Abstract

The existence and stability of three-wave solitons, both (1+1) and (2+1) dimensional, that result from a double-resonance (type I plus type II) parametric interaction in a purely quadratic nonlinear medium are investigated. We demonstrate the existence of a family of stable solitons for a broad parameter range in the double-resonance model. Further, these solitons exhibit multistability, a property that is potentially useful for optical switching applications. We introduce a way to measure the quality of multistability and use this measure to compare the double-resonance model with single-resonance models in χ(2) media. We also discuss the modulational instability of the double-resonance system and present physical estimates of the power required for soliton generation.

© 2000 Optical Society of America

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  1. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
    [CrossRef]
  2. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 6, 1918–1935 (1962).
    [CrossRef]
  3. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).
  4. P. St. J. Russell, “All-optical high gain transistor action using second-order nonlinearities,” Electron. Lett. 29, 1228–1229 (1993); G. Assanto, “Transistor action through nonlinear cascading in type II interactions,” Opt. Lett. 20, 1595–1597 (1995); A. Kobyakov, F. Lederer, O. Bang, and Yu. S. Kivshar, “Nonlinear phase shift and all-optical switching in quasi-phase-matched quadratic media,” Opt. Lett. OPLEDP 23, 506–508 (1998); K. Gallo and G. Assanto, “All-optical diode based on second-harmonic generation in an asymmetric waveguide,” J. Opt. Soc. Am. B JOBPDE 16, 267–269 (1999).
    [CrossRef] [PubMed]
  5. A. V. Buryak and Yu. S. Kivshar, “Solitons due to second harmonic generation,” Phys. Lett. A 197, 407–412 (1995); D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, “Instabilities of solitons governed by quadratic nonlinearities,” Phys. Rev. Lett. 75, 591–595 (1995).
    [CrossRef] [PubMed]
  6. 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]
  7. H. T. Tran, “Self-induced phase-matching and three-wave bright spatial solitons in quadratic media,” Opt. Commun. 118, 581–586 (1995); B. A. Malomed, D. Anderson, and M. Lisak, “Three-wave interaction solitons in a dispersive medium with quadratic nonlinearity,” Opt. Commun. 126, 251–254 (1996).
    [CrossRef]
  8. C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with quadratic nonlinearity,” Prog. Opt. (to be published).
  9. A. D. Boardman, P. Bontemps, and K. Xie, “Vector solitary optical beams controlled with mixed type I–type II second harmonic generation,” Opt. Quantum Electron. 30, 891–906 (1998).
    [CrossRef]
  10. S. Saltiel and Y. Deyanova, “Polarization switching as a result of cascading of two simultaneously phase-matched quadratic processes,” Opt. Lett. 24, 1296–1298 (1999); S. Saltiel, K. Koynov, Y. Deyanova, and Y. S. Kivshar, “Nonlinear phase shift resulting from two-color multistep cascading,” J. Opt. Soc. Am. B 17, 959–965 (2000).
    [CrossRef]
  11. Yu. S. Kivshar, T. J. Alexander, and S. Saltiel, “Spatial optical solitons resulting from multistep cascading,” Opt. Lett. 24, 759–761 (1999); Yu. S. Kivshar, A. A. Sukhorukov, and S. M. Saltiel, “Two color multistep cascading and parametric soliton induced waveguides,” Phys. Rev. E 60, R5056–R5059 (1999).
    [CrossRef]
  12. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi phase matched second harmonic generation: tunings and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992); G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, “42%-efficient single pass cw second harmonic generation in periodically poled lithium niobate,” Opt. Lett. 22, 1834–1836 (1997).
    [CrossRef]
  13. C. B. Clausen, O. Bang, and Yu. S. Kivshar, “Spatial solitons and induced Kerr effects in quasi phase matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997).
    [CrossRef]
