Abstract

Spatiotemporal instabilities in nonlinear Kerr media with arbitrary higher-order dispersions are studied by use of standard linear-stability analysis. A generic expression for instability growth rate that unifies and expands on previous results for temporal, spatial, and spatiotemporal instabilities is obtained. It is shown that all odd-order dispersions contribute nothing to instability, whereas all even-order dispersions not only affect the conventional instability regions but may also lead to the appearance of new instability regions. The role of fourth-order dispersion in spatiotemporal instabilities is studied exemplificatively to demonstrate the generic results. Numerical simulations confirm the obtained analytic results.

© 2002 Optical Society of America

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  1. See, for example, G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).
  2. N. C. Kothari and S. C. Abbi, “Instability growth and filamentation of very intense laser beams in self-focusing media,” Prog. Theor. Phys. 83, 414–442 (1990).
    [CrossRef]
  3. R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
    [CrossRef]
  4. L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
    [CrossRef] [PubMed]
  5. D. Anderson, M. Karlsson, M. Lisak, and A. Sergeev, “Modulational instability dynamics in a spatial focusing and temporal defocusing medium,” Phys. Rev. E 47, 3617–3622 (1993).
    [CrossRef]
  6. E. J. Fonseca, S. B. Cavalcanti, and J. M. Hickmann, “Space-time break-up in the self-focusing of ultrashort pulses,” Opt. Commun. 169, 199–205 (1999).
    [CrossRef]
  7. Xiang Liu, K. Beckwitt, and F. Wise, “Transverse instability of optical spatiotemporal solitons in quadratic media,” Phys. Rev. Lett. 85, 1871–1874 (2000).
    [CrossRef] [PubMed]
  8. N. M. Litchinitser, C. J. McKinstrie, C. Martijn de Sterke, and G. P. Agrawal, “Spatiotemporal instabilities in nonlinear bulk media with Bragg gratings,” J. Opt. Soc. Am. B 18, 45–54 (2001).
    [CrossRef]
  9. V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310 (1966).
  10. V. A. Vysloukh and N. A. Sukhotskova, “Influence of third-order dispersion on the generation of a train of picosecond pulses in fiber waveguides due to self-modulation instability,” Sov. J. Quantum Electron. 17, 1509–1511 (1987).
    [CrossRef]
  11. M. J. Potosek, “Modulation instability in an extended non-linear Schrödinger equation,” Opt. Lett. 12, 921–923 (1987).
    [CrossRef]
  12. S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
    [CrossRef] [PubMed]
  13. A. Höök and M. Karlsson, “Ultrashort solitons at the minimum-dispersion wavelength: effects of fourth-order dispersion,” Opt. Lett. 18, 1388–1390 (1993).
    [CrossRef] [PubMed]
  14. F. Kh. Abdullaev, S. A. Darmanyan, S. Bsichoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
    [CrossRef]
  15. T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
    [CrossRef]
  16. T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
    [CrossRef]
  17. J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Observation of pulse splitting in nonlinear dispersive media,” Phys. Rev. Lett. 77, 3783–3786 (1996).
    [CrossRef] [PubMed]
  18. J. K. Ranka and A. L. Gaeta, “Breakdown of the slowly varying envelope approximation in the self-focusing of ultrashort pulses,” Opt. Lett. 23, 534–536 (1998).
    [CrossRef]
  19. A. A. Zozulya, S. A. Diddams, and T. S. Clement, “Investigations of nonlinear femtosecond pulse propagation with the inclusion of Raman, shock, and third-order phase effects,” Phys. Rev. A 58, 3303–3310 (1998).
    [CrossRef]
  20. M. Trippenbach and Y. B. Band, “Effects of self-steepening and self-frequency shifting on short-pulse splitting in dispersive nonlinear media,” Phys. Rev. A 57, 4791–4803 (1998).
    [CrossRef]
  21. M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Modulational instabilities in dispersion-flattened fibers,” Phys. Rev. E 52, 1072–1080 (1995).
    [CrossRef]

2001 (1)

2000 (2)

Xiang Liu, K. Beckwitt, and F. Wise, “Transverse instability of optical spatiotemporal solitons in quadratic media,” Phys. Rev. Lett. 85, 1871–1874 (2000).
[CrossRef] [PubMed]

