Abstract

Nonlinear evolution of femtosecond pulses in media with weak dispersion and power slightly above the critical for self-focusing in the framework of generalized non-paraxial amplitude equation is analyzed. It is found that this nonlinear non-paraxial regime strongly depends from the initial form of the pulses. In case of long pulse (small transverse and large longitudinal size), the dynamics is closer to nonlinear paraxial dynamics of a laser beam, and the difference consists in large spectral and longitudinal spatial modulation of the long pulse. The non-paraxial terms play an important role on the evolution of light bullets and light disks. In regime of light bullets (relatively equal transverse and longitudinal size) weak self-focusing without pedestal and collapse arrest is obtained. Non-collapsed regime of light disks (pulses with small longitudinal and large transverse size) is also observed. Our results are in good agreement with the recent experiments on nonlinear propagation of femtosecond pulses. For first time is demonstrated that such non-paraxial model can explain effects as spectral broadening, collapse arrest and nonlinear wave guide behavior.

© 2007 Optical Society of America

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References

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  1. G. Méchian, A. Couarion, Y. -B. Andre, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, A. Couarion, R. Sauerbrey, "Long range self-channeling of infrared laser pulse in air: a new propagation regime without ionization," Appl. Phys. B 79, 379 (2004).
    [CrossRef]
  2. N. N. Akhmediev, A. Ankiewicz and J. M. Soto-Crespo, "Does the nonlinear Schredinger Equation describe propagation in nonlinear waveguides?," Opt. Lett. 18, 411 (1993).
    [CrossRef] [PubMed]
  3. G. Fibich and G. C. Papanicolaou. "Self-focusing in the presence of small time dispersion and nonparaxiality," Opt. Lett. 22, 1379 (1997).
    [CrossRef]
  4. L. M. Kovachev, L. I. Pavlov, L. M. Ivanov and D. I. Dakova, "Optical filaments and optical bullets in dispersive nonlinear media," J. Russ. Laser Res. 27, 185 (2006).
    [CrossRef]
  5. D. N. Christodoulides and R. I. Joseph, "Exact radial dependence of the field in a nonlinear dispersive dielectric fiber: bright pulse solutions," Opt. Lett. 9, 229 (1984).
    [CrossRef] [PubMed]
  6. Th. Brabec and F. Krausz, "Nonlinear Optical Pulse Propagation in Single-Cycle Regime," Phys. Rev. Lett. 78, 3282 (1997).
    [CrossRef]
  7. N. N. Akhmediev and A. Ankiewicz, Solitons: nonlinear pulses and beams (Charmanand Hall, 1997).
  8. E. A. Golovchenko, E. M. Dianov, A. M. Prokhorrov, A. N. Pilipetsky, and V. N. Serkin, "Self-action and compression of femtosecond pulses in nonlinear dispersive media," JETP Lett. 45, 91 (1987).
  9. K. D. Moll, A. L. Gaeta and G. Fibich, "Self-similar optical wave collapse: observation of the Townes profile," Phys. Rev. Lett. 90, 203902 (2003).
    [CrossRef] [PubMed]
  10. S. A. Ponomarenko and G. P. Agrawal, "Linear optical bullets," Opt. Commun. 261, 1-4 (2006).
    [CrossRef]

2006

L. M. Kovachev, L. I. Pavlov, L. M. Ivanov and D. I. Dakova, "Optical filaments and optical bullets in dispersive nonlinear media," J. Russ. Laser Res. 27, 185 (2006).
[CrossRef]

S. A. Ponomarenko and G. P. Agrawal, "Linear optical bullets," Opt. Commun. 261, 1-4 (2006).
[CrossRef]

2004

G. Méchian, A. Couarion, Y. -B. Andre, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, A. Couarion, R. Sauerbrey, "Long range self-channeling of infrared laser pulse in air: a new propagation regime without ionization," Appl. Phys. B 79, 379 (2004).
[CrossRef]

2003

K. D. Moll, A. L. Gaeta and G. Fibich, "Self-similar optical wave collapse: observation of the Townes profile," Phys. Rev. Lett. 90, 203902 (2003).
[CrossRef] [PubMed]

1997

G. Fibich and G. C. Papanicolaou. "Self-focusing in the presence of small time dispersion and nonparaxiality," Opt. Lett. 22, 1379 (1997).
[CrossRef]

Th. Brabec and F. Krausz, "Nonlinear Optical Pulse Propagation in Single-Cycle Regime," Phys. Rev. Lett. 78, 3282 (1997).
[CrossRef]

1993

1987

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorrov, A. N. Pilipetsky, and V. N. Serkin, "Self-action and compression of femtosecond pulses in nonlinear dispersive media," JETP Lett. 45, 91 (1987).

