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Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 3 — Feb. 10, 2003
  • pp: 248–258
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Dynamics of space-time self-focusing of a femtosecond relativistic laser pulse in an underdense plasma

Maurizio Lontano and Ivane G. Murusidze  »View Author Affiliations


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

The progress in developing ultrashort-pulse laser sources, based on the technique of chirped pulse amplification (CPA) [1

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

,2

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]

], has made available high peak power (from multiterawatt to petawatt) laser pulses on a femtosecond time scale. A compact CPA laser system produces pulses with peak powers in excess of 100 TW and durations less than 19 fs [3

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]

]. The same ultrafast CPA architecture enables production of optical pulses with petawatt peak powers. With increasing peak powers, pulse durations have been steadily decreased and now high peak power CPA systems are evolving into the 10-fs range [4

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

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.

]. Such systems have made available pulses with peak intensities in the 1020 W/cm2 range, and, in the near future, it is expected to increase by at least two orders of magnitude [6

6. T. Tajima and G. Mourou, “Superstrong field science,” ibid. pp. 423–436.

]. 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.

2. Physical model

Our studies of the propagation of ultrashort, ultraintense laser pulses through an underdense plasma is based on a model which systematically incorporates finite-pulse-length effects (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]

]. Here we outline the principal relations and conditions, on which the model is based. Thus this model is aimed at describing laser pulses, for which dispersion effects play as important a role as diffraction. Such regime of propagation in underdense plasmas, where the ratio of the plasma frequency to the (central) laser frequency, ω p /ω ≪ 1, is a small parameter, naturally implies a pulse whose effective length ℓL is much less than its width, ℓL ≪ℓ, where ℓ scales its transverse (to the propagation direction) dimension. Indeed, when the characteristic diffraction length of an optical pulse, L dif=k2, where k is the wavenumber corresponding to ω, is comparable to its dispersion length L disp=(ω/ω p )2 kL2, i.e., L difL disp, then (ℓL/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

Lωpωε1,
(1)

where ℓ 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.

The plasma is modeled by the relativistic fluid equations for cold electrons. This model gives an adequate description of plasmas within the assumptions that, first, the laser pulse duration is short enough and therefore the ions are assumed to be virtually immobile during the transit of the ultrashort pulse, and, second, it is also assumed that, at relativistic intensities, the electrons are driven to velocities much higher than their thermal velocities.

Within the regime of parameters outlined above, we model the evolution of the circularly polarized laser pulse propagating in the z direction and described by the transverse component of its vector potential

A=12(x±iy)A(r,t)exp(ikziωt)+c.c.,
(2)

where 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-(ωp22))1/2.

From Maxwell’s equations and the relativistic electron fluid equations to the lowest (zeroth-order) in the small parameter ε we derive the following set of coupled equations [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]

,16

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.

]

2ikAζ2vg2Aτζd2Aτ2+2A=
kp2(βg[(1+φ)2(1+A2)γg2]121)A,
(3)
2φτ2=ωp2(βgγg)2(βg(1+φ)[(1+φ)2(1+A2)γg2]121).
(4)

Here we have employed a moving frame of reference introducing the variables ζ=z, τ=t - z / vg , where vg ≡ ∂ω/∂k=kc 2/ω is the group velocity, β g =vg /c, β g =(1-βg2)-1/2. Note that, independent variables in Eqs. (3)–(4) are dimensional, whereas AeA /(mc 2), and φ→eφ /(mc 2) (scalar potential) refer to the dimensionless quantities throughout the paper.

Equation (3) is the wave equation, which governs the evolution of the envelope 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=k2 k/∂ω2=-(k/vg2)∂vg /∂ω, is proportional to the group-velocity dispersion parameter. In our case d=-vg2p /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 ζζ2 A, which is second-order in ε, i.e., the conventional paraxial approximation [17

17. A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986).

] is naturally assumed within the parameter regime outlined by Eq. (1): k -1 ≪ ℓ ≪ ℓ ~ k2. In addition, under the same conditions, the GVD and the diffraction terms contribute equally in the pulse evolution. The first order dispersion (FOD) term extends the applicability of Eq. (3) down to single-cycle pulses. Note that the phase in Eq. (2) ϕ0=kz - ωt varies slowly with ζ. Indeed in our case |∂ζϕ0 |=| (vg - v ϕ)/vg | k=βg2 ε2 kk, 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]

].

Incorporating all relevant finite-pulse-length effects within the paraxial approximation, the wave equation, Eq. (3), will be referred to as the pulsed paraxial wave equation (PPWE). In the sense used by us 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]

,16

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.

] and followed here, the PPWE is an extension of the classical paraxial wave equation designed first for light beams-infinite-length light pulses. Thus within the PPWE we are able to describe ultrashort pulses propagating in underdense plasmas in the regime when dispersion and diffraction effects play an equally important role.

The terms contributing to the nonlinear refraction are on the right-hand side of Eq. (3). It is expressed through the scalar and vector potentials and contains contributions both from the relativistic mass increase of the electrons and from the electron density perturbations due to ponderomotive action of the laser pulse. It is valid for arbitrary large values of A as well as φ, thus giving fully relativistic description of the refractive index at relativistic intensities of the propagating pulse.

