Abstract

Nonlinear unbalanced O waves are localized wave packets that propagate without transversal diffraction broadening and temporal dispersion spreading in nonlinear media with anomalous dispersion. The conical nature of these nonsolitary light bullets allows for an energy refilling mechanism of the nonlinear central core of the wave that renders stationary propagation robust against nonlinear losses.

© 2005 Optical Society of America

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References

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  1. Y. Silberberg, "Collapse of optical pulses," Opt. Lett. 15, 1282-1284 (1990).
    [CrossRef] [PubMed]
  2. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).
  3. Y. F. Chen, K. Beckwitt, F. W. Wise, and B. A. Malomed, "Criteria for the experimental observation ofmultidimensional optical solitons in saturable media," Phys. Rev. E 70, 046610 (2004).
    [CrossRef]
  4. P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91, 093904 (2003); http://focus.aps.org/story/v12/st7.
    [CrossRef] [PubMed]
  5. O. Jedrkiewicz, J. Trull, G. Valiulis, A. Piskarskas, C. Conti, S. Trillo, and P. Di Trapani, "Nonlinear X waves in second-harmonic generation: Experimental results," Phys. Rev. E 68, 026610 (2003).
    [CrossRef]
  6. G. Valiulis, J. Kilius, O. Jedrkiewicz, A. Bramati, S. Minardi, C. Conti, S. Trillo, A. Piskarskas, and P. Di Trapani, in Quantum Electronics and Laser Science, Vol. 57 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001).
  7. C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, O. Jedrkiewicz, and J. Trull, "Nonlinear electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003).
    [CrossRef]
  8. M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, "Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses," Phys. Rev. Lett. 93, 153902 (2004).
    [CrossRef] [PubMed]
  9. J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, "Unified description of nondiffracting X and Y waves," Phys. Rev. E 62, 4261-4275 (2000).
    [CrossRef]
  10. S. Longhi, "Localized subluminal envelope pulses in dispersive media," Opt. Lett. 29, 147-149 (2004).
    [CrossRef] [PubMed]
  11. M. A. Porras and P. Di Trapani, "Localized and stationary light wave modes in dispersive media," Phys. Rev. E 69, 066606 (2004).
    [CrossRef]
  12. J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
    [CrossRef] [PubMed]
  13. P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, "Nonlinear Bessel beams," Opt. Commun. 222, 107-115 (2003).
    [CrossRef]
  14. R. Y. Chiao, E. Garmire, and C. H. Townes, "Trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
    [CrossRef]
  15. I. S. Gradsteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1965).

2004 (4)

Y. F. Chen, K. Beckwitt, F. W. Wise, and B. A. Malomed, "Criteria for the experimental observation ofmultidimensional optical solitons in saturable media," Phys. Rev. E 70, 046610 (2004).
[CrossRef]

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, "Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses," Phys. Rev. Lett. 93, 153902 (2004).
[CrossRef] [PubMed]

S. Longhi, "Localized subluminal envelope pulses in dispersive media," Opt. Lett. 29, 147-149 (2004).
[CrossRef] [PubMed]

M. A. Porras and P. Di Trapani, "Localized and stationary light wave modes in dispersive media," Phys. Rev. E 69, 066606 (2004).
[CrossRef]

2003 (4)

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91, 093904 (2003); http://focus.aps.org/story/v12/st7.
[CrossRef] [PubMed]

O. Jedrkiewicz, J. Trull, G. Valiulis, A. Piskarskas, C. Conti, S. Trillo, and P. Di Trapani, "Nonlinear X waves in second-harmonic generation: Experimental results," Phys. Rev. E 68, 026610 (2003).
[CrossRef]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, O. Jedrkiewicz, and J. Trull, "Nonlinear electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, "Nonlinear Bessel beams," Opt. Commun. 222, 107-115 (2003).
[CrossRef]

2000 (1)

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, "Unified description of nondiffracting X and Y waves," Phys. Rev. E 62, 4261-4275 (2000).
[CrossRef]

1990 (1)

1987 (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, "Trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

Agrawal, G. P.

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

Anderson, D.

P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, "Nonlinear Bessel beams," Opt. Commun. 222, 107-115 (2003).
[CrossRef]

Beckwitt, K.

