Abstract

We investigate the possibility of obtaining temporal bright solitons in three-wave parametric interaction by using the pulse tilt to modify the effective dispersion of the nonlinear crystal. We show that, in the large-beam approximation, the temporal evolution of the tilted pulse can be described by (1+1)-dimensional equations, which account for the tilt by means of effective group-velocity mismatch and dispersion coefficients. Numerical calculations show that the soliton regime is accessible in several experimental conditions. The limit of the (1+1)-dimensional approximation is investigated with the aid of a (2+1)-dimensional model.

© 1999 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  13. R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, and P. Foggi, “A collinearly phase-matched parametric generator/amplifier of visible femtosecond pulses,” IEEE J. Quantum Electron. 34, 459 (1998).
    [CrossRef]
  14. A. Dubietis, G. Valiulis, G. Tamosauskas, R. Danielius, and A. Piskarskas, “Nonlinear second-harmonic pulse compression with tilted pulses,” Opt. Lett. 22, 1071 (1997).
    [CrossRef] [PubMed]
  15. A. Dubietis, G. Valiulis, G. Tamosauskas, R. Danielius, and A. Piskarskas, “Nonlinear pulse compression in the ultraviolet,” Opt. Commun. 144, 55 (1997).
    [CrossRef]
  16. P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett. 81, 570 (1998).
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  23. C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2)(2) cascading,” J. Opt. Soc. Am. B 11, 2434 (1994).
    [CrossRef]
  24. V. E. Zakharov and S. V. Manakov, “On the theory of resonant interaction of wave packets in nonlinear media,” Sov. Phys. JETP 42, 842 (1976).
  25. G. Valiulis and A. Stabinis, “Compression of ultrashort light pulses in nonlinear crystals by frequency mixing due to strong energy exchange,” Lith. Phys. J. 31, 53 (1991).
  26. E. Ibragimov and A. Struthers, “Second-harmonic pulse compression in the soliton regime,” Opt. Lett. 21, 1582 (1996).
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  27. E. Ibragimov and A. Struthers, “Three-wave soliton interaction of ultrashort pulses in quadratic media,” J. Opt. Soc. Am. B 14, 1472 (1997).
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  28. A. V. Buryak and Y. S. Kivshar, “Spatial optical solitons governed by quadratic nonlinearity,” Opt. Lett. 19, 1612 (1994); L. Torner, “Stationary solitary waves with second-order nonlinearities,” Opt. Commun. 114, 136 (1995);
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  29. L. Torner, D. Mihalache, D. Mazilu, E. M. Wright, W. Torruellas, G. I. Stegeman, “Stationary trapping of light beams in bulk second-order nonlinear media,” Opt. Commun. 121, 149 (1995).
    [CrossRef]
  30. L. Torner, D. Mazilu, and D. Mihalache, “Walking solitons in quadratic nonlinear media,” Phys. Rev. Lett. 77, 2455 (1996).
    [CrossRef] [PubMed]

1998 (3)

P. DiTrapani, G. Valiulis, W. Chinaglia, and A. Andreoni, “Quantum-noise space-mode locking and generation of 2D quasi-solitons in X(2) media,” Phys. Rev. Lett. 80, 261 (1998).
[CrossRef]

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, and P. Foggi, “A collinearly phase-matched parametric generator/amplifier of visible femtosecond pulses,” IEEE J. Quantum Electron. 34, 459 (1998).
[CrossRef]

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett. 81, 570 (1998).
[CrossRef]

1997 (4)

1996 (6)

L. Torner, D. Mazilu, and D. Mihalache, “Walking solitons in quadratic nonlinear media,” Phys. Rev. Lett. 77, 2455 (1996).
[CrossRef] [PubMed]

E. Ibragimov and A. Struthers, “Second-harmonic pulse compression in the soliton regime,” Opt. Lett. 21, 1582 (1996).
[CrossRef] [PubMed]