  14. K. Fradkin-Kashi and A. Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quantum Electron. 35, 1649–1655 (1999).
    [CrossRef]
  15. X. Mu, X. Gu, M. V. Makarov, Y. J. Ding, J. Wang, J. Wei, and Y. Liu, “Third-harmonic generation by cascading second-order nonlinear processes in a cerium-doped KTiOPO4 crystal,” Opt. Lett. 25, 117–119 (2000).
    [CrossRef]
  16. S. Trillo and G. Assanto, “Polarization spatial chaos in second-harmonic generation,” Opt. Lett. 19, 1825–1827 (1994).
    [CrossRef] [PubMed]
  17. S. Trillo and P. Ferro, “Modulational instability in second-harmonic generation,” Opt. Lett. 20, 438–440 (1995).
    [CrossRef] [PubMed]
  18. M. Haelterman and A. P. Sheppard, “Extended modulational instability and new type of solitary wave in coupled nonlinear Schrödinger equations,” Phys. Lett. A 185, 265–272 (1994); H. He, P. D. Drummond, and B. A. Malomed, “Modulational stability in dispersive optical systems with cascaded nonlinearity,” Opt. Commun. 123, 394–402 (1996); H. He, A. Arraf, C. M. de Sterke, P. D. Drummond, and B. A. Malomed, “Theory of modulational instability in Bragg gratings with quadratic nonlinearity,” Phys. Rev. E PLEEE8 59, 6064–6078 (1999).
    [CrossRef]
  19. D. V. Skryabin and W. J. Firth, “Modulational instability of solitary waves in nondegenerate three-wave mixing: the role of phase symmetries,” Phys. Rev. Lett. 81, 3379–3382 (1998).
    [CrossRef]
  20. M. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 16, 1020–1028 (1973) [Radiophys. Quantum Electron. 16, 783–789 (1973)]; D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, “Nonlinear theory of oscillating, decaying, and collapsing solitons in the generalized nonlinear Schrödinger equation,” Phys. Rev. E PLEEE8 53, 1940–1953 (1996).
    [CrossRef]
  21. C. Etrich, U. Peschel, F. Lederer, B. A. Malomed, and Y. S. Kivshar, “Origin of the persistent oscillations of solitary waves in nonlinear quadratic media,” Phys. Rev. E 54, 4321–4324 (1996); Y. S. Kivshar, D. E. Pelinovsky, T. Cretegny, and M. Peyrard, “Internal modes of solitary waves,” Phys. Rev. Lett. 80, 5032–5035 (1998).
    [CrossRef]
  22. A. V. Buryak and Yu. S. Kivshar, “Multistability of three wave parametric self trapping,” Phys. Rev. Lett. 78, 3286–3289 (1997).
    [CrossRef]
  23. D. J. Kaup, T. I. Lakoba, and B. A. Malomed, “Asymmetric solitons in mismatched dual-core optical fibers,” J. Opt. Soc. Am. B 14, 1199–1206 (1997); V. E. Zakharov and E. A. Kuznetsov, “Optical solitons and quasisolitons,” JETP 86, 1035–1046 (1998).
    [CrossRef]
  24. L. Torner and E. M. Wright, “Soliton excitations and mutual locking of light beams in bulk quadratic nonlinear crystals,” J. Opt. Soc. Am. B 13, 864–875 (1996).
    [CrossRef]
  25. T. Peschel, U. Peschel, F. Lederer, and B. A. Malomed, “Solitary waves in Bragg gratings with a quadratic nonlinearity,” Phys. Rev. E 55, 4730–4739 (1997); W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Three-wave gap solitons in waveguides with quadratic nonlinearity,” Phys. Rev. E 58, 6708–6722 (1998); E. A. Ostrovskaya, Yu. S. Kivshar, D. V. Skryabin, and W. J. Firth, “Stability of multihump optical solitons,” Phys. Rev. Lett. PRLTAO 83, 296–299 (1999).
    [CrossRef]
  26. B. Bourliaguet, V. Couderc, A. Barthelemy, G. W. Ross, P. G. R. Smith, D. C. Hanna, and C. De Angelis, “Observation of quadratic spatial solitons in periodically poled lithium niobate,” Opt. Lett. 24, 1410–1412 (1999).
    [CrossRef]
  27. 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 soliton waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
    [CrossRef] [PubMed]

2000

1999

1998

D. V. Skryabin and W. J. Firth, “Modulational instability of solitary waves in nondegenerate three-wave mixing: the role of phase symmetries,” Phys. Rev. Lett. 81, 3379–3382 (1998).
[CrossRef]

A. D. Boardman, P. Bontemps, and K. Xie, “Vector solitary optical beams controlled with mixed type I–type II second harmonic generation,” Opt. Quantum Electron. 30, 891–906 (1998).
[CrossRef]

