## Dynamics of space-time self-focusing of a femtosecond relativistic laser pulse in an underdense plasma

Optics Express, Vol. 11, Issue 3, pp. 248-258 (2003)

http://dx.doi.org/10.1364/OE.11.000248

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### Abstract

The propagation of femtosecond, multiterawatt, relativistic laser pulses in a transparent plasma is studied. The spatio-temporal dynamics of ultrashort, high-power laser pulses in underdense plasmas differs dramatically from that of long laser beams. We present the results of numerical studies of these dynamics within a model which systematically incorporates finite pulse length effects (*i.e.*, dispersion) along with diffraction and nonlinear refraction in a strongly nonlinear, relativistic regime. New space-time patterns of self-compression, self-focusing and self-phase-modulation, typical of ultrashort, high-intensity laser pulses, are analyzed. The parameters of our numerical simulations correspond to a new class of high-peak-power (> 100 TW), ultrashort-pulsed laser systems, producing pulses with a duration in the 10 – 20 femtosecond range. Spatio-temporal dynamics of these self-effects and underlying physical mechanisms are discussed.

© 2002 Optical Society of America

## 1. Introduction

1. M. D. Perry and G. Mourou, “Terawatt to petawatt subpicosecond lasers,” Science **64**, 917–924 (1994). [CrossRef]

2. G. A. Mourou, C. P. J. Barty, and M. D. Perry, “Ultrahigh-Intensity Lasers: Physics of the Extreme on a Tabletop,” Phys. Today **51**, 22–28 (1998). [CrossRef]

3. K. Yamakawa, M. Aoyama, T. Kase, Y. Akahane, and H. Takuma, “100-TW sub-20-fs Ti:sapphire laser system operating at a 10-Hz repetition rate,” Opt. Lett. **23**, 1468–1470 (1998). [CrossRef]

4. K. Yamakawa and C. P. J. Barty, “Ultrafast, ultrahigh-peak, and high-average power Ti:sapphire laser system and its application”, IEEE J. Se. Top. Quantum Electron. **6**, 658–675 (2000). [CrossRef]

5. K. Yamakawa, Y. Akahane, M. Aoyama, Y. Fukuda, N. Inoue, J. Ma, and H. Ueda, “Status and future developments of ultrahigh intensity lasers at JAERI,” in *Superstrong Fields in Plasmas*, M. Lontano, G. Mourou, O. Svelto, and T. Tajima, eds., AIP Conference Proceedings, Vol.611 (American Institute of Physics, Melville, New York, 2002), pp. 385–396.

^{20}W/cm

^{2}range, and, in the near future, it is expected to increase by at least two orders of magnitude [6]. At these intensities, the electron motion in the laser fields becomes highly relativistic and nonlinear. These recent achievements are bringing the nonlinear optics of laser pulses in plasmas to a new physical regime of laser-plasma interactions, dominated by (i) strong relativistic nonlinearities and (ii) finite-pulse-length effects.

3. K. Yamakawa, M. Aoyama, T. Kase, Y. Akahane, and H. Takuma, “100-TW sub-20-fs Ti:sapphire laser system operating at a 10-Hz repetition rate,” Opt. Lett. **23**, 1468–1470 (1998). [CrossRef]

5. K. Yamakawa, Y. Akahane, M. Aoyama, Y. Fukuda, N. Inoue, J. Ma, and H. Ueda, “Status and future developments of ultrahigh intensity lasers at JAERI,” in *Superstrong Fields in Plasmas*, M. Lontano, G. Mourou, O. Svelto, and T. Tajima, eds., AIP Conference Proceedings, Vol.611 (American Institute of Physics, Melville, New York, 2002), pp. 385–396.