R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138 (1996).
[CrossRef]

S. Szatmari, P. Simon, and M. Feuerhake, “Group-velocity-dispersion-compensated propagation of short pulses in dispersive media,” Opt. Lett. 21, 1156 (1996).
[CrossRef] [PubMed]

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, and P. Foggi, “Matching of group velocities by spatial walk-off in collinear three-wave interaction with tilted pulses,” Opt. Lett. 21, 973 (1996).
[CrossRef] [PubMed]

1995 (1)

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

1994 (2)

1993 (1)

1991 (2)

G. Valiulis and K. Staliunas, “On the subject of the integrability and soliton solutions of three-wave interaction equations,” Lith. Phys. J. 31, 38 (1991).

G. Valiulis and A. Stabinis, “Compression of ultrashort light pulses in nonlinear crystals by frequency mixing due to strong energy exchange,” Lith. Phys. J. 31, 53 (1991).

1989 (1)

O. E. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. 25, 2464 (1989).
[CrossRef]

1986 (1)

O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229 (1986).
[CrossRef]

1978 (1)

Y. N. Karamzin, A. P. Sukhorukov, and T. S. Filipchuk, “On a new class of coupled solutions in dispersive media with quadratic nonlinearities,” Moscow Univ. Phys. Bull. 19, 91 (1978).

1976 (2)

Y. N. Karamzin and A. P. Sukhorukov, “Mutual focusing of high-power light beams in media with quadratic nonlinearity,” Sov. Phys. JETP 41, 414 (1976).

V. E. Zakharov and S. V. Manakov, “On the theory of resonant interaction of wave packets in nonlinear media,” Sov. Phys. JETP 42, 842 (1976).

1974 (1)

Y. 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).

Andreoni, A.

P. DiTrapani, G. Valiulis, W. Chinaglia, and A. Andreoni, “Quantum-noise space-mode locking and generation of 2D quasi-solitons in X(2) media,” Phys. Rev. Lett. 80, 261 (1998).
[CrossRef]

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, and P. Foggi, “A collinearly phase-matched parametric generator/amplifier of visible femtosecond pulses,” IEEE J. Quantum Electron. 34, 459 (1998).
[CrossRef]

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, and P. Foggi, “Matching of group velocities by spatial walk-off in collinear three-wave interaction with tilted pulses,” Opt. Lett. 21, 973 (1996).
[CrossRef] [PubMed]

Baek, Y.

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

R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138 (1996).
[CrossRef]

Baumann, I.

Caironi, D.

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett. 81, 570 (1998).
[CrossRef]

Chinaglia, W.

P. DiTrapani, G. Valiulis, W. Chinaglia, and A. Andreoni, “Quantum-noise space-mode locking and generation of 2D quasi-solitons in X(2) media,” Phys. Rev. Lett. 80, 261 (1998).
[CrossRef]

Danielius, R.

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, and P. Foggi, “A collinearly phase-matched parametric generator/amplifier of visible femtosecond pulses,” IEEE J. Quantum Electron. 34, 459 (1998).
[CrossRef]

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett. 81, 570 (1998).
[CrossRef]

A. Dubietis, G. Valiulis, G. Tamosauskas, R. Danielius, and A. Piskarskas, “Nonlinear pulse compression in the ultraviolet,” Opt. Commun. 144, 55 (1997).
[CrossRef]

A. Dubietis, G. Valiulis, G. Tamosauskas, R. Danielius, and A. Piskarskas, “Nonlinear second-harmonic pulse compression with tilted pulses,” Opt. Lett. 22, 1071 (1997).
[CrossRef] [PubMed]

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, and P. Foggi, “Matching of group velocities by spatial walk-off in collinear three-wave interaction with tilted pulses,” Opt. Lett. 21, 973 (1996).
[CrossRef] [PubMed]

Di Trapani, P.