Y. F. Chen, K. Beckwitt, F. W. Wise, and B. A. Malomed, "Criteria for the experimental observation ofmultidimensional optical solitons in saturable media," Phys. Rev. E 70, 046610 (2004).
[CrossRef]

Bramati, A.

G. Valiulis, J. Kilius, O. Jedrkiewicz, A. Bramati, S. Minardi, C. Conti, S. Trillo, A. Piskarskas, and P. Di Trapani, in Quantum Electronics and Laser Science, Vol. 57 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001).

Chen, Y. F.

Y. F. Chen, K. Beckwitt, F. W. Wise, and B. A. Malomed, "Criteria for the experimental observation ofmultidimensional optical solitons in saturable media," Phys. Rev. E 70, 046610 (2004).
[CrossRef]

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, "Trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

Conti, C.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91, 093904 (2003); http://focus.aps.org/story/v12/st7.
[CrossRef] [PubMed]

O. Jedrkiewicz, J. Trull, G. Valiulis, A. Piskarskas, C. Conti, S. Trillo, and P. Di Trapani, "Nonlinear X waves in second-harmonic generation: Experimental results," Phys. Rev. E 68, 026610 (2003).
[CrossRef]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, O. Jedrkiewicz, and J. Trull, "Nonlinear electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

G. Valiulis, J. Kilius, O. Jedrkiewicz, A. Bramati, S. Minardi, C. Conti, S. Trillo, A. Piskarskas, and P. Di Trapani, in Quantum Electronics and Laser Science, Vol. 57 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001).

Di Trapani, P.

M. A. Porras and P. Di Trapani, "Localized and stationary light wave modes in dispersive media," Phys. Rev. E 69, 066606 (2004).
[CrossRef]

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, "Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses," Phys. Rev. Lett. 93, 153902 (2004).
[CrossRef] [PubMed]

O. Jedrkiewicz, J. Trull, G. Valiulis, A. Piskarskas, C. Conti, S. Trillo, and P. Di Trapani, "Nonlinear X waves in second-harmonic generation: Experimental results," Phys. Rev. E 68, 026610 (2003).
[CrossRef]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, O. Jedrkiewicz, and J. Trull, "Nonlinear electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91, 093904 (2003); http://focus.aps.org/story/v12/st7.
[CrossRef] [PubMed]

G. Valiulis, J. Kilius, O. Jedrkiewicz, A. Bramati, S. Minardi, C. Conti, S. Trillo, A. Piskarskas, and P. Di Trapani, in Quantum Electronics and Laser Science, Vol. 57 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001).

Dubietis, A.

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, "Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses," Phys. Rev. Lett. 93, 153902 (2004).
[CrossRef] [PubMed]

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Faccio, D.

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, "Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses," Phys. Rev. Lett. 93, 153902 (2004).
[CrossRef] [PubMed]

Fagerholm, J.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, "Unified description of nondiffracting X and Y waves," Phys. Rev. E 62, 4261-4275 (2000).
[CrossRef]

Friberg, A. T.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, "Unified description of nondiffracting X and Y waves," Phys. Rev. E 62, 4261-4275 (2000).
[CrossRef]

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, "Trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

Gradsteyn, I. S.

I. S. Gradsteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1965).

Jedrkiewicz, O.

O. Jedrkiewicz, J. Trull, G. Valiulis, A. Piskarskas, C. Conti, S. Trillo, and P. Di Trapani, "Nonlinear X waves in second-harmonic generation: Experimental results," Phys. Rev. E 68, 026610 (2003).
[CrossRef]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91, 093904 (2003); http://focus.aps.org/story/v12/st7.
[CrossRef] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, O. Jedrkiewicz, and J. Trull, "Nonlinear electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

G. Valiulis, J. Kilius, O. Jedrkiewicz, A. Bramati, S. Minardi, C. Conti, S. Trillo, A. Piskarskas, and P. Di Trapani, in Quantum Electronics and Laser Science, Vol. 57 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001).

Johannisson, P.

P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, "Nonlinear Bessel beams," Opt. Commun. 222, 107-115 (2003).
[CrossRef]

Kilius, J.

