Abstract

The interaction of optical beams in bidispersive nonlinear media with self-focusing nonlinearity is numerically investigated. Under proper conditions, the interaction of spatially separated beams can lead to the creation of two prominent filaments along the other spatial or temporal dimension and, thus, to an effective spatiospatial or spatiotemporal exchange. This X-wave generation-based effect could potentially be exploited in optical realizations of spatial or spatiotemporal filtering.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. J. Rasmussen and K. Rypdal, “Blow-up in nonlinear Schrödinger equations. I. A general review,” Phys. Scr. 33, 481-497 (1986).
    [CrossRef]
  2. L. Bergé, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303, 259-370 (1998).
    [CrossRef]
  3. Y. Kivshar and D. E. Pelinovsky, “Self-focusing and transverse instabilities of solitary waves,” Phys. Rep. 331, 117-194 (2000).
    [CrossRef]
  4. P. Chernev and V. Petrov, “Self-focusing of light pulses in the presence of normal group-velocity dispersion,” Opt. Lett. 17, 172-174 (1992).
    [CrossRef] [PubMed]
  5. G. G. Luther, J. V. Moloney, and A. C. Newell, “Self-focusing threshold in normally dispersive media,” Opt. Lett. 19, 862-864 (1994).
    [CrossRef] [PubMed]
  6. G. G. Luther, J. V. Moloney, A. C. Newell, and E. M. Wright, “Short-pulse conical emission and spectral broadening in normally dispersive media,” Opt. Lett. 19, 789-791 (1994).
    [CrossRef] [PubMed]
  7. 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]
  8. A. G. Litvak, V. A. Mironov, and E. M. Sher, “Regime of wave-packet self-action with normal dispersion of the group velocity,” Phys. Rev. E 61, 891-893 (2000).
    [CrossRef]
  9. 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]
  10. C. Conti “X-wave mediated instability of plane waves in Kerr media,” Phys. Rev. E 68, 016606 (2003).
    [CrossRef]
  11. M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic nonlinear X waves for femtosecond pulse propagation in water,” Phys. Rev. Lett. 92, 253901 (2004).
    [CrossRef] [PubMed]
  12. D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. 96, 193901 (2006).
    [CrossRef] [PubMed]
  13. P. Di Trapani, G. Valiulis, A. Piskarkas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
    [CrossRef] [PubMed]
  14. C. Conti, “Generation and nonlinear dynamics of X waves of the Schrödinger equation,” Phys. Rev. E 70, 046613 (2004).
    [CrossRef]
  15. Y. Kominis, N. Moshonas, P. Papagiannis, K. Hizanidis, and D. N. Christodoulides, “Continuous wave controlled nonlinear X-wave generation,” Opt. Lett. 30, 2924-2926 (2005).
    [CrossRef] [PubMed]
  16. D. N. Christodoulides, N. K. Efremidis, P. Di Trapani, and B. A. Malomed, “Bessel X waves in two- and three-dimensional bidispersive optical systems,” Opt. Lett. 29, 1446-1448 (2004).
    [CrossRef] [PubMed]
  17. N. N. Akhmediev and S. Wabnitz, “Phase detecting of solitons by mixing with continuous wave background in an optical fiber,” J. Opt. Soc. Am. B 9, 236-242 (1992).
    [CrossRef]
  18. Y. Kominis and K. Hizanidis, “Solitary wave interactions with continuous waves,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 16, 1753-1764 (2006).
    [CrossRef]
  19. L. Bergé, M. R. Schmidt, J. J. Rasmussen, P. L. Christiansen, and K. Ø. Rasmussen, “Amalgamation of interacting light beamlets in Kerr-type media,” J. Opt. Soc. Am. B 14, 2550-2562 (1997).
    [CrossRef]
  20. G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518-1523 (1999).
    [CrossRef] [PubMed]
  21. L. Bergé, K. Germaschewski, R. Grauer, and J. J. Rasmussen, “Hyperbolic shock waves of the optical self-focusing with normal group-velocity dispersion,” Phys. Rev. Lett. 89, 153902 (2002).
    [CrossRef] [PubMed]

2006

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

Y. Kominis and K. Hizanidis, “Solitary wave interactions with continuous waves,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 16, 1753-1764 (2006).
[CrossRef]

2005

2004

D. N. Christodoulides, N. K. Efremidis, P. Di Trapani, and B. A. Malomed, “Bessel X waves in two- and three-dimensional bidispersive optical systems,” Opt. Lett. 29, 1446-1448 (2004).
[CrossRef] [PubMed]

M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic nonlinear X waves for femtosecond pulse propagation in water,” Phys. Rev. Lett. 92, 253901 (2004).
[CrossRef] [PubMed]