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[CrossRef]

1999 (1)

E. J. Fonseca, S. B. Cavalcanti, and J. M. Hickmann, “Space-time break-up in the self-focusing of ultrashort pulses,” Opt. Commun. 169, 199–205 (1999).
[CrossRef]

1998 (3)

A. A. Zozulya, S. A. Diddams, and T. S. Clement, “Investigations of nonlinear femtosecond pulse propagation with the inclusion of Raman, shock, and third-order phase effects,” Phys. Rev. A 58, 3303–3310 (1998).
[CrossRef]

M. Trippenbach and Y. B. Band, “Effects of self-steepening and self-frequency shifting on short-pulse splitting in dispersive nonlinear media,” Phys. Rev. A 57, 4791–4803 (1998).
[CrossRef]

J. K. Ranka and A. L. Gaeta, “Breakdown of the slowly varying envelope approximation in the self-focusing of ultrashort pulses,” Opt. Lett. 23, 534–536 (1998).
[CrossRef]

1997 (2)

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

1996 (1)

J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Observation of pulse splitting in nonlinear dispersive media,” Phys. Rev. Lett. 77, 3783–3786 (1996).
[CrossRef] [PubMed]

1995 (1)

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Modulational instabilities in dispersion-flattened fibers,” Phys. Rev. E 52, 1072–1080 (1995).
[CrossRef]

1994 (1)

F. Kh. Abdullaev, S. A. Darmanyan, S. Bsichoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
[CrossRef]

1993 (2)

D. Anderson, M. Karlsson, M. Lisak, and A. Sergeev, “Modulational instability dynamics in a spatial focusing and temporal defocusing medium,” Phys. Rev. E 47, 3617–3622 (1993).
[CrossRef]

A. Höök and M. Karlsson, “Ultrashort solitons at the minimum-dispersion wavelength: effects of fourth-order dispersion,” Opt. Lett. 18, 1388–1390 (1993).
[CrossRef] [PubMed]

1992 (1)

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[CrossRef] [PubMed]

1991 (1)

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef] [PubMed]

1990 (1)

N. C. Kothari and S. C. Abbi, “Instability growth and filamentation of very intense laser beams in self-focusing media,” Prog. Theor. Phys. 83, 414–442 (1990).
[CrossRef]

1987 (2)

V. A. Vysloukh and N. A. Sukhotskova, “Influence of third-order dispersion on the generation of a train of picosecond pulses in fiber waveguides due to self-modulation instability,” Sov. J. Quantum Electron. 17, 1509–1511 (1987).
[CrossRef]

M. J. Potosek, “Modulation instability in an extended non-linear Schrödinger equation,” Opt. Lett. 12, 921–923 (1987).
[CrossRef]

1966 (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310 (1966).

Abbi, S. C.

N. C. Kothari and S. C. Abbi, “Instability growth and filamentation of very intense laser beams in self-focusing media,” Prog. Theor. Phys. 83, 414–442 (1990).
[CrossRef]

Abdullaev, F. Kh.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bsichoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
[CrossRef]

Agrawal, G. P.

N. M. Litchinitser, C. J. McKinstrie, C. Martijn de Sterke, and G. P. Agrawal, “Spatiotemporal instabilities in nonlinear bulk media with Bragg gratings,” J. Opt. Soc. Am. B 18, 45–54 (2001).
[CrossRef]

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Modulational instabilities in dispersion-flattened fibers,” Phys. Rev. E 52, 1072–1080 (1995).
[CrossRef]

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[CrossRef] [PubMed]

Anderson, D.

D. Anderson, M. Karlsson, M. Lisak, and A. Sergeev, “Modulational instability dynamics in a spatial focusing and temporal defocusing medium,” Phys. Rev. E 47, 3617–3622 (1993).
[CrossRef]

Baboiu, D.-M.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Band, Y. B.

M. Trippenbach and Y. B. Band, “Effects of self-steepening and self-frequency shifting on short-pulse splitting in dispersive nonlinear media,” Phys. Rev. A 57, 4791–4803 (1998).
[CrossRef]

Beckwitt, K.

Xiang Liu, K. Beckwitt, and F. Wise, “Transverse instability of optical spatiotemporal solitons in quadratic media,” Phys. Rev. Lett. 85, 1871–1874 (2000).
[CrossRef] [PubMed]

Bespalov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310 (1966).