1997

C. B. Clausen, O. Bang, and Yu. S. Kivshar, “Spatial solitons and induced Kerr effects in quasi phase matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997).
[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]

A. V. Buryak and Yu. S. Kivshar, “Multistability of three wave parametric self trapping,” Phys. Rev. Lett. 78, 3286–3289 (1997).
[CrossRef]

1996

1995

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 soliton waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

S. Trillo and P. Ferro, “Modulational instability in second-harmonic generation,” Opt. Lett. 20, 438–440 (1995).
[CrossRef] [PubMed]

1994

1962

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 6, 1918–1935 (1962).
[CrossRef]

1961

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Arie, A.

K. Fradkin-Kashi and A. Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quantum Electron. 35, 1649–1655 (1999).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 6, 1918–1935 (1962).
[CrossRef]

Assanto, G.

Bang, O.

C. B. Clausen, O. Bang, and Yu. S. Kivshar, “Spatial solitons and induced Kerr effects in quasi phase matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997).
[CrossRef]

Barthelemy, A.

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 6, 1918–1935 (1962).
[CrossRef]

Boardman, A. D.

A. D. Boardman, P. Bontemps, and K. Xie, “Vector solitary optical beams controlled with mixed type I–type II second harmonic generation,” Opt. Quantum Electron. 30, 891–906 (1998).
[CrossRef]

Bontemps, P.

A. D. Boardman, P. Bontemps, and K. Xie, “Vector solitary optical beams controlled with mixed type I–type II second harmonic generation,” Opt. Quantum Electron. 30, 891–906 (1998).
[CrossRef]

Bourliaguet, B.

Buryak, A. V.

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]

A. V. Buryak and Yu. S. Kivshar, “Multistability of three wave parametric self trapping,” Phys. Rev. Lett. 78, 3286–3289 (1997).
[CrossRef]

Clausen, C. B.

C. B. Clausen, O. Bang, and Yu. S. Kivshar, “Spatial solitons and induced Kerr effects in quasi phase matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997).
[CrossRef]

Couderc, V.

De Angelis, C.

Ding, Y. J.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 6, 1918–1935 (1962).
[CrossRef]

Ferro, P.

Firth, W. J.

D. V. Skryabin and W. J. Firth, “Modulational instability of solitary waves in nondegenerate three-wave mixing: the role of phase symmetries,” Phys. Rev. Lett. 81, 3379–3382 (1998).
[CrossRef]

Fradkin-Kashi, K.

K. Fradkin-Kashi and A. Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quantum Electron. 35, 1649–1655 (1999).
[CrossRef]

Franken, P. A.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Gu, X.

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 soliton waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Hanna, D. C.

Hill, A. E.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Kivshar, Yu. S.

C. B. Clausen, O. Bang, and Yu. S. Kivshar, “Spatial solitons and induced Kerr effects in quasi phase matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997).
[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]

A. V. Buryak and Yu. S. Kivshar, “Multistability of three wave parametric self trapping,” Phys. Rev. Lett. 78, 3286–3289 (1997).
[CrossRef]

Liu, Y.

Makarov, M. V.

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 soliton waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Mu, X.

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 6, 1918–1935 (1962).
[CrossRef]

Peters, C. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Ross, G. W.

Skryabin, D. V.

D. V. Skryabin and W. J. Firth, “Modulational instability of solitary waves in nondegenerate three-wave mixing: the role of phase symmetries,” Phys. Rev. Lett. 81, 3379–3382 (1998).
[CrossRef]

Smith, P. G. R.

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 soliton waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Torner, L.

L. Torner and E. M. Wright, “Soliton excitations and mutual locking of light beams in bulk quadratic nonlinear crystals,” J. Opt. Soc. Am. B 13, 864–875 (1996).
[CrossRef]

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 soliton waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

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 soliton waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Trillo, S.

Van Stryland, E. W.

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 soliton waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Wang, J.

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 soliton waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

Wei, J.

Weinreich, G.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Wright, E. M.

Xie, K.

A. D. Boardman, P. Bontemps, and K. Xie, “Vector solitary optical beams controlled with mixed type I–type II second harmonic generation,” Opt. Quantum Electron. 30, 891–906 (1998).
[CrossRef]

IEEE J. Quantum Electron.

