bergé, O. Bang, J. J. Rasmussen, and V. K. Mezentsev, “Self-focusing and soliton-like structures in materials with competing quadratic and cubic nonlinearities,” Phys. Rev. E (to be published).

Rubenshik, A. M.

A. A. Kanashov and A. M. Rubenshik, “On diffraction and dispersion effect on three wave interaction,” Physica D 4, 122 (1981).
[Crossref]

Rypdal, K.

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

Schiek, R.

C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2):χ(2) cascading,” J. Opt. Soc. Am. B 11, 2434 (1994).
[Crossref]

X. Baek, R. Schiek, and G. I. Stegeman, “Nonlinear guided waves and their application,” in Nonlinear Guided Wave Phenomena, Vol. 6 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 24.

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118 (1971) [Sov. Phys. JETP 34, 62 (1972)].

Shatah, J.

M. Grillakis, J. Shatah, and W. Strauss, “Stability theory of solitary waves in the presence of symmetry, I,” J. Funct. Anal. 74, 160 (1987).
[Crossref]

Sheik-Bahae, M.

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

Steblina, V. V.

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

Stegeman, G.

Stegeman, G. I.

W. E. 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 (1995).
[Crossref] [PubMed]

L. Torner, C. R. Menyuk, W. E. Torruellas, and G. I. Stegeman, “Two-dimensional solitons with second-order nonlinearities,” Opt. Lett. 20, 13 (1995).
[Crossref] [PubMed]

M. L. Sundheimer, Ch. Bosshard, E. W. Van Stryland, G. I. Stegeman, and J. D. Bierlein, “Large nonlinear phase modulation in quasi-phase-matched KTP waveguides as a result of cascaded second-order processes,” Opt. Lett. 18, 13 (1993).
[Crossref]

X. Baek, R. Schiek, and G. I. Stegeman, “Nonlinear guided waves and their application,” in Nonlinear Guided Wave Phenomena, Vol. 6 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 24.

Strauss, W.

M. Grillakis, J. Shatah, and W. Strauss, “Stability theory of solitary waves in the presence of symmetry, I,” J. Funct. Anal. 74, 160 (1987).
[Crossref]

Sukhorukov, A. P.

Yu. N. Karamzin and A. P. Sukhorukov, “Nonlinear interaction of diffracted light beams in a medium with quadratic nonlinearity: mutual focusing of beams and limitation on the efficiency of optical frequency converters,” JETP Lett. 20, 339 (1974).

Sundheimer, M. L.

Telegin, L. S.

L. S. Telegin and A. S. Chirkin, “Interaction in frequency doubling of ultrashort laser pulses,” Kvant. Elektron. (Moscow) 9, 2086 (1982) [Sov. J. Quantum Electron. 12, 1354 (1982)].
[Crossref]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (fortran) (Cambridge U. Press, Cambridge, 1989).

Torner, L.

L. Torner, C. R. Menyuk, W. E. Torruellas, and G. I. Stegeman, “Two-dimensional solitons with second-order nonlinearities,” Opt. Lett. 20, 13 (1995).
[Crossref] [PubMed]

W. E. 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 (1995).
[Crossref] [PubMed]

C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2):χ(2) cascading,” J. Opt. Soc. Am. B 11, 2434 (1994).
[Crossref]

Torruellas, W. E.

W. E. 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 (1995).
[Crossref] [PubMed]

L. Torner, C. R. Menyuk, W. E. Torruellas, and G. I. Stegeman, “Two-dimensional solitons with second-order nonlinearities,” Opt. Lett. 20, 13 (1995).
[Crossref] [PubMed]

Trillo, S.

Vakhitov, M. G.

M. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Radiophys. Quantum Electron. 16, 783 (1975).
[Crossref]

Van Stryland, E. W.

Vanherzeele, H.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (fortran) (Cambridge U. Press, Cambridge, 1989).

Wabnitz, S.

Wang, Z.

W. E. 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 (1995).
[Crossref] [PubMed]

Wyller, J.