14. S. V. Bulanov, I. N. Inovenkov, V. I. Kirsanov, N. M. Naumova, and A. S. Sakharov, “Nonlinear depletion of ultrashort and relativistically strong laser pulses in an underdense plasma,” Phys. Fluids B **4**, 1935–1942 (1992). [CrossRef]

## 2. Physical model

*i.e.*, dispersion) self-consistently with diffraction and noinlinear refraction. The detailed derivation of the basic equations is given in [15

15. D. Farina, M. Lontano, I. G. Murusidze, and S. V. Mikeladze, “Hydrodynamic approach to the interaction of a relativistic ultrashort laser pulse with an underdense plasma,” Phys. Rev. E **63**, 056409(10) (2001). [CrossRef]

_{p}/ω ≪ 1, is a small parameter, naturally implies a pulse whose effective length ℓ

_{L}is much less than its width, ℓ

_{L}≪ℓ

_{┴}, where ℓ

_{┴}scales its transverse (to the propagation direction) dimension. Indeed, when the characteristic diffraction length of an optical pulse,

*L*

_{dif}=

*k*

*k*is the wavenumber corresponding to ω, is comparable to its dispersion length

*L*

_{disp}=(ω/ω

_{p})

^{2}

*k*

*L*

_{dif}≈

*L*

_{disp}, then (ℓ

_{L}/ℓ

_{┴})

^{2}≈(ω

_{p}/ω)

^{2}≪ 1. In addition, a pulse with duration on a femtosecond time scale, for λ ≈ 1 µm radiation, will contain a few optical cycles, which makes it necessary to take into account also the first-order dispersion effects. Thus we have the following scalings

_{‖}is the length scale over which the laser pulse changes significantly due to propagation effects (diffraction, dispersion and nonlinear refraction). According to these scalings the laser pulse may be identified as a thin, wide disk of light with a sharp intensity gradient along the propagation direction and a smooth transverse profile.

*z*direction and described by the transverse component of its vector potential

*A*(

**r**,

*t*) is the pulse envelope, generally a complex valued function, ω is the central frequency, and

*k*is the corresponding wavenumber in the plasma,

*kc*=ω(1-(

^{2}))

^{1/2}.

15. D. Farina, M. Lontano, I. G. Murusidze, and S. V. Mikeladze, “Hydrodynamic approach to the interaction of a relativistic ultrashort laser pulse with an underdense plasma,” Phys. Rev. E **63**, 056409(10) (2001). [CrossRef]

16. I. G. Murusidze, G. I. Suramlishvili, and M. Lontano, “Spatiotemporal self-focusing and splitting of a femtosecond, multiterawatt, relativistic laser pulse in an underdense plasma,” in *Superstrong Fields in Plasmas*, M. Lontano, G. Mourou, O. Svelto, and T. Tajima, eds., AIP Conference Proceedings, Vol.611 (American Institute of Physics, Melville, New York, 2002), pp. 177–184.

*z*, τ=

*t*-

*z*/

*v*

_{g}, where

*v*

_{g}≡ ∂ω/∂

*k*=

*kc*

^{2}/ω is the group velocity, β

_{g}=

*v*

_{g}/

*c*, β

_{g}=(1-

^{-1/2}. Note that, independent variables in Eqs. (3)–(4) are dimensional, whereas

*A*→

*eA*/(

*mc*

^{2}), and φ→

*e*φ /(

*mc*

^{2}) (scalar potential) refer to the dimensionless quantities throughout the paper.

*A*(

**r**

_{┴}, ζ, τ) of the ultrashort, relativistically intense laser pulse propagating in an underdense plasma. The second and the third terms on its left-hand side account for finite pulse duration. They represent the first- and second-order dispersion effects, respectively. The coefficient of the third term,

*d*=

*k*∂

^{2}

*k*/∂ω

^{2}=-(

*k*/

*v*

_{g}/∂ω, is proportional to the group-velocity dispersion parameter. In our case

*d*=-

_{p}/ω)

^{2}, i.e., the third term in Eq. (3) accounts for negative or “anomalous” GVD, as is typical of transparent plasmas. The fourth term is the diffraction term. The only term neglected on the left of Eq. (3) is the term

*A*, which is second-order in ε, i.e., the conventional paraxial approximation [17] is naturally assumed within the parameter regime outlined by Eq. (1):