C. Conti, “Generation and nonlinear dynamics of X waves of the Schrödinger equation,” Phys. Rev. E 70, 046613 (2004).
[CrossRef]

2003

P. Di Trapani, G. Valiulis, A. Piskarkas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef] [PubMed]

C. Conti “X-wave mediated instability of plane waves in Kerr media,” Phys. Rev. E 68, 016606 (2003).
[CrossRef]

2002

L. Bergé, K. Germaschewski, R. Grauer, and J. J. Rasmussen, “Hyperbolic shock waves of the optical self-focusing with normal group-velocity dispersion,” Phys. Rev. Lett. 89, 153902 (2002).
[CrossRef] [PubMed]

2000

A. G. Litvak, V. A. Mironov, and E. M. Sher, “Regime of wave-packet self-action with normal dispersion of the group velocity,” Phys. Rev. E 61, 891-893 (2000).
[CrossRef]

Y. Kivshar and D. E. Pelinovsky, “Self-focusing and transverse instabilities of solitary waves,” Phys. Rep. 331, 117-194 (2000).
[CrossRef]

1999

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518-1523 (1999).
[CrossRef] [PubMed]

1998

L. Bergé, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303, 259-370 (1998).
[CrossRef]

1997

1996

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]

1994

1992

1986

J. J. Rasmussen and K. Rypdal, “Blow-up in nonlinear Schrödinger equations. I. A general review,” Phys. Scr. 33, 481-497 (1986).
[CrossRef]

Agrawal, G. P.

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]

Akhmediev, N. N.

Bergé, L.

L. Bergé, K. Germaschewski, R. Grauer, and J. J. Rasmussen, “Hyperbolic shock waves of the optical self-focusing with normal group-velocity dispersion,” Phys. Rev. Lett. 89, 153902 (2002).
[CrossRef] [PubMed]

L. Bergé, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303, 259-370 (1998).
[CrossRef]

L. Bergé, M. R. Schmidt, J. J. Rasmussen, P. L. Christiansen, and K. Ø. Rasmussen, “Amalgamation of interacting light beamlets in Kerr-type media,” J. Opt. Soc. Am. B 14, 2550-2562 (1997).
[CrossRef]

Bragheri, F.

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

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]

Chernev, P.

Christiansen, P. L.

Christodoulides, D. N.

Conti, C.

C. Conti, “Generation and nonlinear dynamics of X waves of the Schrödinger equation,” Phys. Rev. E 70, 046613 (2004).
[CrossRef]

P. Di Trapani, G. Valiulis, A. Piskarkas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef] [PubMed]

C. Conti “X-wave mediated instability of plane waves in Kerr media,” Phys. Rev. E 68, 016606 (2003).
[CrossRef]

Couairon, A.

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

Di Trapani, P.

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

D. N. Christodoulides, N. K. Efremidis, P. Di Trapani, and B. A. Malomed, “Bessel X waves in two- and three-dimensional bidispersive optical systems,” Opt. Lett. 29, 1446-1448 (2004).
[CrossRef] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarkas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef] [PubMed]

Dubietis, A.

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

Efremidis, N. K.

Faccio, D.

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

Gaeta, A. L.

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]

Germaschewski, K.

L. Bergé, K. Germaschewski, R. Grauer, and J. J. Rasmussen, “Hyperbolic shock waves of the optical self-focusing with normal group-velocity dispersion,” Phys. Rev. Lett. 89, 153902 (2002).
[CrossRef] [PubMed]

Grauer, R.

L. Bergé, K. Germaschewski, R. Grauer, and J. J. Rasmussen, “Hyperbolic shock waves of the optical self-focusing with normal group-velocity dispersion,” Phys. Rev. Lett. 89, 153902 (2002).
[CrossRef] [PubMed]

Hizanidis, K.

Y. Kominis and K. Hizanidis, “Solitary wave interactions with continuous waves,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 16, 1753-1764 (2006).
[CrossRef]

Y. Kominis, N. Moshonas, P. Papagiannis, K. Hizanidis, and D. N. Christodoulides, “Continuous wave controlled nonlinear X-wave generation,” Opt. Lett. 30, 2924-2926 (2005).
[CrossRef] [PubMed]

Jedrkiewicz, O.

P. Di Trapani, G. Valiulis, A. Piskarkas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef] [PubMed]

Kivshar, Y.

Y. Kivshar and D. E. Pelinovsky, “Self-focusing and transverse instabilities of solitary waves,” Phys. Rep. 331, 117-194 (2000).
[CrossRef]

Kolesik, M.

M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic nonlinear X waves for femtosecond pulse propagation in water,” Phys. Rev. Lett. 92, 253901 (2004).
[CrossRef] [PubMed]

Kominis, Y.