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[Crossref] [PubMed]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118 (1971) [Sov. Phys. JETP 34, 62 (1972)].

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[Crossref]

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

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Kvant. Elektron. (Moscow) (1)

L. S. Telegin and A. S. Chirkin, “Interaction in frequency doubling of ultrashort laser pulses,” Kvant. Elektron. (Moscow) 9, 2086 (1982) [Sov. J. Quantum Electron. 12, 1354 (1982)].
[Crossref]

Opt. Commun. (2)

A. G. Kalocsai and J. W. Haus, “Self-modulation effects in quadratically nonlinear materials,” Opt. Commun. 97, 239 (1993).
[Crossref]

S. Trillo, A. V. Buryak, and Y. S. Kivshar, “Modulational instabilities and optical solitons due to competition of χ(2) and χ(3) nonlinearities,” Opt. Commun. 122, 200 (1996).
[Crossref]

Opt. Lett. (7)

Opt. Spectrosc. (USSR) (1)

N. R. Belashenkov, S. V. Gagarskii, and M. V. Inochkin, “Nonlinear refraction of light on second-harmonic generation,” Opt. Spectrosc. (USSR) 66, 806 (1989) [Opt. Spektrosk. 66, 1383 (1989)]

Phys. Rev. A (2)

L. Bergé, V. K. Mezentsev, J. J. Rasmussen, and J. Wyller, “Formation of stable solitons in quadratic nonlinear media,” Phys. Rev. A 52, R28 (1995).
[Crossref] [PubMed]

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

Phys. Rev. Lett. (2)

D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, “Instability of solitons governed by quadratic nonlinearities,” Phys. Rev. Lett. 75, 591 (1995).
[Crossref] [PubMed]

W. E. 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 (1995).
[Crossref] [PubMed]

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[Crossref]

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[Crossref]

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L. A. Ostrovskii, “Self-action of light in crystals,” Pis’ma Zh. Eksp. Teor. Fiz. 5, 331 (1967) [JETP Lett. 5, 272 (1967)].

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[Crossref]

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[Crossref]

Zh. Eksp. Teor. Fiz. (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118 (1971) [Sov. Phys. JETP 34, 62 (1972)].

Other (7)

A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, 1992), Chap. 2.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

X. Baek, R. Schiek, and G. I. Stegeman, “Nonlinear guided waves and their application,” in Nonlinear Guided Wave Phenomena, Vol. 6 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 24.

L. Bergé, O. Bang, J. J. Rasmussen, and V. K. Mezentsev, “Self-focusing and soliton-like structures in materials with competing quadratic and cubic nonlinearities,” Phys. Rev. E (to be published).

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (fortran) (Cambridge U. Press, Cambridge, 1989).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1980), Chap. 12.

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Figures (3)

Fig. 1
Fig. 1

Ratio of the amplitudes Aw=w(0) and Av=v(0) as functions of the eigenvalue λ for the GS solution to Eqs. (4.7). The parameters are r¯1=2r¯2=-1 and 4η1=η2=1, with α given at each curve.

Fig. 2
Fig. 2

Bifurcation point λbif as function of α. The parameters are r¯1=2r¯2=-1 and 4η1=η2=1.

Fig. 3
Fig. 3

Total power N as function of the eigenvalue λ for the GS solution to Eqs. (4.7). The parameters are r¯1=2r¯2=-1 and 4η1=η2=1, with α given at each curve.