*k*

^{-1}≪ ℓ

_{┴}≪ ℓ

_{‖}~

*k*

_{0}=

*kz*- ω

*t*varies slowly with ζ. Indeed in our case |∂

_{ζ}ϕ

_{0}|=| (

*v*

_{g}-

*v*

_{ϕ})/

*v*

_{g}|

*k*=

^{2}

*k*≪

*k*, which is also one of the prerequisites for a consistent extension of the wave equation down to single-cycle regime [18

18. T. Brabec and F. Krausz, “Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,” Phys. Rev. Lett. **78**, 3282–3285 (1997). [CrossRef]

15. D. Farina, M. Lontano, I. G. Murusidze, and S. V. Mikeladze, “Hydrodynamic approach to the interaction of a relativistic ultrashort laser pulse with an underdense plasma,” Phys. Rev. E **63**, 056409(10) (2001). [CrossRef]

16. I. G. Murusidze, G. I. Suramlishvili, and M. Lontano, “Spatiotemporal self-focusing and splitting of a femtosecond, multiterawatt, relativistic laser pulse in an underdense plasma,” in *Superstrong Fields in Plasmas*, M. Lontano, G. Mourou, O. Svelto, and T. Tajima, eds., AIP Conference Proceedings, Vol.611 (American Institute of Physics, Melville, New York, 2002), pp. 177–184.

*A*as well as φ, thus giving fully relativistic description of the refractive index at relativistic intensities of the propagating pulse.

_{g}

*p*

_{z}=γ - φ - 1 and

*n*(β

_{g}-

*p*

_{z}/ γ)=β

_{g}, where

*p*

_{z}(

*f*

*p*

_{z}/

*mc*) is the normalized longitudinal momentum of the electrons,

*n*(→

*n*/

*n*

_{0}) is the electron density normalized to its unperturbed initial value, and the relativistic factor γ=(1+

*A*|

^{2})

^{1/2}[15

**63**, 056409(10) (2001). [CrossRef]

*A*|>1, the longitudinal momentum is also relativistic, and is on the order of the transverse momentum, i.e., |

*p*

_{z}|

_{max}~ |

*A*|

_{max}. Thus the relativistic axial motion of electrons ponderomotively excited by the laser pulse, equally with the transverse oscillations of the electrons in the laser field, contributes to the relativistic reduction (depression) of the local plasma frequency (ω

_{p}∝γ

^{-1/2}), hence, to the corresponding increase in the local refractive index [15

**63**, 056409(10) (2001). [CrossRef]

*k*

_{p}ℓ

_{┴}≫1, where

*k*

_{p}=ω

_{p}/

*c*) [15

**63**, 056409(10) (2001). [CrossRef]

19. I. G. Murusidze and L. N. Tsintsadze, “Generation of large amplitude plasma wakefields with low phase velocities by an intense short laser pulse,” J. Plasma Phys. **48**, 391–395 (1992). [CrossRef]

*a*

_{0}≡

*eA*

_{0}/(

*mc*

^{2}) > 1, which implies

*I*λ

^{2}>α

_{Π}·1.37 × 10

^{18}W cm

^{-2}µm

^{2}where

*I*is the laser intensity and λ is the wavelength, and α

_{Π}=1,2 for linear and circular polarization, respectively. At these laser parameters the plasma response, in general, can be excited with relativistic amplitudes, i.e. | φ |>1.

_{p}, ℓ

_{L}< λ

_{p},

*i.e.*our pulse is so short that it does not see even a single plasma wavelength. Therefore its coupling with plasma waves cannot be described in terms of the conventional wave-wave interaction. The shortness of the pulse can suppress parametric Raman processes and significantly limit the number of instabilities. However, the ultrashort pulse experiences a new type of instabilities, which are the subject of our studies in the following section.

