Abstract

We demonstrate the existence of stable optical light bullets in nonlinear dissipative media for both cases of normal and anomalous chromatic dispersion. The prediction is based on direct numerical simulations of the (3+1)-dimensional complex cubic-quintic Ginzburg-Landau equation. We do not impose conditions of spherical or cylindrical symmetry. Regions of existence of stable bullets are determined in the parameter space. Beyond the domain of parameters where stable bullets are found, unstable bullets can be transformed into “rockets” i.e. bullets elongated in the temporal domain. A few examples of the interaction between two optical bullets are considered using spatial and temporal interaction planes.

© 2006 Optical Society of America

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References

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  1. Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. 15, 1282 (1990).
    [CrossRef] [PubMed]
  2. B. A. Malomed, D. Mihalache, F. Wise, L. Torner, “Spatiotemporal solitons,” J. Opt. B. 7, R53 (2005).
    [CrossRef]
  3. N. N. Rosanov, “Spatial Hysteresis and Optical Patterns,” Springer, Berlin Heidelberg, 2002 (see section 6.6 and references therein);
  4. N. Rosanov, “Solitons in laser systems with absorption,” chapter in “Dissipative solitons,” N. Akhmediev, A. Ankiewicz eds., (Springer-Verlag, Berlin2005).
    [CrossRef]
  5. Ph. Grelu, J. M. Soto-Crespo, N. Akhmediev, “Light bullets and dynamic pattern formation in nonlinear dissipative systems,” Opt. Express 13, 9352–9630 (2005). http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-23-9352
    [CrossRef] [PubMed]
  6. C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, “Nonlinear Electromagnetic X Waves,” Phys. Rev. Lett., 90, 170406 (2003).
    [CrossRef] [PubMed]
  7. P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett., 91, 093904 (2003).
    [CrossRef] [PubMed]
  8. C. Zhou, M. Yu, X. He, “X-wave solutions of complex Ginzburg-Landau equations,” Phys. Rev. E 73, 026209 (2006).
    [CrossRef]
  9. M. Fermann, V. Kruglov, B. Thomsen, J. Dudley, J. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett., 84, 6010 (2000).
    [CrossRef] [PubMed]
  10. J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev E 55, 4783 (1997).
    [CrossRef]
  11. “Dissipative solitons,” edited by N. Akhmediev, A. Ankiewicz, (Springer-Verlag, Berlin2005).
  12. N. N. Akhmediev, A. Ankiewicz, “Solitons: nonlinear pulses and beams”, (Chapman & Hall, London, 1997).

2006 (1)

C. Zhou, M. Yu, X. He, “X-wave solutions of complex Ginzburg-Landau equations,” Phys. Rev. E 73, 026209 (2006).
[CrossRef]

2005 (2)

2003 (2)

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, “Nonlinear Electromagnetic X Waves,” Phys. Rev. Lett., 90, 170406 (2003).
[CrossRef] [PubMed]

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

2000 (1)

M. Fermann, V. Kruglov, B. Thomsen, J. Dudley, J. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett., 84, 6010 (2000).
[CrossRef] [PubMed]

1997 (1)

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev E 55, 4783 (1997).
[CrossRef]

1990 (1)

Afanasjev, V. V.

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev E 55, 4783 (1997).
[CrossRef]

Akhmediev, N.

Akhmediev, N. N.

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev E 55, 4783 (1997).
[CrossRef]

N. N. Akhmediev, A. Ankiewicz, “Solitons: nonlinear pulses and beams”, (Chapman & Hall, London, 1997).

Ankiewicz, A.

N. N. Akhmediev, A. Ankiewicz, “Solitons: nonlinear pulses and beams”, (Chapman & Hall, London, 1997).

Conti, C.

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

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, “Nonlinear Electromagnetic X Waves,” Phys. Rev. Lett., 90, 170406 (2003).
[CrossRef] [PubMed]

Di Trapani, P.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, “Nonlinear Electromagnetic X Waves,” Phys. Rev. Lett., 90, 170406 (2003).
[CrossRef] [PubMed]

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

Dudley, J.

M. Fermann, V. Kruglov, B. Thomsen, J. Dudley, J. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett., 84, 6010 (2000).
[CrossRef] [PubMed]

Fermann, M.

M. Fermann, V. Kruglov, B. Thomsen, J. Dudley, J. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett., 84, 6010 (2000).
[CrossRef] [PubMed]

Grelu, Ph.