K. Fradkin-Kashi and A. Arie, “Multiple-wavelength quasi-phase-matched nonlinear interactions,” IEEE J. Quantum Electron. 35, 1649–1655 (1999).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Opt. Quantum Electron.

A. D. Boardman, P. Bontemps, and K. Xie, “Vector solitary optical beams controlled with mixed type I–type II second harmonic generation,” Opt. Quantum Electron. 30, 891–906 (1998).
[CrossRef]

Phys. Rev.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 6, 1918–1935 (1962).
[CrossRef]

Phys. Rev. Lett.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

C. B. Clausen, O. Bang, and Yu. S. Kivshar, “Spatial solitons and induced Kerr effects in quasi phase matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997).
[CrossRef]

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 soliton waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995).
[CrossRef] [PubMed]

D. V. Skryabin and W. J. Firth, “Modulational instability of solitary waves in nondegenerate three-wave mixing: the role of phase symmetries,” Phys. Rev. Lett. 81, 3379–3382 (1998).
[CrossRef]

A. V. Buryak and Yu. S. Kivshar, “Multistability of three wave parametric self trapping,” Phys. Rev. Lett. 78, 3286–3289 (1997).
[CrossRef]

Other

D. J. Kaup, T. I. Lakoba, and B. A. Malomed, “Asymmetric solitons in mismatched dual-core optical fibers,” J. Opt. Soc. Am. B 14, 1199–1206 (1997); V. E. Zakharov and E. A. Kuznetsov, “Optical solitons and quasisolitons,” JETP 86, 1035–1046 (1998).
[CrossRef]

M. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 16, 1020–1028 (1973) [Radiophys. Quantum Electron. 16, 783–789 (1973)]; D. E. Pelinovsky, V. V. Afanasjev, and Yu. S. Kivshar, “Nonlinear theory of oscillating, decaying, and collapsing solitons in the generalized nonlinear Schrödinger equation,” Phys. Rev. E PLEEE8 53, 1940–1953 (1996).
[CrossRef]

C. Etrich, U. Peschel, F. Lederer, B. A. Malomed, and Y. S. Kivshar, “Origin of the persistent oscillations of solitary waves in nonlinear quadratic media,” Phys. Rev. E 54, 4321–4324 (1996); Y. S. Kivshar, D. E. Pelinovsky, T. Cretegny, and M. Peyrard, “Internal modes of solitary waves,” Phys. Rev. Lett. 80, 5032–5035 (1998).
[CrossRef]

T. Peschel, U. Peschel, F. Lederer, and B. A. Malomed, “Solitary waves in Bragg gratings with a quadratic nonlinearity,” Phys. Rev. E 55, 4730–4739 (1997); W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Three-wave gap solitons in waveguides with quadratic nonlinearity,” Phys. Rev. E 58, 6708–6722 (1998); E. A. Ostrovskaya, Yu. S. Kivshar, D. V. Skryabin, and W. J. Firth, “Stability of multihump optical solitons,” Phys. Rev. Lett. PRLTAO 83, 296–299 (1999).
[CrossRef]

H. T. Tran, “Self-induced phase-matching and three-wave bright spatial solitons in quadratic media,” Opt. Commun. 118, 581–586 (1995); B. A. Malomed, D. Anderson, and M. Lisak, “Three-wave interaction solitons in a dispersive medium with quadratic nonlinearity,” Opt. Commun. 126, 251–254 (1996).
[CrossRef]

C. Etrich, F. Lederer, B. A. Malomed, T. Peschel, and U. Peschel, “Optical solitons in media with quadratic nonlinearity,” Prog. Opt. (to be published).

M. Haelterman and A. P. Sheppard, “Extended modulational instability and new type of solitary wave in coupled nonlinear Schrödinger equations,” Phys. Lett. A 185, 265–272 (1994); H. He, P. D. Drummond, and B. A. Malomed, “Modulational stability in dispersive optical systems with cascaded nonlinearity,” Opt. Commun. 123, 394–402 (1996); H. He, A. Arraf, C. M. de Sterke, P. D. Drummond, and B. A. Malomed, “Theory of modulational instability in Bragg gratings with quadratic nonlinearity,” Phys. Rev. E PLEEE8 59, 6064–6078 (1999).
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).