Brabec, T.

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[CrossRef]

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

Bsichoff, S.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bsichoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
[CrossRef]

Cao, X. D.

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[CrossRef] [PubMed]

Cavalcanti, S. B.

E. J. Fonseca, S. B. Cavalcanti, and J. M. Hickmann, “Space-time break-up in the self-focusing of ultrashort pulses,” Opt. Commun. 169, 199–205 (1999).
[CrossRef]

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef] [PubMed]

Christiansen, P. L.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bsichoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
[CrossRef]

Clement, T. S.

A. A. Zozulya, S. A. Diddams, and T. S. Clement, “Investigations of nonlinear femtosecond pulse propagation with the inclusion of Raman, shock, and third-order phase effects,” Phys. Rev. A 58, 3303–3310 (1998).
[CrossRef]

Cressoni, J. C.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef] [PubMed]

da Cruz, H. R.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef] [PubMed]

Darmanyan, S. A.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bsichoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
[CrossRef]

de Sterke, C. Martijn

Diddams, S. A.

A. A. Zozulya, S. A. Diddams, and T. S. Clement, “Investigations of nonlinear femtosecond pulse propagation with the inclusion of Raman, shock, and third-order phase effects,” Phys. Rev. A 58, 3303–3310 (1998).
[CrossRef]

Fonseca, E. J.

E. J. Fonseca, S. B. Cavalcanti, and J. M. Hickmann, “Space-time break-up in the self-focusing of ultrashort pulses,” Opt. Commun. 169, 199–205 (1999).
[CrossRef]

Fuerst, R. A.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Gaeta, A. L.

J. K. Ranka and A. L. Gaeta, “Breakdown of the slowly varying envelope approximation in the self-focusing of ultrashort pulses,” Opt. Lett. 23, 534–536 (1998).
[CrossRef]

J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Observation of pulse splitting in nonlinear dispersive media,” Phys. Rev. Lett. 77, 3783–3786 (1996).
[CrossRef] [PubMed]

Gouveia-Neto, A. S.

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef] [PubMed]

Hickmann, J. M.

E. J. Fonseca, S. B. Cavalcanti, and J. M. Hickmann, “Space-time break-up in the self-focusing of ultrashort pulses,” Opt. Commun. 169, 199–205 (1999).
[CrossRef]

Höök, A.

Karlsson, M.

A. Höök and M. Karlsson, “Ultrashort solitons at the minimum-dispersion wavelength: effects of fourth-order dispersion,” Opt. Lett. 18, 1388–1390 (1993).
[CrossRef] [PubMed]

D. Anderson, M. Karlsson, M. Lisak, and A. Sergeev, “Modulational instability dynamics in a spatial focusing and temporal defocusing medium,” Phys. Rev. E 47, 3617–3622 (1993).
[CrossRef]

Kothari, N. C.

N. C. Kothari and S. C. Abbi, “Instability growth and filamentation of very intense laser beams in self-focusing media,” Prog. Theor. Phys. 83, 414–442 (1990).
[CrossRef]

Krausz, F.

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[CrossRef]

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

Lawrence, B.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Liou, L. W.

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[CrossRef] [PubMed]

Lisak, M.

D. Anderson, M. Karlsson, M. Lisak, and A. Sergeev, “Modulational instability dynamics in a spatial focusing and temporal defocusing medium,” Phys. Rev. E 47, 3617–3622 (1993).
[CrossRef]

Litchinitser, N. M.

Liu, Xiang

Xiang Liu, K. Beckwitt, and F. Wise, “Transverse instability of optical spatiotemporal solitons in quadratic media,” Phys. Rev. Lett. 85, 1871–1874 (2000).
[CrossRef] [PubMed]

McKinstrie, C. J.

N. M. Litchinitser, C. J. McKinstrie, C. Martijn de Sterke, and G. P. Agrawal, “Spatiotemporal instabilities in nonlinear bulk media with Bragg gratings,” J. Opt. Soc. Am. B 18, 45–54 (2001).
[CrossRef]

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Modulational instabilities in dispersion-flattened fibers,” Phys. Rev. E 52, 1072–1080 (1995).
[CrossRef]

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[CrossRef] [PubMed]

Potosek, M. J.