G. Valiulis, J. Kilius, O. Jedrkiewicz, A. Bramati, S. Minardi, C. Conti, S. Trillo, A. Piskarskas, and P. Di Trapani, in Quantum Electronics and Laser Science, Vol. 57 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001).

Lisak, M.

P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, "Nonlinear Bessel beams," Opt. Commun. 222, 107-115 (2003).
[CrossRef]

Longhi, S.

Malomed, B. A.

Y. F. Chen, K. Beckwitt, F. W. Wise, and B. A. Malomed, "Criteria for the experimental observation ofmultidimensional optical solitons in saturable media," Phys. Rev. E 70, 046610 (2004).
[CrossRef]

Marklund, M.

P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, "Nonlinear Bessel beams," Opt. Commun. 222, 107-115 (2003).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Minardi, S.

G. Valiulis, J. Kilius, O. Jedrkiewicz, A. Bramati, S. Minardi, C. Conti, S. Trillo, A. Piskarskas, and P. Di Trapani, in Quantum Electronics and Laser Science, Vol. 57 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001).

Parola, A.

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, "Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses," Phys. Rev. Lett. 93, 153902 (2004).
[CrossRef] [PubMed]

Piskarskas, A.

O. Jedrkiewicz, J. Trull, G. Valiulis, A. Piskarskas, C. Conti, S. Trillo, and P. Di Trapani, "Nonlinear X waves in second-harmonic generation: Experimental results," Phys. Rev. E 68, 026610 (2003).
[CrossRef]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91, 093904 (2003); http://focus.aps.org/story/v12/st7.
[CrossRef] [PubMed]

G. Valiulis, J. Kilius, O. Jedrkiewicz, A. Bramati, S. Minardi, C. Conti, S. Trillo, A. Piskarskas, and P. Di Trapani, in Quantum Electronics and Laser Science, Vol. 57 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001).

Porras, M. A.

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, "Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses," Phys. Rev. Lett. 93, 153902 (2004).
[CrossRef] [PubMed]

M. A. Porras and P. Di Trapani, "Localized and stationary light wave modes in dispersive media," Phys. Rev. E 69, 066606 (2004).
[CrossRef]

Ryzhik, I. M.

I. S. Gradsteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1965).

Salo, J.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, "Unified description of nondiffracting X and Y waves," Phys. Rev. E 62, 4261-4275 (2000).
[CrossRef]

Salomaa, M. M.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, "Unified description of nondiffracting X and Y waves," Phys. Rev. E 62, 4261-4275 (2000).
[CrossRef]

Silberberg, Y.

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, "Trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

Trillo, S.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, O. Jedrkiewicz, and J. Trull, "Nonlinear electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91, 093904 (2003); http://focus.aps.org/story/v12/st7.
[CrossRef] [PubMed]

O. Jedrkiewicz, J. Trull, G. Valiulis, A. Piskarskas, C. Conti, S. Trillo, and P. Di Trapani, "Nonlinear X waves in second-harmonic generation: Experimental results," Phys. Rev. E 68, 026610 (2003).
[CrossRef]

G. Valiulis, J. Kilius, O. Jedrkiewicz, A. Bramati, S. Minardi, C. Conti, S. Trillo, A. Piskarskas, and P. Di Trapani, in Quantum Electronics and Laser Science, Vol. 57 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001).

Trull, J.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, O. Jedrkiewicz, and J. Trull, "Nonlinear electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

O. Jedrkiewicz, J. Trull, G. Valiulis, A. Piskarskas, C. Conti, S. Trillo, and P. Di Trapani, "Nonlinear X waves in second-harmonic generation: Experimental results," Phys. Rev. E 68, 026610 (2003).
[CrossRef]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91, 093904 (2003); http://focus.aps.org/story/v12/st7.
[CrossRef] [PubMed]

Valiulis, G.