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, and P. Foggi, “A collinearly phase-matched parametric generator/amplifier of visible femtosecond pulses,” IEEE J. Quantum Electron. 34, 459 (1998).
[CrossRef]

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett. 81, 570 (1998).
[CrossRef]

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, and P. Foggi, “Matching of group velocities by spatial walk-off in collinear three-wave interaction with tilted pulses,” Opt. Lett. 21, 973 (1996).
[CrossRef] [PubMed]

DiTrapani, P.

P. DiTrapani, G. Valiulis, W. Chinaglia, and A. Andreoni, “Quantum-noise space-mode locking and generation of 2D quasi-solitons in X(2) media,” Phys. Rev. Lett. 80, 261 (1998).
[CrossRef]

Drummond, P. D.

Dubietis, A.

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett. 81, 570 (1998).
[CrossRef]

A. Dubietis, G. Valiulis, G. Tamosauskas, R. Danielius, and A. Piskarskas, “Nonlinear second-harmonic pulse compression with tilted pulses,” Opt. Lett. 22, 1071 (1997).
[CrossRef] [PubMed]

A. Dubietis, G. Valiulis, G. Tamosauskas, R. Danielius, and A. Piskarskas, “Nonlinear pulse compression in the ultraviolet,” Opt. Commun. 144, 55 (1997).
[CrossRef]

Feuerhake, M.

Filipchuk, T. S.

Y. N. Karamzin, A. P. Sukhorukov, and T. S. Filipchuk, “On a new class of coupled solutions in dispersive media with quadratic nonlinearities,” Moscow Univ. Phys. Bull. 19, 91 (1978).

Foggi, P.

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, and P. Foggi, “A collinearly phase-matched parametric generator/amplifier of visible femtosecond pulses,” IEEE J. Quantum Electron. 34, 459 (1998).
[CrossRef]

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, and P. Foggi, “Matching of group velocities by spatial walk-off in collinear three-wave interaction with tilted pulses,” Opt. Lett. 21, 973 (1996).
[CrossRef] [PubMed]

Hagan, D. J.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

Ibragimov, E.

Karamzin, Y. N.

Y. N. Karamzin, A. P. Sukhorukov, and T. S. Filipchuk, “On a new class of coupled solutions in dispersive media with quadratic nonlinearities,” Moscow Univ. Phys. Bull. 19, 91 (1978).

Y. N. Karamzin and A. P. Sukhorukov, “Mutual focusing of high-power light beams in media with quadratic nonlinearity,” Sov. Phys. JETP 41, 414 (1976).

Y. 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).

Manakov, S. V.

V. E. Zakharov and S. V. Manakov, “On the theory of resonant interaction of wave packets in nonlinear media,” Sov. Phys. JETP 42, 842 (1976).

Martinez, O. E.

O. E. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. 25, 2464 (1989).
[CrossRef]

O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229 (1986).
[CrossRef]

Mazilu, D.

L. Torner, D. Mazilu, and D. Mihalache, “Walking solitons in quadratic nonlinear media,” Phys. Rev. Lett. 77, 2455 (1996).
[CrossRef] [PubMed]

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

Menyuk, C. R.

Mihalache, D.

L. Torner, D. Mazilu, and D. Mihalache, “Walking solitons in quadratic nonlinear media,” Phys. Rev. Lett. 77, 2455 (1996).
[CrossRef] [PubMed]

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

Piskarskas, A.

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett. 81, 570 (1998).
[CrossRef]

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, and P. Foggi, “A collinearly phase-matched parametric generator/amplifier of visible femtosecond pulses,” IEEE J. Quantum Electron. 34, 459 (1998).
[CrossRef]

A. Dubietis, G. Valiulis, G. Tamosauskas, R. Danielius, and A. Piskarskas, “Nonlinear second-harmonic pulse compression with tilted pulses,” Opt. Lett. 22, 1071 (1997).
[CrossRef] [PubMed]

A. Dubietis, G. Valiulis, G. Tamosauskas, R. Danielius, and A. Piskarskas, “Nonlinear pulse compression in the ultraviolet,” Opt. Commun. 144, 55 (1997).
[CrossRef]

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, and P. Foggi, “Matching of group velocities by spatial walk-off in collinear three-wave interaction with tilted pulses,” Opt. Lett. 21, 973 (1996).
[CrossRef] [PubMed]

Schiek, R.