Equations (112)

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izϕ1-r¯12ϕ1+η1|ϕ1|2ϕ1=0,
izϕ1-r¯12ϕ1+ϕ1*ϕ2=0,
izϕ2-r¯22ϕ2-αϕ2+½ϕ12=0,
2E-μ0t2D=(·E),
D(t)0E(t)+-tχ(1)(t-t1)·E(t1)dt1+-tχ(2)(t-t1;t-t2):E(t1)E(t2)dt1dt2+-tχ(3)(t-t1;t-t2;t-t3)E(t1)×E(t2)E(t3)dt1dt2dt3,
E(x, y, z, t)E1(x, y, z, t)exp(iθ1)eˆ1+E2(x, y, z, t)exp(iθ2)eˆ2+c.c.,
D(x, y, z, t)D1(x, y, z, t)exp(iθ1)dˆ1+D2(x, y, z, t)exp(iθ2)dˆ2+c.c.,
(2Ep+i2kpzEp-kp2Ep)-μ0(t2Dp-i2ωptDp
-ωp2Dp)=0
Dp(1)=0[˜pEp+i˜ptEp-½ ˜pt2Ep-i ˜pt3Ep+124˜pt4Ep+],
˜p=1+χ˜p(1)np2+iαpnpc/ωp,
np2=1+Re(χ˜p(1)),αp=ωpnpcIm(χ˜p(1)).
χ˜p(1)=10-[eˆp·χ(1)(t)·eˆp]exp(iωpt)dt.
D1(2)=2[δ˜1E1*E2+iδ˜1aE2tE1*+iδ˜1bE1*tE2-½ δ˜1aaE2t2E1*-½ δ˜1bbE1*t2E2-δ˜1ab(tE1*)(tE2)+]exp(-iΔkz),
D2(2)=[δ˜2E12+iδ˜2atE12-δ˜2aaE1t2E1-δ˜2ab(tE1)2+]exp(iΔkz).
χ˜p(2)(ωa, ωb)=-[eˆp·χ(2)(t, t):eˆ1eˆ3-p]×exp(iωat+iωbt)dtdt.
D1(3)=3(χ˜11(3)|E1|2+2χ˜12(3)|E2|2)E1+,
D2(3)=3(χ˜22(3)|E2|2+2χ˜21(3)|E1|2)E2+,
χ˜pq(3)=-[eˆp·χ(3)(t, t, t)eˆqeˆqeˆp]×exp[iωq(t-t)+iωpt]dtdtdt,
Ep(x, y, z, t)=κAp(x¯, y¯, z¯, t¯),
(x¯, y¯, z¯)=k1(x, y, z),t¯=t/τ0,
τ02=|k1/k1|,κ=k12/(μ0ω12χ2),
χ2=5.7×10-23C/V2,
χ˜p(2)/χ2χ˜pq(3)/χ31-1,
Ap(x¯, y¯, z¯, t¯)=Ap(1)(X, Y, Z, T)+2Ap(2)(X, Y, Z, T)+,
(X, Y)=1/2(x¯, y¯),Z=z¯,T=t¯
i2kpk1(ZAp(1)+βpTAp(1))+k122Ap(1)=0,
Ap(1)(X, Y, Z, T)=Fp(X, Y)Bp(1)(Z, T)
k122Fp+ηp2Fp=0,Fp2dXdY=1,
i2kpk1(ZBp(1)+βpTBp(1))-ηp2Bp(1)=0,
kp2=(npωp/c)2-ηp2,
Bp(1)(Z, T)=Ψp(ζ, τ),ζ=Z,τ=T-β1Z,