Y. Kominis and K. Hizanidis, “Solitary wave interactions with continuous waves,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 16, 1753-1764 (2006).
[CrossRef]

Y. Kominis, N. Moshonas, P. Papagiannis, K. Hizanidis, and D. N. Christodoulides, “Continuous wave controlled nonlinear X-wave generation,” Opt. Lett. 30, 2924-2926 (2005).
[CrossRef] [PubMed]

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]

Litvak, A. G.

A. G. Litvak, V. A. Mironov, and E. M. Sher, “Regime of wave-packet self-action with normal dispersion of the group velocity,” Phys. Rev. E 61, 891-893 (2000).
[CrossRef]

Luther, G. G.

Malomed, B. A.

McKinstrie, C. J.

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]

Mironov, V. A.

A. G. Litvak, V. A. Mironov, and E. M. Sher, “Regime of wave-packet self-action with normal dispersion of the group velocity,” Phys. Rev. E 61, 891-893 (2000).
[CrossRef]

Moloney, J. V.

Moshonas, N.

Newell, A. C.

Papagiannis, P.

Pelinovsky, D. E.

Y. Kivshar and D. E. Pelinovsky, “Self-focusing and transverse instabilities of solitary waves,” Phys. Rep. 331, 117-194 (2000).
[CrossRef]

Petrov, V.

Piskarkas, A.

P. Di Trapani, G. Valiulis, A. Piskarkas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef] [PubMed]

Porras, M. A.

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

Ranka, J. K.

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]

Rasmussen, J. J.

L. Bergé, K. Germaschewski, R. Grauer, and J. J. Rasmussen, “Hyperbolic shock waves of the optical self-focusing with normal group-velocity dispersion,” Phys. Rev. Lett. 89, 153902 (2002).
[CrossRef] [PubMed]

L. Bergé, M. R. Schmidt, J. J. Rasmussen, P. L. Christiansen, and K. Ø. Rasmussen, “Amalgamation of interacting light beamlets in Kerr-type media,” J. Opt. Soc. Am. B 14, 2550-2562 (1997).
[CrossRef]

J. J. Rasmussen and K. Rypdal, “Blow-up in nonlinear Schrödinger equations. I. A general review,” Phys. Scr. 33, 481-497 (1986).
[CrossRef]

Rasmussen, K. Ø.

Rypdal, K.

J. J. Rasmussen and K. Rypdal, “Blow-up in nonlinear Schrödinger equations. I. A general review,” Phys. Scr. 33, 481-497 (1986).
[CrossRef]

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]

Schmidt, M. R.

Segev, M.

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518-1523 (1999).
[CrossRef] [PubMed]

Sher, E. M.

A. G. Litvak, V. A. Mironov, and E. M. Sher, “Regime of wave-packet self-action with normal dispersion of the group velocity,” Phys. Rev. E 61, 891-893 (2000).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518-1523 (1999).
[CrossRef] [PubMed]

Trillo, S.

P. Di Trapani, G. Valiulis, A. Piskarkas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef] [PubMed]

Trull, J.

P. Di Trapani, G. Valiulis, A. Piskarkas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef] [PubMed]

Valiulis, G.

P. Di Trapani, G. Valiulis, A. Piskarkas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef] [PubMed]

Wabnitz, S.

Wright, E. M.

M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic nonlinear X waves for femtosecond pulse propagation in water,” Phys. Rev. Lett. 92, 253901 (2004).
[CrossRef] [PubMed]

G. G. Luther, J. V. Moloney, A. C. Newell, and E. M. Wright, “Short-pulse conical emission and spectral broadening in normally dispersive media,” Opt. Lett. 19, 789-791 (1994).
[CrossRef] [PubMed]

Int. J. Bifurcation Chaos Appl. Sci. Eng.

Y. Kominis and K. Hizanidis, “Solitary wave interactions with continuous waves,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 16, 1753-1764 (2006).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Phys. Rep.

L. Bergé, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303, 259-370 (1998).
[CrossRef]

Y. Kivshar and D. E. Pelinovsky, “Self-focusing and transverse instabilities of solitary waves,” Phys. Rep. 331, 117-194 (2000).
[CrossRef]

Phys. Rev. A

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

C. Conti “X-wave mediated instability of plane waves in Kerr media,” Phys. Rev. E 68, 016606 (2003).
[CrossRef]

A. G. Litvak, V. A. Mironov, and E. M. Sher, “Regime of wave-packet self-action with normal dispersion of the group velocity,” Phys. Rev. E 61, 891-893 (2000).
[CrossRef]

C. Conti, “Generation and nonlinear dynamics of X waves of the Schrödinger equation,” Phys. Rev. E 70, 046613 (2004).
[CrossRef]

Phys. Rev. Lett.