Simon, P.

Sohler, W.

Solcia, C.

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, and P. Foggi, “A collinearly phase-matched parametric generator/amplifier of visible femtosecond pulses,” IEEE J. Quantum Electron. 34, 459 (1998).
[CrossRef]

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, and P. Foggi, “Matching of group velocities by spatial walk-off in collinear three-wave interaction with tilted pulses,” Opt. Lett. 21, 973 (1996).
[CrossRef] [PubMed]

Stabinis, A.

G. Valiulis and A. Stabinis, “Compression of ultrashort light pulses in nonlinear crystals by frequency mixing due to strong energy exchange,” Lith. Phys. J. 31, 53 (1991).

Staliunas, K.

G. Valiulis and K. Staliunas, “On the subject of the integrability and soliton solutions of three-wave interaction equations,” Lith. Phys. J. 31, 38 (1991).

Stegeman, G. I.

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

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138 (1996).
[CrossRef]

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

Struthers, A.

Sukhorukov, A. P.

Y. N. Karamzin, A. P. Sukhorukov, and T. S. Filipchuk, “On a new class of coupled solutions in dispersive media with quadratic nonlinearities,” Moscow Univ. Phys. Bull. 19, 91 (1978).

Y. N. Karamzin and A. P. Sukhorukov, “Mutual focusing of high-power light beams in media with quadratic nonlinearity,” Sov. Phys. JETP 41, 414 (1976).

Y. 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).

Szatmari, S.

Tamosauskas, G.

A. Dubietis, G. Valiulis, G. Tamosauskas, R. Danielius, and A. Piskarskas, “Nonlinear second-harmonic pulse compression with tilted pulses,” Opt. Lett. 22, 1071 (1997).
[CrossRef] [PubMed]

A. Dubietis, G. Valiulis, G. Tamosauskas, R. Danielius, and A. Piskarskas, “Nonlinear pulse compression in the ultraviolet,” Opt. Commun. 144, 55 (1997).
[CrossRef]

Torner, L.

G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

L. Torner, D. Mazilu, and D. Mihalache, “Walking solitons in quadratic nonlinear media,” Phys. Rev. Lett. 77, 2455 (1996).
[CrossRef] [PubMed]

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

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

Torruellas, W.

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

Valiulis, G.

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius, and A. Piskarskas, “Observation of temporal solitons in second-harmonic generation with tilted pulses,” Phys. Rev. Lett. 81, 570 (1998).
[CrossRef]

P. DiTrapani, G. Valiulis, W. Chinaglia, and A. Andreoni, “Quantum-noise space-mode locking and generation of 2D quasi-solitons in X(2) media,” Phys. Rev. Lett. 80, 261 (1998).
[CrossRef]

A. Dubietis, G. Valiulis, G. Tamosauskas, R. Danielius, and A. Piskarskas, “Nonlinear pulse compression in the ultraviolet,” Opt. Commun. 144, 55 (1997).
[CrossRef]

A. Dubietis, G. Valiulis, G. Tamosauskas, R. Danielius, and A. Piskarskas, “Nonlinear second-harmonic pulse compression with tilted pulses,” Opt. Lett. 22, 1071 (1997).
[CrossRef] [PubMed]

G. Valiulis and A. Stabinis, “Compression of ultrashort light pulses in nonlinear crystals by frequency mixing due to strong energy exchange,” Lith. Phys. J. 31, 53 (1991).

G. Valiulis and K. Staliunas, “On the subject of the integrability and soliton solutions of three-wave interaction equations,” Lith. Phys. J. 31, 38 (1991).