O. Jedrkiewicz, J. Trull, G. Valiulis, A. Piskarskas, C. Conti, S. Trillo, and P. Di Trapani, "Nonlinear X waves in second-harmonic generation: Experimental results," Phys. Rev. E 68, 026610 (2003).
[CrossRef]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91, 093904 (2003); http://focus.aps.org/story/v12/st7.
[CrossRef] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, O. Jedrkiewicz, and J. Trull, "Nonlinear electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

G. Valiulis, J. Kilius, O. Jedrkiewicz, A. Bramati, S. Minardi, C. Conti, S. Trillo, A. Piskarskas, and P. Di Trapani, in Quantum Electronics and Laser Science, Vol. 57 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001).

Wise, F. W.

Y. F. Chen, K. Beckwitt, F. W. Wise, and B. A. Malomed, "Criteria for the experimental observation ofmultidimensional optical solitons in saturable media," Phys. Rev. E 70, 046610 (2004).
[CrossRef]

Opt. Commun. (1)

P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, "Nonlinear Bessel beams," Opt. Commun. 222, 107-115 (2003).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. E (4)

Y. F. Chen, K. Beckwitt, F. W. Wise, and B. A. Malomed, "Criteria for the experimental observation ofmultidimensional optical solitons in saturable media," Phys. Rev. E 70, 046610 (2004).
[CrossRef]

O. Jedrkiewicz, J. Trull, G. Valiulis, A. Piskarskas, C. Conti, S. Trillo, and P. Di Trapani, "Nonlinear X waves in second-harmonic generation: Experimental results," Phys. Rev. E 68, 026610 (2003).
[CrossRef]

M. A. Porras and P. Di Trapani, "Localized and stationary light wave modes in dispersive media," Phys. Rev. E 69, 066606 (2004).
[CrossRef]

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, "Unified description of nondiffracting X and Y waves," Phys. Rev. E 62, 4261-4275 (2000).
[CrossRef]

Phys. Rev. Lett. (5)

J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

R. Y. Chiao, E. Garmire, and C. H. Townes, "Trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964).
[CrossRef]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, O. Jedrkiewicz, and J. Trull, "Nonlinear electromagnetic X-waves," Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Di Trapani, "Nonlinear unbalanced Bessel beams: stationary conical waves supported by nonlinear losses," Phys. Rev. Lett. 93, 153902 (2004).
[CrossRef] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91, 093904 (2003); http://focus.aps.org/story/v12/st7.
[CrossRef] [PubMed]

Other (3)

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

G. Valiulis, J. Kilius, O. Jedrkiewicz, A. Bramati, S. Minardi, C. Conti, S. Trillo, A. Piskarskas, and P. Di Trapani, in Quantum Electronics and Laser Science, Vol. 57 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001).

I. S. Gradsteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1965).

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Figures (5)

Fig. 1
Fig. 1

(a) Gray-scale plot of the amplitude A of the linear O wave of Eq. (9) of axial wave vector shift δ in a linear medium with anomalous dispersion k 0 < 0 . The radial transversal coordinate is r = ( x 2 + y 2 ) 1 2 . (b) Spatiotemporal spectrum of the linear O wave is ideally a Dirac delta over a sphere of radius 2 k 0 δ . K = ( k x 2 + k y 2 ) is the radial coordinate in the transversal frequency plane, and Ω represents the detuning from the carrier frequency.

Fig. 2
Fig. 2

(a) Spatiotemporal radial profile of the amplitude A of the linear O wave of Eq. (9) and of the nonlinear O wave in self-focusing Kerr media ( n 2 > 0 ) with k 0 n 2 a 0 2 δ n 0 = 2 and its asymptotic fitting to the out-of-phase O wave of relation (16) with α exp ( i β ) = 0.674 exp ( 0.84 i ) . Amplitudes are normalized to a 0 . (b) Spatiotemporal spectral density A ̂ 2 , in arbitrary units, of the linear O wave of Eq. (9) and the nonlinear O wave in (a). In the evaluation of the spectra, the Dirac delta singularity in the spectral density was avoided by apodizing the radial profile with a broad Gaussian function. (c) Spatiotemporal radial profile of the amplitude A of the nonlinear O wave in self-defocusing Kerr media ( n 2 < 0 ) with k 0 n 2 a 0 2 δ n 0 = 0.85 and its asymptotic fitting to relation (16) with α exp ( i β ) = 1.805 exp ( 1.57 i ) . (d) Spatiotemporal spectral density A ̂ 2 , in arbitrary units, of the nonlinear O wave in (c).