Ranka, J. K.

J. K. Ranka and A. L. Gaeta, “Breakdown of the slowly varying envelope approximation in the self-focusing of ultrashort pulses,” Opt. Lett. 23, 534–536 (1998).
[CrossRef]

J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Observation of pulse splitting in nonlinear dispersive media,” Phys. Rev. Lett. 77, 3783–3786 (1996).
[CrossRef] [PubMed]

Schirmer, R. W.

J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Observation of pulse splitting in nonlinear dispersive media,” Phys. Rev. Lett. 77, 3783–3786 (1996).
[CrossRef] [PubMed]

Sergeev, A.

D. Anderson, M. Karlsson, M. Lisak, and A. Sergeev, “Modulational instability dynamics in a spatial focusing and temporal defocusing medium,” Phys. Rev. E 47, 3617–3622 (1993).
[CrossRef]

Sørensen, M. P.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bsichoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
[CrossRef]

Stegeman, G. I.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Sukhotskova, N. A.

V. A. Vysloukh and N. A. Sukhotskova, “Influence of third-order dispersion on the generation of a train of picosecond pulses in fiber waveguides due to self-modulation instability,” Sov. J. Quantum Electron. 17, 1509–1511 (1987).
[CrossRef]

Talanov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310 (1966).

Torruellas, W. E.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Trillo, S.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Trippenbach, M.

M. Trippenbach and Y. B. Band, “Effects of self-steepening and self-frequency shifting on short-pulse splitting in dispersive nonlinear media,” Phys. Rev. A 57, 4791–4803 (1998).
[CrossRef]

Vysloukh, V. A.

V. A. Vysloukh and N. A. Sukhotskova, “Influence of third-order dispersion on the generation of a train of picosecond pulses in fiber waveguides due to self-modulation instability,” Sov. J. Quantum Electron. 17, 1509–1511 (1987).
[CrossRef]

Wabnitz, S.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Wise, F.

Xiang Liu, K. Beckwitt, and F. Wise, “Transverse instability of optical spatiotemporal solitons in quadratic media,” Phys. Rev. Lett. 85, 1871–1874 (2000).
[CrossRef] [PubMed]

Yu, M.

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Modulational instabilities in dispersion-flattened fibers,” Phys. Rev. E 52, 1072–1080 (1995).
[CrossRef]

Zozulya, A. A.

A. A. Zozulya, S. A. Diddams, and T. S. Clement, “Investigations of nonlinear femtosecond pulse propagation with the inclusion of Raman, shock, and third-order phase effects,” Phys. Rev. A 58, 3303–3310 (1998).
[CrossRef]

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

JETP Lett. (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310 (1966).

Opt. Commun. (2)

E. J. Fonseca, S. B. Cavalcanti, and J. M. Hickmann, “Space-time break-up in the self-focusing of ultrashort pulses,” Opt. Commun. 169, 199–205 (1999).
[CrossRef]

F. Kh. Abdullaev, S. A. Darmanyan, S. Bsichoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (4)

A. A. Zozulya, S. A. Diddams, and T. S. Clement, “Investigations of nonlinear femtosecond pulse propagation with the inclusion of Raman, shock, and third-order phase effects,” Phys. Rev. A 58, 3303–3310 (1998).
[CrossRef]

M. Trippenbach and Y. B. Band, “Effects of self-steepening and self-frequency shifting on short-pulse splitting in dispersive nonlinear media,” Phys. Rev. A 57, 4791–4803 (1998).
[CrossRef]

S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, “Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation,” Phys. Rev. A 43, 6162–6165 (1991).
[CrossRef] [PubMed]

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202–4208 (1992).
[CrossRef] [PubMed]

Phys. Rev. E (2)

D. Anderson, M. Karlsson, M. Lisak, and A. Sergeev, “Modulational instability dynamics in a spatial focusing and temporal defocusing medium,” Phys. Rev. E 47, 3617–3622 (1993).
[CrossRef]

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Modulational instabilities in dispersion-flattened fibers,” Phys. Rev. E 52, 1072–1080 (1995).
[CrossRef]

Phys. Rev. Lett. (4)