i2kpk1(ZAp(2)+βpTAp(2))+k122Ap(2)+ηp2Ap(2)
=Gp(Fp, Ψp),
 FpGp(Fp, Ψp)dXdY=0,
iζΨ1+iΛ1Ψ1-r1τ2Ψ1+ρ1Ψ1*Ψ2 exp(-iΔβζ/2)
+(γ11|Ψ1|2+2γ12|Ψ2|2)Ψ1=0,
iζΨ2+iΛ2Ψ2-r2τ2Ψ2+σρ2Ψ12 exp(iΔβζ/2)
+2σ(γ22|Ψ2|2+2γ21|Ψ1|2)Ψ2=iΔβτΨ2.
Λp=αp2k1,ρp=χ˜p(2)Q(2)χ2,
γpq=3χ˜pq(3)Qpq(3)2χ3,Δβ=Δkk1,
Δβ=k1-k2k1τ0,rp=sign(k1kp)2|k1/kp|,
Q(2)= F12F2dXdY,Qpq(3)= Fq2Fp2dXdY.
iζΨ1+iΛ1Ψ1-r1τ2Ψ1+ρ1Ψ1*Ψ2 exp(-iΔβζ)
+(γ11|Ψ1|2+2γ12|Ψ2|2)Ψ1=0,
iζΨ2+iΛ2Ψ2-r2τ2Ψ2+ρ2Ψ12 exp(iΔβζ)
+2(γ22|Ψ2|2+2γ21|Ψ1|2)Ψ2=iΔβτΨ2,
χ˜p(2)/χ2χ˜pq(3)/χ3|k1-k2||k1k1|,
2k1-k2k1αp2k12,
Ap(x¯, y¯, z¯, t¯)=Fp(X, Y)[Ψp(ζ, τ)+3/2Ψp(2)(ζ, τ)+],
(X, Y)=1/4(x¯, y¯),ζ=z¯,
τ=1/2(t¯-β1z¯).
iζΨ1-r1τ2Ψ1+ρ1Ψ1*Ψ2 exp(-iΔβζ)
=1/2G1(1)+G1(2),
iζΨ2-r2τ2Ψ2+ρ2Ψ12 exp(iΔβζ)-iΔβτΨ2
=1/2G2(1)+G2(2),
|k1-k2||k1k1|1/2,2k1-k2k1,αp2k12.
G1(1)=β1ζτΨ1+is1τ3Ψ1-i(ρ1aΨ2τΨ1*+ρ1bΨ1*τΨ2)exp(-iΔβζ),
G2(1)=½β1ζτΨ2+i½s2τ3Ψ2-a1τ2Ψ2-iρ2aΨ1τΨ1 exp(iΔβζ).
sp=kpkp+3kpkp6k12τ02,a1=β1Δβ12,
ρpf=(2δ˜p+ωpδ˜pf)Q(2)ω1τ0χ2,
G1(2)=-½ζ2Ψ1-v1τ4Ψ1-(γ11|Ψ1|2+2γ12|Ψ2|2)Ψ1-iΛ1Ψ1+[ρ1aaΨ2τ2Ψ1*+ρ1bbΨ1*τ2Ψ2+2ρ1ab(τΨ1*)×(τΨ2)]exp(-iΔβζ),
G2(2)=-¼ζ2Ψ2-½v2τ4Ψ2-2(γ22|Ψ2|2+2γ21|Ψ1|2)Ψ2-iΛ2Ψ2+a2τ2Ψ2+½[ρ2aaΨ1τ2Ψ1+ρ2ab(τΨ1)2-2b1Ψ12]exp(iΔβζ),
a2=(Δβ)24,
vp=kpkp+4kpkp+3(kp)224k12τ04,b1=ρ2Δkk2,
ρpfg=(2δ˜p+2ωpδ˜pf+2ωpδ˜pg+ωp2δ˜pfg)Q(2)2ω12τ02χ2,
2Ep+i2(kpz+κpx)Ep-(kp2+κp2)Ep+μ0ωp2Dp
=0,
D1=0˜1E1+2χ˜1(2)E1*E2 exp(-iΔkz)+3(χ˜11(3)|E1|2+2χ˜12(3)|E2|2)E1,
D2=0˜2E2+χ˜2(2)E12 exp(iΔkz)+3(χ˜22(3)|E2|2+2χ˜21(3)|E1|2)E2,