P. St. J. Russell, “All-optical high gain transistor action using second-order nonlinearities,” Electron. Lett. 29, 1228–1229 (1993); G. Assanto, “Transistor action through nonlinear cascading in type II interactions,” Opt. Lett. 20, 1595–1597 (1995); A. Kobyakov, F. Lederer, O. Bang, and Yu. S. Kivshar, “Nonlinear phase shift and all-optical switching in quasi-phase-matched quadratic media,” Opt. Lett. OPLEDP 23, 506–508 (1998); K. Gallo and G. Assanto, “All-optical diode based on second-harmonic generation in an asymmetric waveguide,” J. Opt. Soc. Am. B JOBPDE 16, 267–269 (1999).
[CrossRef] [PubMed]

A. V. Buryak and Yu. S. Kivshar, “Solitons due to second harmonic generation,” Phys. Lett. A 197, 407–412 (1995); D. E. Pelinovsky, A. V. Buryak, and Yu. S. Kivshar, “Instabilities of solitons governed by quadratic nonlinearities,” Phys. Rev. Lett. 75, 591–595 (1995).
[CrossRef] [PubMed]

S. Saltiel and Y. Deyanova, “Polarization switching as a result of cascading of two simultaneously phase-matched quadratic processes,” Opt. Lett. 24, 1296–1298 (1999); S. Saltiel, K. Koynov, Y. Deyanova, and Y. S. Kivshar, “Nonlinear phase shift resulting from two-color multistep cascading,” J. Opt. Soc. Am. B 17, 959–965 (2000).
[CrossRef]

Yu. S. Kivshar, T. J. Alexander, and S. Saltiel, “Spatial optical solitons resulting from multistep cascading,” Opt. Lett. 24, 759–761 (1999); Yu. S. Kivshar, A. A. Sukhorukov, and S. M. Saltiel, “Two color multistep cascading and parametric soliton induced waveguides,” Phys. Rev. E 60, R5056–R5059 (1999).
[CrossRef]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi phase matched second harmonic generation: tunings and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992); G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, “42%-efficient single pass cw second harmonic generation in periodically poled lithium niobate,” Opt. Lett. 22, 1834–1836 (1997).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Q(β) diagram for various values of χ1e, χ2=0.1, Δ=-1, and r=-1. Note the appearance, and then gradual disappearance, of multistability. Filled circle, the unstable solution pictured in Fig. 2; arrows, the stable-soliton solution into which it evolves under the action of perturbations: to the right after increase of power and to the left after decrease of power.

Fig. 2
Fig. 2

Unstable-soliton solution for β=1.24, Δ=-1, r=-1, χ1e=0.3, and χ2=0.1 (a) switched up to the right-hand stable branch with a small initial increase of power and (b) switched down to the left-hand stable branch with a small initial decrease in power.

Fig. 3
Fig. 3

Parameter regions of soliton existence in the degenerate double-resonance system (χ1e=0). Stable solitons exist in a broad parameter range. Inset, power Q versus nonlinear phase velocity shift β for Δ=-10 and r=1. Far left, the quasi-soliton branch.

Fig. 4
Fig. 4

Q(β) diagram for χ2=0.1, χ1e=0.3, Δ=-1, and r=-1. Note (open circle) that, as β1, Q does not vanish but rather attains a finite value. This result shows the threshold power necessary for the soliton’s existence at given values of the parameters.

Fig. 5
Fig. 5

(a) Initial and (b) final soliton profiles after collision and fusion. The value of β jumps from 1.1 to approximately 2.3. χ2=0.1, χ1e=0.4, Δ=-1, r=-1, C=0.5, δ=π/20.

Fig. 6
Fig. 6

Illustration of how the quality measure J is calculated for (a) typical and (b) defective [i.e., ΔQQmax(1)-Qmin(2)] Q(β) diagrams. (a) Solitons of the double-resonance system [Eqs. (3)] at χ1e=0.41, χ2=0.04, Δd=-1.0, r=-1. (b) The usual type II model [Eqs. (11)] at Δ=-1 and |Qim|=6.6.