## 3. Space-time self-focusing and splitting of laser pulse

*A*=

*A*

_{0}exp(-τ

^{2}/

*r*

^{2}/

*r*≡ (

*x*

^{2}+

*y*

^{2})

^{1/2}, and the amplitude

*A*

_{0}is a real constant. Having no initial phase varying with τ, the pulse is transform-limited [20],

*i.e.*with no frequency modulation at the input plane. The radial dependence of the pulse corresponds to the fundamental Gaussian mode with a spot size

*w*

_{0}and a planar wave front (with infinite radius of curvature) [17]. Though axial symmetry of the pulse’s transverse profile is maintained during the entire time of the simulation evolution, however, in the course of the propagation the pulse develops transverse profiles with higher order axial symmetry as we go from the front to the back of the pulse. Simulation results described here represent a laser pulse with the following dimensionless parameters: the pulse amplitude

*A*

_{0}=3; the ratio of the laser spot size to the plasma skin depth

*w*

_{0}

*k*

_{p}=25; the ratio of the initial effective pulse duration to the spot size is defined by

*v*

_{g}τ

_{0}/

*w*

_{0}=0.1. The preformed plasma is assumed to be underdense with

*n / n*

_{c}=(ω

_{p}/ω)

^{2}=0.01, where

*n*

_{c}is the critical plasma density. For λ=800 nm laser wavelength, the above dimensionless parameters imply a laser pulse with a peak intensity

*I*

_{0}=3.85 × 10

^{19}W/cm

^{2}, peak power

*P*

_{0}=612.7 TW, spot size

*w*

_{0}=31.8 µm, duration (length) τ

_{FWHM}=12.6 fs (3.77 µm) full width at half maximum of the intensity profile. The vacuum Rayleigh length for this pulse is

*Z*

_{R}=3.98 mm. The plasma density

*n*=1.74×10

^{19}cm

^{-3}and the corresponding plasma wavelength λ

_{p}=8 µm (

*P*

_{cr}≈ 1.7 TW [7

7. P. Sprangle, C.-M. Tang, and E. Esarey, “Relativistic self-focusing of short-pulse radiation beams in plasmas,” IEEE Trans. Plasma Sci. **PS-15**, 145–153 (1987). [CrossRef]

8. G.-Z. Sun, E. Ott, Y. C. Lee, and P. Guzdar, “Self-focusing of short intense pulses in plasmas,” Phys. Fluids **30**, 526–532 (1987). [CrossRef]

*x*, and

*y*. Radiative (vanishing reflection coefficient) boundary conditions are applied [15

**63**, 056409(10) (2001). [CrossRef]

16. I. G. Murusidze, G. I. Suramlishvili, and M. Lontano, “Spatiotemporal self-focusing and splitting of a femtosecond, multiterawatt, relativistic laser pulse in an underdense plasma,” in *Superstrong Fields in Plasmas*, M. Lontano, G. Mourou, O. Svelto, and T. Tajima, eds., AIP Conference Proceedings, Vol.611 (American Institute of Physics, Melville, New York, 2002), pp. 177–184.

*A*| is mapped in (

*z*′,

*r*′) plane, where

*z*′=

*k*

_{p}(

*z*- ν

_{g}

*t*) and

*r*′=

*k*

_{p}

*r*are the dimensionless longitudinal (in the moving frame) and radial coordinates in

*r*′ axis is ten times larger than that of the

*z*′ axis. The propagation distances into the plasma are shown at the right upper and lower corners of each plot in units of the Rayleigh length and in micromiters, respectively. At the initial stage up to a propagation distance ≈160 µm (0.04

*Z*

_{R}) the pulse almost preserves its proportions in spite of the contraction of its radial and longitudinal dimensions due to SF and SC. A noticeable increase in the peak amplitude is already evident in Fig. 1(c). Here the pulse’s peak is shifted forward and leading edge has become steeper. In the following stage modulations of both the longitudinal (time) and radial profiles become evident. At this stage the trailing half of the pulse develops noticeable modulations of the time profile which become well pronounced after propagating 318 µm (0.08

*Z*

_{R}), Fig. 1(d). Becoming deeper with the intensity dropping almost to zero between successive peaks, these modulations eventually lead to pulse splitting into a sequence of pulses, which follow each other arranged in decreasing order of their amplitudes, Fig. 1(e), 1(f). The initial stage of this splitting is shown also in Fig. 4(a), and the further dynamics of the time profile modulations is shown in Figs. 5(a) and 5(b). Eventually one can identify three separate peaks with relativistic amplitudes at the split back of the pulse.