Xiang Liu, K. Beckwitt, and F. Wise, “Transverse instability of optical spatiotemporal solitons in quadratic media,” Phys. Rev. Lett. 85, 1871–1874 (2000).
[CrossRef] [PubMed]

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear medium,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, “Observation of pulse splitting in nonlinear dispersive media,” Phys. Rev. Lett. 77, 3783–3786 (1996).
[CrossRef] [PubMed]

Prog. Theor. Phys. (1)

N. C. Kothari and S. C. Abbi, “Instability growth and filamentation of very intense laser beams in self-focusing media,” Prog. Theor. Phys. 83, 414–442 (1990).
[CrossRef]

Rev. Mod. Phys. (1)

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[CrossRef]

Sov. J. Quantum Electron. (1)

V. A. Vysloukh and N. A. Sukhotskova, “Influence of third-order dispersion on the generation of a train of picosecond pulses in fiber waveguides due to self-modulation instability,” Sov. J. Quantum Electron. 17, 1509–1511 (1987).
[CrossRef]

Other (1)

See, for example, G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).

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Figures (5)

Fig. 1
Fig. 1

Spatiotemporal instability regions in the (K, Ω) plane in the normal GVD regime for various combinations of the signs of the nonlinear refractive coefficient and FOD. The left- and right-hand columns correspond, respectively, to self-focusing and self-defocusing media, and the first, second, and third rows correspond, respectively, to no FOD, positive FOD (PFOD), and negative FOD (NFOD). Solid curves enclosing the instability region are contours of zero gain. The dotted curve in each case indicates the contour of maximum gain. The dashed lines in (c) and (f) show the spatial frequency thresholds.

Fig. 2
Fig. 2

Same as in Fig. 1 except that the GVD is anomalous.

Fig. 3
Fig. 3

Field intensity distribution as a function of both the transverse coordinate and time at a normalized propagation distance Z=1.6 with the initial seeding at K=8.94 and Ω=12.2.

Fig. 4
Fig. 4

Temporal and spectral distribution of the field intensity at a normalized propagation distance Z=0.8 in the self-focusing medium with anomalous GVD and positive FOD with the initial seeding at K=0, Ω1=3.36, and Ω2=9.4. The dotted curve shows the input intensity distribution.

Fig. 5
Fig. 5

Same as in Fig. 4 except the initial seeding is at K=0.4, Ω1=2.713, and Ω2=9.63.

Equations (22)

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β(ω)=β0+m=1 βmm!(ω-ω0)m,
ξA=12β02A+iDˆA+iβ0 n2n0(|A|2A),
zU=i2T2U+iD˜U+iμNnl(|U|2U),
U(X, Y, Z, T)=U0[1+a(X, Y, Z, T)]exp(iI0Z),
Zu=-12T2v-j=1(-1)j(δ2jT2jv+δ2j+1T2j+1u),
Zv=12T2u+j=1(-1)j(δ2jT2ju-δ2j+1T2j+1v)+2I0u.
u˜(KX, KY, Ω, Z)=-u(X, Y, T, Z)exp[i(KXX+KYY+ΩT)]dXdYdT,
v˜(KX, KY, Ω, Z)=-v(X, Y, T, Z)exp[i(KXX+KYY+ΩT)]dXdYdT,
zu˜zv˜=m11m12m21m22u˜v˜,
g2=-G(G-μΩc2/2),
G=K2/2-σΩ2/2-j=2δ2jΩ2j,
Km2-σΩm2-2j=2δ2jΩm2j=12μΩc2,gmax=Ωc24.
G(G-μΩc2/2)<0.
0<G<Ωc2/2,
-Ωc2/2<G<0.
j=12N(Ω2-Ωj2)=0,
kth=1/8|δ4|Ωc2
kth=(1-8|δ4|Ωc2)/(8|δ4|Ωc2)
1+1+8|δ4|Ωc2(k2-1)4|δ4|Ωc2
<w2<1+1+8|δ4|Ωc2k24|δ4|Ωc2,
U(X, T, Z=0)=U0[1+0.05 sin(KX)]×[1+0.01 sin(ΩT)].
U(X, T, Z=0)=U0[1+0.05 sin(KX)][1+0.01 sin(Ω1T)+0.01 sin(Ω2T)].

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