Fig. 3
Fig. 3

Spatiotemporal (a) radial profile of the amplitude A (normalized to a 0 ) and (b) radial spectral density A ̂ 2 (in arbitrary units) of nonlinear O waves in media with four-photon absorption ( M = 4 ) and with increasing β ( M ) a 0 2 M 2 2 δ = 2 and 3 (lower and higher solid curves, respectively), and their asymptotic fittings to the unbalanced O waves of relation (23) (dotted–dashed curves), with α out = 0.51 and 0.25, α in = 1.68 and 2.40, and β = 0.06 π and 0.0225 π , all respectively. The dashed curves correspond to the linear O wave.

Fig. 4
Fig. 4

Boundary of the regions of existence of nonlinear O waves in media with Kerr self-focusing and self-defocusing nonlinearities and four-photon absorption ( M = 4 ) as given by the approximate Eq. (28) (solid curve) and numerically evaluated (filled circles) from Eqs. (29, 30). Each nonlinear O wave is defined by a pair γ = k 0 n 2 a 0 2 δ n 0 , η = β ( M ) a 0 2 M 2 2 δ .

Fig. 5
Fig. 5

(a) Spatiotemporal radial profile of the amplitude A of the nonlinear O wave with self-focusing Kerr strength γ = k 0 n 2 a 0 2 δ n 0 = 2 and four-photon absorption ( M = 4 ) and strength η = β ( M ) a 0 2 M 2 2 δ = 6 , and its asymptotic fitting to the unbalanced O wave of relation (23), with α out = 0.20 , α in = 1.39 , and β = 0.84 . The dashed curve is the linear O wave. Amplitudes are normalized to a 0 . (b) Spatiotemporal spectral density A ̂ 2 of the nonlinear O wave in (a), in arbitrary units. (c) Spatiotemporal radial profile of the amplitude A of the nonlinear O wave with self-defocusing Kerr strength γ = k 0 n 2 a 0 2 δ n 0 = 0.85 and four-photon absorption with η = β ( M ) a 0 2 M 2 2 δ = 0.7 , and its asymptotic fitting to the unbalanced O wave with α out = 1.56 , α in = 2.08 , and β = 1.60 . (d) Spatiotemporal spectral density A ̂ 2 , in arbitrary units, of the nonlinear O wave in (c).

Equations (40)