Werner, M. J.

Wright, E. M.

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

Zakharov, V. E.

V. E. Zakharov and S. V. Manakov, “On the theory of resonant interaction of wave packets in nonlinear media,” Sov. Phys. JETP 42, 842 (1976).

IEEE J. Quantum Electron. (2)

O. E. Martinez, “Achromatic phase matching for second harmonic generation of femtosecond pulses,” IEEE J. Quantum Electron. 25, 2464 (1989).
[CrossRef]

R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, and P. Foggi, “A collinearly phase-matched parametric generator/amplifier of visible femtosecond pulses,” IEEE J. Quantum Electron. 34, 459 (1998).
[CrossRef]

J. Opt. Soc. Am. B (3)

JETP Lett. (1)

Y. 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).

Lith. Phys. J. (2)

G. Valiulis and K. Staliunas, “On the subject of the integrability and soliton solutions of three-wave interaction equations,” Lith. Phys. J. 31, 38 (1991).

G. Valiulis and A. Stabinis, “Compression of ultrashort light pulses in nonlinear crystals by frequency mixing due to strong energy exchange,” Lith. Phys. J. 31, 53 (1991).

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Figures (9)

Fig. 1
Fig. 1

(a) Pulse-front tilt produced by a grating. γ0 is the incidence angle, θ0 is the reflection angle, ψ is the pulse-front tilt angle, and τ0 is the initial pulse width. (b) Tilted pulse in a birefringent crystal.

Fig. 2
Fig. 2

Definition of the effective GVD by means of geometrical parameters of the tilted pulse.

Fig. 3
Fig. 3

Effects of the pulse-front tilt on the crystal dispersion. SH generation in type I BBO. (a) Solid curve, tilt angle for exact GVM compensation (Lv1=); dashed curve, absolute values of pulse-splitting lengths Lv1 for untilted 1-ps pulses. Absolute values of dispersive lengths of (b) the SH and (c) the fundamental 1-ps pulses. Solid curves, untilted pulses; dashed curves, pulses tilted by the angle given in (a).

Fig. 4
Fig. 4

Soliton formation in the case of GVD0: degenerate parametric amplification in a type I LiIO3 crystal in phase matching (Δk=0). Pump wavelength, λ3=527.5 nm; input-seed intensity, I1=10-5 GW/cm2; inset, intensity profiles of interacting pulses in the soliton regime.

Fig. 5
Fig. 5

Nondegenerate case. Idler and signal wavelengths, λ1=950 nm and λ2=1186 nm, respectively; other parameters as in Fig. 4.

Fig. 6
Fig. 6

Soliton formation in the case of GVD0: SH generation in type I BBO crystal in phase matching (Δk=0). Fundamental wavelength, λ1=800 nm; inset as in Fig. 4.

Fig. 7
Fig. 7

Soliton formation in the case of g3=0: SH in type I BBO crystal in phase matching (Δk=0). Fundamental wavelength, λ1=1055 nm; inset as in Fig. 4.

Fig. 8
Fig. 8

Soliton formation in the case of large phase mismatch (Δk=-150 cm-1): SH generation in BBO type I crystal. Fundamental wavelength, λ1=527.5 nm; inset as in Fig. 4.

Fig. 9
Fig. 9

Comparison between 1-D and 2-D calculations. Frequency doubling of 527.5-nm pulses in 7-mm type I phase-matching BBO crystal (I1=I2=6 GW/cm2). All other parameters as in Fig. 8.