Ap(x¯, y¯, z¯, t¯)=Fp(Y)[Ψp(ζ, τ)+2Ψp(2)(ζ, τ)+],Y=1/2y¯,ζ=2z¯,
τ=(x¯-ν1z¯),νp=κp/kp.
(1+2νp2)kp2=(npωp/c)2-ηp2,
iζΨ1+iΛ1Ψ1+½τ2Ψ1+ρ1Ψ1*Ψ2 exp(-iΔβζ)
+(γ11|Ψ1|2+2γ12|Ψ2|2)Ψ1=0, 
iζΨ2+iΛ2Ψ2+¼τ2Ψ2+ρ2Ψ12 exp(iΔβζ)
+2(γ22|Ψ2|2+2γ21|Ψ1|2)Ψ2=iΔντΨ2,
κpkpχ˜p(2)/χ2χ˜pq(3)/χ3,2k1-k2k1αp2k12.
Ap(x¯, y¯, z¯, t¯)=Fp(Y)[Ψp(ζ, τ)+3/2Ψp(2)(ζ, τ)+],
Y=1/4y¯,ζ=z¯,
τ=1/2(x¯-1/2ν1z¯),1/2νp=κp/kp.
iζΨ1+½τ2Ψ1+ρ1Ψ1*Ψ2 exp(-iΔβζ)=G1,
iζΨ2+¼τ2Ψ2+ρ2Ψ12 exp(iΔβζ)-iΔντΨ2=G2,
κpkp1/2,2k1-k2k1,αp2k12.
G1=-(γ11|Ψ1|2+2γ12|Ψ2|2)Ψ1-iΛ1Ψ1-½ζ2Ψ1+ν1ζτΨ1-½ν12τ2Ψ1,
G2=-2(γ22|Ψ2|2+2γ21|Ψ1|2)Ψ2-iΛ2Ψ2-¼ζ2Ψ2+½ν1ζτΨ2-¼(ν12+b2)τ2Ψ2,
χ˜1(2)=χ˜2(2),χ˜12(3)=χ˜21(3)
iζΨ1-r1τ2Ψ1+ρ1Ψ1*Ψ2 exp(-iΔβζ)+(γ11|Ψ1|2
+2γ12|Ψ2|2)Ψ1=0,
iζΨ2-iΔβτΨ2-r2τ2Ψ2+ρ2Ψ12 exp(iΔβζ)
+2(γ22|Ψ2|2+2γ21|Ψ1|2)Ψ2=0,
Ψp(ζ, τ)=Ψ¯p(ζ, τ)exp[i(βpζ+pΩτ)],
β2=Δβ+2β1
Ψ¯p(ζ, τ)=κpϕp(ξ, θ),ξ=κ3ζ,θ=κ4(τ+κ5ζ)
κ5=2r1Ω,β1=r1Ω2,
Ω=Δβ/(2r1-4r2)0forr12r2forr1=2r2,Δβ=0
κ12=κ32/2ρ12,κ2=κ3/ρ1,κ3=ρ12/2γ12,
κ42=2|κ3|,
iξϕ1-r¯1θ2ϕ1+ϕ1*ϕ2+η1|ϕ1|2ϕ1
+|ϕ2|2ϕ1=0,
iξϕ2-r¯2θ2ϕ2-αϕ2+½ϕ12+η2|ϕ2|2ϕ2+|ϕ1|2ϕ2
=0,
r¯p=2rp sign(χ˜12(3)),α=2γ12(Δβ-ΩΔβ)/ρ12,
η1=γ11/4γ12,η2=γ22/γ12.
H=[α|ϕ2|2-Re(ϕ12ϕ2*)-½η1|ϕ1|4-½η2|ϕ2|4-|ϕ1|2|ϕ2|2-r¯1|θϕ1|2-r¯2|θϕ2|2]dθ,
N=(|ϕ1|2+2|ϕ2|2)dθ,
P= Im(ϕ1ξϕ1*+ϕ2ξϕ2*)dθ,
ϕ1(ξ, θ)=w(θ)exp(iλξ),ϕ2(ξ, θ)=v(θ)exp(i2λξ),
w-λw+wv+¼w3+v2w=0,
½v-βv+½w2+v3+w2v=0,
HGS=T(θ)+U(θ)=pw22mw+pv22mv+U(θ).
U(θ)=½(-λw2-βv2+w2v+w4+½v4+w2v2).

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