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

z A = i 2 k 0 2 A i k 0 2 τ 2 A + i f ( A 2 ) A g ( A 2 ) A ,
E = A ( x , y , z , τ ) exp ( i ω 0 t + i k 0 z ) ,
R = ( x 2 + y 2 + ξ 2 ) 1 2 = ( x 2 + y 2 τ 2 k 0 k 0 ) 1 2
z a 2 = 1 k 0 1 R 2 R ( R 2 R φ a 2 ) 2 g ( a 2 ) a 2 ,
z φ = 1 2 k 0 [ 1 R 2 a R ( R 2 R a ) ( R φ ) 2 ] + f ( a 2 ) .
φ ( R , z ) = φ ( R ) δ z ,
a + 2 a R + 2 k 0 δ a ( ϕ ) 2 a + 2 k 0 f ( a 2 ) a = 0 ,
( 1 R 2 ) ( R 2 ϕ a 2 ) = 2 k 0 g ( a 2 ) a 2 ,
a ( 0 ) = a 0 , a ( 0 ) = 0 , ϕ ( 0 ) = 0 , lim r a = 0 ,
A ( R , z ) = a 0 sin ( 2 k 0 δ R ) 2 k 0 δ R exp ( i δ z )
A ̂ ( k x , k y , Ω , z ) = d τ d x d y A ( x , y , τ , z ) exp [ i ( Ω τ k x x k y y ) ] ,
A ̂ ( K , z ) = 4 π k 0 k 0 0 d R R 2 A ( R , z ) sin K R K R
K = ( k x 2 + k y 2 k 0 k 0 Ω 2 ) 1 2
E = A ( R , z ) exp ( i ω 0 t + i k 0 z ) = k 0 8 δ 2 δ k 0 2 δ k 0 d Ω a 0 J 0 [ 2 k 0 ( δ + k 0 Ω 2 2 ) r ] exp [ i ( δ + k 0 Ω 2 2 ) z ] exp [ ( k 0 + k 0 Ω + k 0 Ω 2 2 ) z ] exp [ i ( ω 0 + Ω ) t ] ,
A = a 0 2 [ exp ( i 2 k 0 δ R ) i 2 k 0 δ R + exp ( i 2 k 0 δ R ) i 2 k 0 δ R ] × exp ( i δ z )
a + 2 a R + 2 k 0 δ a 0 ,
A a 0 2 { α exp [ i ( 2 k 0 δ R + β ) ] i 2 k 0 δ R + α exp [ i ( 2 k 0 δ R + β ) ] i 2 k 0 δ R } × exp ( i δ z ) ,
Δ R = β ( 2 k 0 δ ) 1 2
a + 2 a R + 2 k 0 [ δ + f ( a 0 2 ) ] a 0
a a 0 sin { 2 k 0 [ δ + f ( a 0 2 ) ] R } 2 k 0 [ δ + f ( a 0 2 ) ] R
δ > min { 0 , f ( a 0 2 ) }
4 π R 2 ( 1 k 0 ϕ a 2 ) = 0 R 2 g ( a 2 ) a 2 4 π R 2 d R N R ,
ϕ k 0 N 4 π R 2 a 2
a + 2 a R + 2 k 0 δ a ( k 0 N 4 π R 2 ) 2 1 a 3 0
b 2 k 0 δ b + ( k 0 N 4 π ) 2 1 b 3 ,
A a 0 2 { α out exp [ i ( 2 k 0 δ R + β ) ] i 2 k 0 δ R + α in exp [ i ( 2 k 0 δ R + β ) ] i 2 k 0 δ R } exp ( i δ z ) ,
π a 0 2 k 0 α in 2 α out 2 2 k 0 δ = N
C = 2 α in α out α in 2 + α out 2 .
δ > d M β ( M ) a 0 2 M 2 ,
d M = 0.312 , 0.176 , 0.137 , 0.116 , 0.102
a + 2 a R + 2 k 0 [ δ + f ( a 0 2 ) ] a 0 ,
a a 0 sin ( 2 k 0 [ δ + f ( a 0 2 ) ] R ) 2 k 0 [ δ + f ( a 0 2 ) ] R for R 0 .
δ > min { 0 , d M β ( M ) a 0 2 M 2 f ( a 0 2 ) }
a ̃ + 2 a ̃ ρ + a ̃ ( ϕ ) 2 a ̃ + γ a ̃ 3 = 0 ,
1 ρ 2 ( ρ 2 ϕ a ̃ 2 ) = η a ̃ 2 M ,
γ > 2 d M η 1 .
exp [ i 2 k 0 δ ( r 2 τ 2 k 0 k 0 ) ] i 2 k 0 δ ( r 2 τ 2 k 0 k 0 ) = k 0 8 δ 2 δ k 0 2 δ k 0 d Ω H 0 ( 1 ) [ 2 k 0 ( δ + k 0 Ω 2 2 ) r ] exp ( i τ Ω ) ,
exp [ i 2 k 0 δ ( r 2 τ 2 k 0 k 0 ) ] i 2 k 0 δ ( r 2 τ 2 k 0 k 0 ) = k 0 8 δ 2 δ k 0 2 δ k 0 d Ω H 0 ( 2 ) [ 2 k 0 ( δ + k 0 Ω 2 2 ) r ] exp ( i τ Ω ) ,
E = A ( R , z ) exp ( i ω 0 t + i k 0 z ) = k 0 8 δ 2 δ k 0 2 δ k 0 d Ω A UBB exp [ ( k 0 + k 0 Ω + k 0 Ω 2 2 ) z ] × exp [ i ( ω 0 + Ω ) t ] ,
A UBB = a 0 2 { α out exp ( i β ) H 0 ( 1 ) [ 2 k 0 ( δ + k 0 Ω 2 2 ) r ] + α in exp ( i β ) H 0 ( 2 ) [ 2 k 0 ( δ + k 0 Ω 2 2 ) r ] } × exp [ i ( δ + k 0 Ω 2 2 ) z ]

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