*w*

_{eff}is the radial variance [17] normalized to its initial value

*w*

_{0}/√2, and τ

_{eff}is the root mean square (rms) duration [20] corresponding to the pulse’s on-axis time profile normalized to its initial value, which for our input pulse is τ

_{0}/2. Figure 2(a) clearly illustrates that the rates of SF and SC remain very close to each other up to the time when the pulse has traveled 437.7 µm (0.11

*Z*

_{R}), thus the initial ratio of the pulse length to its width is preserved. Figure 2(b) shows the increase in the laser peak intensity

*I*∝|

*E*

*A*

*E*|~

*c*

^{-1}| (ω+δω) ‖

*A*|, where δω is the frequency shift associated with SPM induced phase shifts. Large frequency downshifts in the focal region, which will be analyzed below, explain the difference between the growth rates.

_{ref}=[1 -

*n*/(

*n*

_{c}γ)]

^{1/2}, induced by the laser pulse through the relativistic mass increase and through the ponderomotively driven electron density perturbation. Initially, when the pulse enters the plasma the relativistic mass increase of the electrons results from their quiver motion in the laser field. However, later on the ponderomotively driven longitudinal electron motion becomes highly relativistic also within the focal region. Thus, the ponderomotive effect also contributes to the relativistic increase in electron mass. Virtually the transverse and longitudinal motions are coupled through the γ factor. As to the electron density perturbation, its effect is to decrease (increase) n

_{ref}at the leading (trailing) edge. Indeed, the electron density compression shown in Figs. 3(a) and 3(b) acts as a diverging lens diffracting the relatively low intensity part of the pulse’s leading edge. It is created by the front of the pulse, which ponderomotively “snowplows” plasma electrons forward. However, the effect of the density compression is counterbalanced by the relativistic factor in the body of the leading half of the pulse and the net effect is a local increase of the on axis value of the refractive index relative to its off-axis value. This light-induced “lensing” effectively counterbalances the diffractive spreading and the pulse self-focuses.

*A*| and phase ϕ (the latter is presented through the corresponding frequency shift, δω=-∂ϕ/∂τ) of the complex envelope

*A*=|

*A*| exp(

*i*ϕ) of the laser pulse for two propagation distances. This numerical full-field(amplitude + phase) analysis, in analogy with the frequency-resolved optical gating [21

21. R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Inst. **68**, 3277–3295 (1997). [CrossRef]

**0**) are shown in Figs. 4(b) and 4(f). This frequency downshift is located mainly at the leading half of the pulse with a minimum before the main peak. This downshift becomes deeper with time that is also clearly indicated in the axial profiles of the frequency chirp in Figs. 4(c) and 4(g), where they are shown along with the pulse intensity profiles. The longitudinal (time) and radial variations of the frequency shift reflects the complex interplay between the relativistic and ponderomotive effects contributing to the nonlinear index of refraction, which through SPM originates these frequency modulations. First note that the relativistic factor, γ, which is dominant in the shaping of the refractive index, exhibits clearly pronounced longitudinal asymmetry. Figure 3(c) shows the longitudinal profile of γ with the profiles of its ingredients

*A*|

^{2}. The origin of the asymmetric longitudinal profile of γ, hence that of n

_{ref}, is the longitudinal relativistic motion of the plasma electrons, ponderomotively driven by the pulse forward and backward from the focal region as is seen from Fig. 3(c). This asymmetry becomes evident at early stages of the propagation before the time profile modulations at the trailing edge become noticeable. Respectively, we observe an asymmetric broadening of the spectrum: the frequency downshift at the leading half is not followed by symmetric upshift at the trailing half, as one would expect if there were no strongly relativistic plasma response [10

10. W. B. Mori, “The physics of the nonlinear optics of plasmas at relativistic intensities for short-pulse lasers,” IEEE J. Quantum Electron. **33**, 1942–1953 (1997). [CrossRef]

22. I. Watts, M. Zepf, E. L. Clark, M. Tatarakis, K. Krushelnik, A. E. Dangor, R Alott, R. J. Clarke, D. Neely, and P. N. Norreys, “Measurement of relativistic self-phase-modulation in plasma,” Phys. Rev. E66, 036409 (2002). [CrossRef]