1984

Appl. Phys. B

G. Méchian, A. Couarion, Y. -B. Andre, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, A. Couarion, R. Sauerbrey, "Long range self-channeling of infrared laser pulse in air: a new propagation regime without ionization," Appl. Phys. B 79, 379 (2004).
[CrossRef]

J. Russ. Laser Res.

L. M. Kovachev, L. I. Pavlov, L. M. Ivanov and D. I. Dakova, "Optical filaments and optical bullets in dispersive nonlinear media," J. Russ. Laser Res. 27, 185 (2006).
[CrossRef]

JETP Lett.

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorrov, A. N. Pilipetsky, and V. N. Serkin, "Self-action and compression of femtosecond pulses in nonlinear dispersive media," JETP Lett. 45, 91 (1987).

Opt. Commun.

S. A. Ponomarenko and G. P. Agrawal, "Linear optical bullets," Opt. Commun. 261, 1-4 (2006).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

Th. Brabec and F. Krausz, "Nonlinear Optical Pulse Propagation in Single-Cycle Regime," Phys. Rev. Lett. 78, 3282 (1997).
[CrossRef]

K. D. Moll, A. L. Gaeta and G. Fibich, "Self-similar optical wave collapse: observation of the Townes profile," Phys. Rev. Lett. 90, 203902 (2003).
[CrossRef] [PubMed]

Other

N. N. Akhmediev and A. Ankiewicz, Solitons: nonlinear pulses and beams (Charmanand Hall, 1997).

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Figures (5)

Fig. 1.
Fig. 1.

Nonlinear dynamics of the intensity profile of a Gaussian beam governed by 2D paraxial Eq. (6) under initial conditions ψ(x, y, t =0)=exp(-(x 2+y 2)/2), and γ=2.25. The beam is focused to a minimal diameter with large pedestal on distance 2–3zdiff .

Fig. 2.
Fig. 2.

View of the spot of long Gaussian pulse (surfaces | ψ(x, y, z =0, t =0; π/15; 2π/15; 3π/15) |2 (x, y plane, y-vertical, x-horizontal)) governed by nonparaxial Eq. (5) under initial conditions ψ(x, y, z, t =0)=exp(-(x 2+y 2+δ 2 z 2)/2), α=40; δ 2=1/81; γ=2.25. The self-focusing regime is similar to self-focusing of a laser beam. Compare with Fig. 1.

Fig. 3.
Fig. 3.

View of surfaces | ψ(x, y=0, z , t =0; π/15; 2π/15; 3π/15) |2 (x, z -plane, x-vertical, z -horizontal) at the same long Gaussian pulse governed by Eq. (5) and under the same initial conditions as on Fig. 2. Large spatial longitudinal modulation is observed.

Fig. 4.
Fig. 4.

Nonlinear evolution of a Gaussian light bullet governed by Eq. (5) under initial conditions ψ(x, y, z, t =0)=exp(-(x 2+y 2+δ 2 z ′2)/2), α=40; δ2=1; γ=2.25. The surfaces | ψ(x, y=0, z, t =0; 2π; 4π; 6π) |2 are plotted. Significant increase at the self-focusing distance z≃18zdiff and a collapse without pedestal (Townes profile) are observed.

Fig. 5.
Fig. 5.

Nonlinear evolution of a light disk governed by Eq. (5) under initial conditions ψ(x, y, z, t =0)=exp(-(x 2+y 2+2z ′2)/2),α=40; δ2=81; γ=2.25. The surfaces | ψ(x, y=0, z , t =0; 3π; 6π; 9π) |2 are presented (x-vertical, z -horizontal). The optical disk preserves its shape in the numerical experiment over a distance 28zdiff .

Equations (8)

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i [ A t + v A z + ( n 2 + k 0 v 2 n 2 ω ) ( A 2 A ) t ] =
v 2 k 0 Δ A v 2 ( k + 1 k 0 v 2 ) 2 A t 2 + k 0 v n 2 2 A 2 A ,
2 i α δ 2 ( ψ t + γ 1 ( ( ψ 2 ψ ) t ( ψ 2 ψ ) z ) ) = Δ ψ β 2 ψ z 2
( β + δ 2 ) ( 2 ψ t 2 2 2 ψ t z ) + γ ψ 2 ψ ,
α = k 0 z 0 ; δ 2 = r 2 z 0 2 ; β = z dif z disp ; γ = k 0 2 r 0 2 n 2 A 0 2 ;
γ 1 = n 2 A 0 2 ( n 2 + ( k 0 v 2 ) ( n 2 ω ) ) ; z dif = k 0 r 2 ; z disp = t 0 2 k .
2 i α δ 2 ψ t = Δ ψ δ 2 ( 2 ψ t 2 2 2 ψ t z ) + γ ψ 2 ψ .
2 i ψ t = Δ ψ + γ ψ 2 ψ .

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