Equations (56)

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A(x, t)=bmA0(px, t-k0qx),
p=-cos γ0cos θ0,q=-mλ022πcd cos θ0,k0=2πnλ0,
ψ=arctand(ct)dxt-k0qx=const.=arctan(ck0q).
A(x, t)=bmA0(px, t-x/c tan ψ)=bmA0(px, t-x/V),V=c/tan ψ.
xs=x cos β-z sin β,
zs=z cos β+x sin β.
Azs+1uAt-ig22At2+i2k02Axs2=0,
A0pxs, t-xsutan γ,
A0t-xsutan γ.
zs=xs tan β.
A(xs, t)|z=0=A0t-xsu(tan γ+tan β).
A(xs, t, zs)=12π-+-A0(t1)×expiωt1-t-xsutan γ-zsu-i2gsω2(zs-xs tan β)dt1dω,
gs=g-1k0tan γ+tan βu2.
A(x, t, z)=12π-+-A0(t1)expiωt1-t-xuTtan α-zuT-i2gTω2z×dt1dω,
uT=u cos β(1+tan β tan α),
gT=1cos βg-1k0tan(α-β)+tan βu2.
Az+1uAt-i2g2At2=0,
A(x, t, z)=12π-+-A0(t1)expiωt1-t-zu-i2gω2zdt1dω.
A(x, t)|z=0=A0t-xuTtan α,
t-xuTtan α-zuT
tan ψc=tan αuT
tan ψ=cutan αcos β(1+tan β tan α).
1LgT=1Lg-1LdT,
LdT=k0Θ24 ln 2,Θ=τ0utan α.
I=I0 exp-4 ln 2 [t-(x/u)tan α]2τ02=I0 exp-4 ln 2tτ0-xΘ2.
A1z+1u1A1t-i2g12A1t2+β1A1x+i2k012A1x2
=-iσ1A2*A3 exp(-iΔkz),
A2z+1u2A2t-i2g22A2t2+β2A2x+i2k022A2x2
=-iσ2A1*A3 exp(-iΔkz),
A3z+1u3A3t-i2g32A3t2+β3A3x+i2k032A3x2
=-iσ3A1A2 exp(iΔkz),
A1z+1u1A1t-i2g12A1t2=-iσ1A2*A3 exp(-iΔkz),
A2z+1u2A2t-i2g22A2t2=-iσ2A1*A3 exp(-iΔkz),
A3z+1u3A3t-i2g32A3t2=-iσ3A1A2 exp(iΔkz),
A1=b1 sech2η-vzτexp[i(Γ1z+δ1η)],
A2=b2 sech2η-vzτexp[i(Γ2z+δ2η)],
A3=b3 sech2η-vzτexp{i[(Γ1+Γ2+Δk)z+(δ1+δ2)η]},
b1=3τ2g2g3σ2σ31/2,b2=3τ2g1g3σ1σ31/2,
b3=-3τ2g1g2σ1σ21/2,η=t-z/u3,
v=v13/g1+v23/g21/g1+1/g2-1/g3,δj=v-vj3gj,
Γj=2gjτ2-vj3δj-12gjδj2,j=1, 2,
τ2=2(g1+g2-g3)v132g2+v232g1-v122g32(g1g3+g2g3-g1g2)-Δk,
vj3=1/uj-1/u3,v12=v13-v23.
A1z+1u1A1t-i2g12A1t2=-iσ1A1*A3 exp(-iΔkz),
A3z+1u3A3t=-i2σ1A12 exp(iΔkz).
iA1z+12g12A1η2=σ1|A1|2A1,
A1z+v31A1η=-iσ1+Δk2A1,
A1=B(η-v31z)exp-izσ1+Δk2,
A1(η, z)=b sechη-v31zτexp[i(θη-Γz)],
b=-g1τ2σ11/2,=-2σ1g1/τ2+v312/g1+Δk,
θ=v31g1,Γ=σ1+Δk2+v312g1,
iA1z+12g12A1η2=-2σ12Δk|A1|2A1,
A3=-σ3ΔkA12×exp(iΔkz),
A1(η, z)=1τg1Δk2σ121/2 sechη-z/usolτ×exp[i(θη-Γz)],
θ=1g1usol,Γ=12g1usol2-g12τ2.
LdTilted=k0Φ24 ln 2ΘΦ=LdΘΦ,

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