14. S. V. Bulanov, I. N. Inovenkov, V. I. Kirsanov, N. M. Naumova, and A. S. Sakharov, “Nonlinear depletion of ultrashort and relativistically strong laser pulses in an underdense plasma,” Phys. Fluids B **4**, 1935–1942 (1992). [CrossRef]

*Superstrong Fields in Plasmas*, M. Lontano, G. Mourou, O. Svelto, and T. Tajima, eds., AIP Conference Proceedings, Vol.611 (American Institute of Physics, Melville, New York, 2002), pp. 177–184.

*z*′,

*r*′) plane) shape acquired by the back of the main pulse as well as by the split pulses shown in Figs. 1(d)–1(f), 4(e) correlates with typical radial distribution of the frequency chirp shown in Figs 4(d) and 4(h). Note that in Fig. 4(d) the frequency in the center is upshifted only relatively to the off-axis spectrum, whereas in Fig. 4(h) this difference becomes “absolute” with blue-shifted (δω >

**0**) spectrum at the center and red-shifted one (δω<

**0**) off-axis. These transverse modulations of the frequency chirp also are attributed to the FOD effect. On the other hand, under anomalous GVD the blue-shifted parts of the pulse acquire higher group velocities than that of the red-shifted parts. Thus the combined action of GVD, FOD and SPM not only affect the time (longitudinal) profile of the pulse) by SC and by splitting of the pulse, but it also initiates an axial transport of energy, which changes its radial profile.

9. E. Esarey, P. Sprangle, J. Krall, and A. Ting, “Self-focusing and guiding of short laser pulses in ionizing gases and plasmas,” IEEE J. Quantum Electron. **33**, 1879–1914 (1997). [CrossRef]

12. P. Spangle, B. Hafizi, and J. R. Peñano, “Laser pulse modulation instabilities in plasma channels,” Phys. Rev. E **61**, 4381–4393 (2000). [CrossRef]

_{L}≪ ℓ

_{┴}). Therefore the further development is only of mathematical value.

## 4. Conclusion

5. K. Yamakawa, Y. Akahane, M. Aoyama, Y. Fukuda, N. Inoue, J. Ma, and H. Ueda, “Status and future developments of ultrahigh intensity lasers at JAERI,” in *Superstrong Fields in Plasmas*, M. Lontano, G. Mourou, O. Svelto, and T. Tajima, eds., AIP Conference Proceedings, Vol.611 (American Institute of Physics, Melville, New York, 2002), pp. 385–396.

22. I. Watts, M. Zepf, E. L. Clark, M. Tatarakis, K. Krushelnik, A. E. Dangor, R Alott, R. J. Clarke, D. Neely, and P. N. Norreys, “Measurement of relativistic self-phase-modulation in plasma,” Phys. Rev. E66, 036409 (2002). [CrossRef]

## Acknowledgments

## References

1. | M. D. Perry and G. Mourou, “Terawatt to petawatt subpicosecond lasers,” Science |

2. | G. A. Mourou, C. P. J. Barty, and M. D. Perry, “Ultrahigh-Intensity Lasers: Physics of the Extreme on a Tabletop,” Phys. Today |

3. | K. Yamakawa, M. Aoyama, T. Kase, Y. Akahane, and H. Takuma, “100-TW sub-20-fs Ti:sapphire laser system operating at a 10-Hz repetition rate,” Opt. Lett. |

4. | K. Yamakawa and C. P. J. Barty, “Ultrafast, ultrahigh-peak, and high-average power Ti:sapphire laser system and its application”, IEEE J. Se. Top. Quantum Electron. |

5. | K. Yamakawa, Y. Akahane, M. Aoyama, Y. Fukuda, N. Inoue, J. Ma, and H. Ueda, “Status and future developments of ultrahigh intensity lasers at JAERI,” in |

6. | T. Tajima and G. Mourou, “Superstrong field science,” |

7. | P. Sprangle, C.-M. Tang, and E. Esarey, “Relativistic self-focusing of short-pulse radiation beams in plasmas,” IEEE Trans. Plasma Sci. |