Fig. 7
Fig. 7

(a) Domain of soliton multistability in the plane (Q, Qim) for the (2+1)D version of Eqs. (11) at Δ=-1. The area where unstable solitons exist is hatched. The thin vertical line shows the line of constant Qim. (b) Corresponding region of soliton existence–stability for Eqs. (11) in (βu, βv) parameter space. The curve labeled Qim(βv, βu)=const corresponds to the vertical line in (a). Parameterization variable β˜ is defined as the path length along this curve, starting from the edge point (βv=0.0, βu=1.0).

Fig. 8
Fig. 8

(a) Invariant Q versus effective soliton propagation constant β˜ for Eqs. (11) at Δ=-1 and several values of Qim. (b) Invariant Q versus β for Eqs. (3) at χ2=0.1, Δd=-1.0, r=-1, and several values of χ1e.

Tables (1)

Tables Icon

Table 1 Comparison of Absolute Maximum of Multistability Figure of Merit Jmax for the Type II Single-Resonance System with Typical Maximum of Multistability Figure of Merit Jmax for Fixed Δd =-1.0 for the Double-Resonance System of Eqs. (3)

Equations (42)

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2ik1E1z+2E1+χ1oE1*E3 exp(iΔk1z)
+χ2E2*E3 expiΔk1+Δk22z=0,
2ik2E2z+2E2+χ1eE2*E3 exp(iΔk2z)
+χ2E1*E3 expiΔk1+Δk22z=0,
2ik3E3z+2E3+χ1oE122 exp(-iΔk1z)
+χ1eE222 exp(-iΔk2z)
+χ2E1E2 exp-iΔk1+Δk22z=0,
iσvz+2v-σβ˜v+v*w+χ2u*w=0,
i(2-σ)uz+2u-(2-σ)α˜u
+χ1eu*w+χ2v*w=0,
2iwz+2w-γ˜w+v22+χ1eu22+χ2vu=0,
2k1o-k2=Δk1=nπ/L,
2k1e-k2=Δk2=mπ/L,
iVZ+2V-βV+V*W+χ2U*W=0,
iUZ+2U-αU+χ1eU*W+χ2V*W=0,
2iWZ+2W-γW+V22+χ1eU22+χ2VU=0,
-βV+V*W+χ2U*W=0,
-αU+χ1eU*W+χ2V*W=0,
-γW+V22+χ1eU22+χ2VU=0.
V=vo+v1(Z)exp(iΩX)+v2(Z)exp(-iΩX),
U=uo+u1(Z)exp(iΩX)+u2(Z)exp(-iΩX),
W=wo+w1(Z)exp(iΩX)+w2(Z)exp(-iΩX)
Lˆ=-sΩ2-βwo0χ2woA0-wosΩ2+β-χ2wo00-A0χ2wo-α-sΩ2χ1woB0-χ2wo0-χ1wosΩ2+α0-BA/20B/20-(sΩ2+γ)/200-A/20-B/20(sΩ2+γ)/2.
V=a sech2(x/λ),U=b sech2(x/λ),
W=c sech2(x/λ),
Q=-+-+(4|W|2+|V|2+|U|2)dXdY.
iVZ+1RVR-βV+χ1oV*W+χ2U*W=0,
iUZ+1RUR-αU+χ1eU*W+χ2V*W=0,
2iWZ+1RWR-γW+χ1oV22+χ1eU22+χ2VU=0.
V(r, Z)=Vo(r-CZ)exp(iC·r/2-i|C|2Z/4),
U(r, Z)=Uo(r-CZ)exp(iC·r/2-i|C|2Z/4),
W(r, Z)=Wo(r-CZ)exp(iC·r-i|C|2Z/2),
J=2Δβ1ΔQΔβ2Qmax×100.
iVZ+2V-βvV+U*W=0,
iUZ+2U-βuU+V*W=0,
2iWZ+2W-2(βv+βu+Δ)W+VU=0,
Qim=-+-+(|V|2-|U|2)dXdY.
Qimβ˜=Qimβuβuβ˜+Qimβvβvβ˜=0,
Qimβu=-Qimβvβvβ˜βuβ˜.
QβvQimβu-QβuQimβv=0.
P=-+-+(|E1|2+|E2|2+|E3|2)dxdy,
I=Pπro2=Qπχ1o2ro41.1×10-10 GW cm2ro4.

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