Harvey, J.

M. Fermann, V. Kruglov, B. Thomsen, J. Dudley, J. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett., 84, 6010 (2000).
[CrossRef] [PubMed]

He, X.

C. Zhou, M. Yu, X. He, “X-wave solutions of complex Ginzburg-Landau equations,” Phys. Rev. E 73, 026209 (2006).
[CrossRef]

Jedrkiewicz, O.

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

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, “Nonlinear Electromagnetic X Waves,” Phys. Rev. Lett., 90, 170406 (2003).
[CrossRef] [PubMed]

Kruglov, V.

M. Fermann, V. Kruglov, B. Thomsen, J. Dudley, J. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett., 84, 6010 (2000).
[CrossRef] [PubMed]

Malomed, B. A.

B. A. Malomed, D. Mihalache, F. Wise, L. Torner, “Spatiotemporal solitons,” J. Opt. B. 7, R53 (2005).
[CrossRef]

Mihalache, D.

B. A. Malomed, D. Mihalache, F. Wise, L. Torner, “Spatiotemporal solitons,” J. Opt. B. 7, R53 (2005).
[CrossRef]

Piskarskas, A.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, “Nonlinear Electromagnetic X Waves,” Phys. Rev. Lett., 90, 170406 (2003).
[CrossRef] [PubMed]

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

Rosanov, N.

N. Rosanov, “Solitons in laser systems with absorption,” chapter in “Dissipative solitons,” N. Akhmediev, A. Ankiewicz eds., (Springer-Verlag, Berlin2005).
[CrossRef]

Rosanov, N. N.

N. N. Rosanov, “Spatial Hysteresis and Optical Patterns,” Springer, Berlin Heidelberg, 2002 (see section 6.6 and references therein);

Silberberg, Y.

Soto-Crespo, J. M.

Ph. Grelu, J. M. Soto-Crespo, N. Akhmediev, “Light bullets and dynamic pattern formation in nonlinear dissipative systems,” Opt. Express 13, 9352–9630 (2005). http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-23-9352
[CrossRef] [PubMed]

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev E 55, 4783 (1997).
[CrossRef]

Thomsen, B.

M. Fermann, V. Kruglov, B. Thomsen, J. Dudley, J. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett., 84, 6010 (2000).
[CrossRef] [PubMed]

Torner, L.

B. A. Malomed, D. Mihalache, F. Wise, L. Torner, “Spatiotemporal solitons,” J. Opt. B. 7, R53 (2005).
[CrossRef]

Trillo, S.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, “Nonlinear Electromagnetic X Waves,” Phys. Rev. Lett., 90, 170406 (2003).
[CrossRef] [PubMed]

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

Trull, J.

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

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, “Nonlinear Electromagnetic X Waves,” Phys. Rev. Lett., 90, 170406 (2003).
[CrossRef] [PubMed]

Valiulis, G.

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

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, “Nonlinear Electromagnetic X Waves,” Phys. Rev. Lett., 90, 170406 (2003).
[CrossRef] [PubMed]

Wabnitz, S.

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev E 55, 4783 (1997).
[CrossRef]

Wise, F.

B. A. Malomed, D. Mihalache, F. Wise, L. Torner, “Spatiotemporal solitons,” J. Opt. B. 7, R53 (2005).
[CrossRef]

Yu, M.

C. Zhou, M. Yu, X. He, “X-wave solutions of complex Ginzburg-Landau equations,” Phys. Rev. E 73, 026209 (2006).
[CrossRef]

Zhou, C.

C. Zhou, M. Yu, X. He, “X-wave solutions of complex Ginzburg-Landau equations,” Phys. Rev. E 73, 026209 (2006).
[CrossRef]

J. Opt. B. (1)

B. A. Malomed, D. Mihalache, F. Wise, L. Torner, “Spatiotemporal solitons,” J. Opt. B. 7, R53 (2005).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev E (1)

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev E 55, 4783 (1997).
[CrossRef]

Phys. Rev. E (1)

C. Zhou, M. Yu, X. He, “X-wave solutions of complex Ginzburg-Landau equations,” Phys. Rev. E 73, 026209 (2006).
[CrossRef]

Phys. Rev. Lett. (3)