8. | G.-Z. Sun, E. Ott, Y. C. Lee, and P. Guzdar, “Self-focusing of short intense pulses in plasmas,” Phys. Fluids |

9. | E. Esarey, P. Sprangle, J. Krall, and A. Ting, “Self-focusing and guiding of short laser pulses in ionizing gases and plasmas,” IEEE J. Quantum Electron. |

10. | W. B. Mori, “The physics of the nonlinear optics of plasmas at relativistic intensities for short-pulse lasers,” IEEE J. Quantum Electron. |

11. | B. Hafizi, A. Ting, P. Sprangle, and R. F. Hubbard, “Relativistic focusing and ponderomotive channeling of intense laser beams,” Phys. Rev. E |

12. | P. Spangle, B. Hafizi, and J. R. Peñano, “Laser pulse modulation instabilities in plasma channels,” Phys. Rev. E |

13. | E. Esarey, C. B. Schroeder, B. A. Shadwick, J. S. Wurtele, and W. P. Leemans, “Nonlinear Theory of Nonparaxial Laser Pulse Propagation in Plasma Channels,” Phys. Rev. Lett. |

14. | S. V. Bulanov, I. N. Inovenkov, V. I. Kirsanov, N. M. Naumova, and A. S. Sakharov, “Nonlinear depletion of ultrashort and relativistically strong laser pulses in an underdense plasma,” Phys. Fluids B |

15. | D. Farina, M. Lontano, I. G. Murusidze, and S. V. Mikeladze, “Hydrodynamic approach to the interaction of a relativistic ultrashort laser pulse with an underdense plasma,” Phys. Rev. E |

16. | I. G. Murusidze, G. I. Suramlishvili, and M. Lontano, “Spatiotemporal self-focusing and splitting of a femtosecond, multiterawatt, relativistic laser pulse in an underdense plasma,” in |

17. | A. E. Siegman, |

18. | T. Brabec and F. Krausz, “Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,” Phys. Rev. Lett. |

19. | I. G. Murusidze and L. N. Tsintsadze, “Generation of large amplitude plasma wakefields with low phase velocities by an intense short laser pulse,” J. Plasma Phys. |

20. | S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, |

21. | R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Inst. |

22. | I. Watts, M. Zepf, E. L. Clark, M. Tatarakis, K. Krushelnik, A. E. Dangor, R Alott, R. J. Clarke, D. Neely, and P. N. Norreys, “Measurement of relativistic self-phase-modulation in plasma,” Phys. Rev. E66, 036409 (2002). [CrossRef] |

**OCIS Codes**

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(190.5940) Nonlinear optics : Self-action effects

(260.5950) Physical optics : Self-focusing

(350.0350) Other areas of optics : Other areas of optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: December 23, 2002

Revised Manuscript: February 3, 2003

Published: February 10, 2003

**Citation**

Maurizio Lontano and Ivane Murusidze, "Dynamics of space-time self-focusing of a femtosecond relativistic laser pulse in an underdense plasma," Opt. Express **11**, 248-258 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-3-248