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]

M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic nonlinear X waves for femtosecond pulse propagation in water,” Phys. Rev. Lett. 92, 253901 (2004).
[CrossRef] [PubMed]

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. 96, 193901 (2006).
[CrossRef] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarkas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef] [PubMed]

L. Bergé, K. Germaschewski, R. Grauer, and J. J. Rasmussen, “Hyperbolic shock waves of the optical self-focusing with normal group-velocity dispersion,” Phys. Rev. Lett. 89, 153902 (2002).
[CrossRef] [PubMed]

Phys. Scr.

J. J. Rasmussen and K. Rypdal, “Blow-up in nonlinear Schrödinger equations. I. A general review,” Phys. Scr. 33, 481-497 (1986).
[CrossRef]

Science

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518-1523 (1999).
[CrossRef] [PubMed]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Isointensity plots at 80% of both the beams intensity. (a) Appearance of two prominent filaments on the temporal axis after the interaction of two spatially localized beams [the same case is also presented in Fig. 4i]; (b) two initially temporally localized beams with displacement Δ t = 4 lead the output main filaments on the temporal axis; (c) two light beams simultaneously incident at an angle at two spatially separated ports of a conceptual device lead to a temporal sequence of two light pulses emanating from the single center port at the end of the device.

Fig. 2
Fig. 2

Output intensity profile at z = 5 (15 mm), with a CW = 0.1 , φ 1 = φ 2 , φ = 0 , and Δ x = 4 . The input power was initially set at (a) N i = 0.9 N c , (b) N i = 1.85 N c , and (c) N i = 3 N c .

Fig. 3
Fig. 3

Output intensity profile at z = 5 (15 mm) for initial spatial separations: (a) Δ x = 3 , (b) Δ x = 4 , (c) Δ x = 5 . In all cases, a CW = 0.2 , φ 1 = φ 2 , φ = 0 , and N i = 1.5 N c .

Fig. 4
Fig. 4

Output intensity profile, at z = 5 , in the presence of a CW, with a CW = 0.2 , for various values of its phase ( φ ) and for various values of the wave packets initial transverse wavenumber difference ( Δ k x ) ; (a),(d),(g) φ = 0 , (b),(e),(h) φ = π / 2 , (c),(f),(i) φ = π ; (a),(b),(c) Δ k x = 0 , (d),(e),(f) Δ k x = 0.1 , (g),(h),(i) Δ k x = 0.2 . In all cases, initial values were set to N i = 2 N c , Δ x = 4 , and φ 1 = φ 2 .

Fig. 5
Fig. 5

(a)–(c) Spatial and (d)–(f) temporal intensity profiles during propagation for Δ k x = 0.2 , a CW = 0.2 , N i = 2 N c , φ 1 = φ 2 , Δ x = 4 , and (a),(d) φ = 0 , (b),(e) φ = π / 2 , (c),(f) φ = π .

Fig. 6
Fig. 6

(a) Movement of one of the two major filaments after the splitting and their clear emergence for various cases of initial wavenumber mismatch and CW phase values. Horizontal axis depicts the normalized propagation distance and vertical axis is the normalized time. Thick and thin full curves represent the cases of Figs. 4a, 4d, thick and thin dotted curves represent the cases of Figs. 4c, 4i, and the dashed curve represents the case of Fig. 4h, respectively. (b) Peak intensity evolution for the same cases.

Fig. 7
Fig. 7

Numerical investigation of (a) the output temporal displacement of the two prominent filaments versus initial spatial separation ( Δ x 0 ) and (b) their respective intensity, for various values of the initial individual beam mass (diamonds, N = 1.5 N c ; stars and black dots N = 2 N c ; squares N = 3 N c ), while a CW = 0.2 , φ = π , and Δ k x = 0.2 , except for the stars where Δ k x = 0 .

Fig. 8
Fig. 8

Frequency displacement Ω versus spatial dimension for the input ( z = 0 ) and output ( z = 5 ) at (a) and (b), respectively. The horizontal axis always represents the spatial ( x ) dimension and is measured in w 0 . The vertical axis is the frequency displacement and is in normalized units Ω 0 = 2 π / t 0 = 140   THz . The white line represents the CW.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

i u z + 2 u x 2 2 u t 2 + | u | 2 u = 0 ,
u 1 , 2 = A   exp [ ( x x 0 ) 2 + ( t t 0 ) 2 2 ± i Δ k x x + i ϕ 1 , 2 ] ,
u CW = A a CW   exp ( i φ ) ,
N i + Δ N i = N i [ 1 + 4 a CW   cos ( Δ ϕ i + Δ k x Δ x / 2 ) exp ( | Δ k x | 2 / 2 ) ] ,

Metrics