M. Fermann, V. Kruglov, B. Thomsen, J. Dudley, J. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett., 84, 6010 (2000).
[CrossRef] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, “Nonlinear Electromagnetic X Waves,” Phys. Rev. Lett., 90, 170406 (2003).
[CrossRef] [PubMed]

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

Other (4)

N. N. Rosanov, “Spatial Hysteresis and Optical Patterns,” Springer, Berlin Heidelberg, 2002 (see section 6.6 and references therein);

N. Rosanov, “Solitons in laser systems with absorption,” chapter in “Dissipative solitons,” N. Akhmediev, A. Ankiewicz eds., (Springer-Verlag, Berlin2005).
[CrossRef]

“Dissipative solitons,” edited by N. Akhmediev, A. Ankiewicz, (Springer-Verlag, Berlin2005).

N. N. Akhmediev, A. Ankiewicz, “Solitons: nonlinear pulses and beams”, (Chapman & Hall, London, 1997).

Supplementary Material (1)

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Figures (11)

Fig. 1
Fig. 1

(a) A region of existence of optical bullets in (εD/β)-plane. (b) The same region of existence in the (ε,β/D)-plane. The latter representation clearly separates the regions for normal (red) and anomalous (blue) dispersions.

Fig. 2.
Fig. 2.

Evolution of the energy Q for the five sets of parameters indicated in Fig. 1(a) by thick dots of the same color.

Fig. 3.
Fig. 3.

Spatial (upper row) and temporal (lower row) profiles of stable optical bullets. The parameters used to make these plots correspond to the location of the thick dots of the corresponding color in Fig. 1(a).

Fig. 4.
Fig. 4.

Pulse evolution for the two sets of equation parameters where optical bullets are unstable. The instability that develops in these examples is in the time domain. The parameters are given in each plot.

Fig .5.
Fig .5.

(a) Growth of the energy Q along with propagation distance z for the elongating optical bullets. (b) This movie shows the process of elongation and the instability at the end of the process. The vertical axis in the movie is t while the horizontal axis is x (or y) [Media 1].

Fig. 6.
Fig. 6.

(a) Soliton profiles at t=0 for (a) z=30 and (b) z=25. The two spatial profiles are identical. The instability develops along the t axis only. (c) Field intensity integrated along (x,y) coordinates vs. z. We can see here the instability at the two ends of the rocket.

Fig. 7.
Fig. 7.

Evolution of the temporal profile of the “rocket” at (x,y)=(0,0).

Fig. 8.
Fig. 8.

(a) Trajectories in the spatial interaction plane showing the interaction of two bullets initially separated in space (along x) and with a certain relative phase difference. The initial condition is denoted by the black dots. ρR is the spatial separation between the maxima of the two bullets, while θ is phase difference between them.

Fig. 9.
Fig. 9.

(a) Trajectories in the temporal interaction plane showing the interaction of two bullets with a variety of initial separations and phase differences. The parameters of the CCGLE are given in the plot. (b) An example of the interaction of two optical bullets when they merge into one. This example corresponds to the blue trajectory in (a).

Fig. 10.
Fig. 10.

Two examples of collisions of optical bullets with non-zero velocity in space. In the case (a), the bullets repel each other and survive the collision. In case (b), only one of the bullets survives after the collision.

Fig. 11.
Fig. 11.

Evolution of a pair of optical bullets for the case of anomalous dispersion. The plot in (a) represents the spatial interaction plane while the plot in (b) is the temporal interaction plane. All simulations are done for the same set of equation parameters, as written in the figure.

Equations (6)

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

z + D 2 ψ tt + 1 2 ψ xx + 1 2 ψ yy + ψ 2 ψ + v ψ 4 ψ = iδψ + ψ 2 ψ + ψ tt + ψ 4 ψ .
Q ( z ) = ψ ( x , y , t , z ) 2 dtdxdy .
I t z = ψ ( x , y , t , z ) 2 dxdy
ψ ( x , y , t , z = 0 ) = ψ 0 ( x ρ R 2 , y , t ) + ψ 0 ( x + ρ R 2 , y , t ) e
ψ ( x , y , t , z = 0 ) = ψ 0 ( x , y , t ρ T 2 ) + ψ 0 ( x , y , t + ρ T 2 ) e ,
ψ ( x , y , t , z = 0 ) = ψ 0 ( x ρ R 2 , y , t ) e ikx + ψ 0 ( x + ρ R 2 , y , t ) e ikx + ,

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