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### References

- M. D. Perry and G. Mourou, �??Terawatt to petawatt subpicosecond lasers,�?? Science 64, 917-924 (1994). [CrossRef]
- G. A. Mourou, C. P. J. Barty and M. D. Perry, �??Ultrahigh-Intensity Lasers: Physics of the Extreme on a Tabletop,�?? Phys. Today 51, 22-28 (1998). [CrossRef]
- K. Yamakawa, M. Aoyama, T. Kase, Y. Akahane, and, H. Takuma, �??100-TW sub-20-fs Ti:sapphire laser system operating at a 10-Hz repetition rate,�?? Opt. Lett. 23, 1468-1470 (1998). [CrossRef]
- K. Yamakawa, C. P. J. Barty, �??Ultrafast, ultrahigh-peak, and high-average power Ti:sapphire laser system and its application,�?? IEEE J. Se. Top. Quantum Electron. 6, 658-675 (2000). [CrossRef]
- K. Yamakawa, Y. Akahane, M. Aoyama, Y. Fukuda, N. Inoue, J. Ma, and H. Ueda, �??Status and future developments of ultrahigh intensity lasers at JAERI,�?? in Superstrong Fields in Plasmas, M. Lontano, G. Mourou, O. Svelto, T. Tajima, eds., AIP Conference Proceedings, Vol.611 (American Institute of Physics, Melville, New York, 2002), pp. 385-396.
- T. Tajima and G. Mourou, �??Superstrong field science,�?? ibid. pp. 423-436.
- P. Sprangle, C.-M. Tang, E. Esarey, �??Relativistic self-focusing of short-pulse radiation beams in plasmas,�?? IEEE Trans. Plasma Sci. PS-15, 145-153 (1987). [CrossRef]
- G.-Z. Sun, E. Ott, Y. C. Lee, and P. Guzdar, �??Self-focusing of short intense pulses in plasmas,�?? Phys. Fluids 30, 526-532 (1987). [CrossRef]
- E. Esarey, P. Sprangle, J. Krall, A. Ting, �??Self-focusing and guiding of short laser pulses in ionizing gases and plasmas,�?? IEEE J. Quantum Electron. 33, 1879-1914 (1997). [CrossRef]
- W. B. Mori, �??The physics of the nonlinear optics of plasmas at relativistic intensities for short-pulse lasers,�?? IEEE J. Quantum Electron. 33, 1942-1953 (1997). [CrossRef]
- B. Hafizi, A. Ting, P. Sprangle, and R. F. Hubbard, �??Relativistic focusing and ponderomotive channeling of intense laser beams,�?? Phys. Rev. E 62, 4120-4125 (2000). [CrossRef]
- P. Spangle, B. Hafizi, and J. R. Peñano, �??Laser pulse modulation instabilities in plasma channels,�?? Phys. Rev. E 61, 4381-4393 (2000). [CrossRef]
- E. Esarey, C. B. Schroeder, B. A. Shadwick, J. S. Wurtele, and W. P. Leemans, �??Nonlinear Theory of Nonparaxial Laser Pulse Propagation in Plasma Channels,�?? Phys. Rev. Lett. 84, 3081-3084 (2000). [CrossRef] [PubMed]
- S. V. Bulanov, I. N. Inovenkov, V. I. Kirsanov, N. M. Naumova, and A. S. Sakharov, �??Nonlinear depletion of ultrashort and relativistically strong laser pulses in an underdense plasma,�?? Phys. Fluids B 4, 1935-1942 (1992) . [CrossRef]
- D. Farina, M. Lontano, I. G. Murusidze, S. V. Mikeladze, �??Hydrodynamic approach to the interaction of a relativistic ultrashort laser pulse with an underdense plasma,�?? Phys. Rev. E 63, 056409(10) (2001). [CrossRef]
- I. G. Murusidze, G. I. Suramlishvili, M. Lontano, "Spatiotemporal self-focusing and splitting of a femtosecond, multiterawatt, relativistic laser pulse in an underdense plasma," in Superstrong Fields in Plasmas, M. Lontano, G. Mourou, O. Svelto, T. Tajima, eds., AIP Conference Proceedings, Vol.611 (American Institute of Physics, Melville, New York, 2002), pp. 177-184.
- A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986).
- T. Brabec and F. Krausz, �??Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,�?? Phys. Rev. Lett. 78, 3282-3285 (1997). [CrossRef]
- I. G. Murusidze, L. N. Tsintsadze, �??Generation of large amplitude plasma wakefields with low phase velocities by an intense short laser pulse,�?? J. Plasma Phys. 48, 391-395 (1992). [CrossRef]
- S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (AIP, New York, 1992).
- R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, D. J. Kane �??Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating," Rev. Sci. Inst. 68, 3277-3295 (1997). [CrossRef]
- I. Watts, M. Zepf, E. L. Clark, M. Tatarakis, K. Krushelnik, A. E. Dangor, R Alott, R. J. Clarke, D. Neely, P. N. Norreys, �??Measurement of relativistic self-phase-modulation in plasma,�?? Phys. Rev. E 66, 03640 (2002). [CrossRef]

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