Equation (4), describing the dynamic plasma response to the laser pulse, is the Poisson’s equation, the right hand-side of which is expressed in terms of the vector and scalar potentials. The scalar potential,φ, accounts for the electrostatic field, which results from electron density perturbations ponderomotively driven by the laser pulse. Integrating the leading-order (zeroth-order in ε) fluid equations, we derive the following relations β gpz =γ - φ - 1 and ng - pz / γ)=β g , where pz (f pz / 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+pz2+| A |2)1/2 [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]

]. At relativistic laser amplitudes, | A |>1, the longitudinal momentum is also relativistic, and is on the order of the transverse momentum, i.e., | pz |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

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]

]. The hierarchy of scales implied by Eq. (1) naturally leads to the plasma response, which is described self-consistently within the quasistatic approximation in the wide pulse limit (kp≫1, where kpp /c) [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]

,19

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]

]. The equation for the plasma response is valid for arbitrarily large relativistic laser intensities, when the amplitude of the normalized vector potential of the laser field, the so-called laser strength parameter, a 0eA 0 /(mc 2) > 1, which implies Iλ2Π·1.37 × 1018 W cm-2 µm2 where I is the laser intensity and λ is the wavelength, and αΠ=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.

For the laser-plasma parameters within the range of our interest it is also typical that the laser pulse length is less than the plasma wavelength, λ p , ℓL < 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

In the numerical simulations, based on Eqs. (3)-(4), we model the propagation of an ultrashort, relativistically intense, circularly polarized laser pulse focused at the edge of a preformed underdense plasma. At the input plane ζ=0 the laser field is given by A=A 0 exp(-τ2/τ02 - r 2/w02), where 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

20. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (AIP, New York, 1992).

], 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

17. A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986).

]. 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 kp =25; the ratio of the initial effective pulse duration to the spot size is defined byvg τ0/w 0=0.1. The preformed plasma is assumed to be underdense with n / nc =(ω p /ω)2=0.01, where nc 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 × 1019 W/cm2, peak power P 0=612.7 TW, spot size w 0=31.8 µm, duration (length) τFWHM=12.6 fs (3.77 µm) full width at half maximum of the intensity profile. The vacuum Rayleigh length for this pulse is ZR =3.98 mm. The plasma density n=1.74×1019 cm-3 and the corresponding plasma wavelength λ p =8 µm (kp1=1.27 µm). Note that the peak power is well above the critical power for relativistic self-focusing, which for our parameters is 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

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]

]. The numerical method is based on a finite-difference scheme with first-order accuracy in ζ and second-order accuracy in τ, x, and y. Radiative (vanishing reflection coefficient) boundary conditions are applied [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]

]. It should be noted that we have simulated the laser propagation within this model for various laser and plasma parameters. The results presented here for the parameters discussed above typify the behavior and space-time patterns revealed in these simulations [16

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.

].

Fig. 1. The sequence of contour plots shows the evolution of the laser field amplitude | A | (in mc 2 / e units) during its propagation through an underdense plasma. The initial Gaussian pulse has A 0=3, w 0 k p =25, k p ν g τ0=2.5 . | A | is mapped in (z′, r′) plane, where z′=k p (z - ν gt) and r′=kpr are the dimensionless longitudinal (in the moving frame) and radial coordinates in kp1 units, respectively.Propagation distances into the plasma are shown in the right upper and lower corners of each plot in units of the Rayleigh length and in micrometers, respectively.

The sequence of contour plots in Fig. 1 shows the dynamics of the simultaneous SF and SC of the pulse and its subsequent splitting. Here | A | is mapped in (z′,r′) plane, where z′=kp (z - ν gt) and r′=kpr are the dimensionless longitudinal (in the moving frame) and radial coordinates in kp1 units, respectively. Note that the scale of the 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 (0.04ZR ) 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 (0.08ZR ), 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.

The dynamics of the pulse’s effective width and length with the propagation distance is shown in Fig. 2(a). Here w eff is the radial variance [17

17. A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986).

] normalized to its initial value w 0/√2, and τ eff is the root mean square (rms) duration [20

20. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (AIP, New York, 1992).

] corresponding to the pulse’s on-axis time profile normalized to its initial value, which for our input pulse is τ0/2. 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 (0.11ZR ), 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|max2 by a factor of three after the pulse has traveled ≈ 0.4 mm through the plasma, while | A |max2 is increased almost by an order of magnitude. Such high rate of energy concentration is a result of the simultaneous SF and SC of the pulse. Note that |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.

Fig. 2. (a) - The effective width w eff and the effective duration τ eff of the laser pulse are shown vs. the propagation distance (in micrometers). Underlined quantities refer to their normalized values. (b) - Dynamics of the peak values of the squared amplitudes of the laser vector potential and of the electric field (normalized to their initial values) during the pulse propagation.
Fig. 3. The axial (longitudinal) asymmetry of the plasma response is illustrated. (a) - Surface plot of the electron density within the focal region. The corresponding on-axis profile of the density (magenta) along with the intensity profile (blue) is shown in (b). (c) - Axial profiles of the relativistic factor,γ (green), the square of the longitudinal momentum, pz2 (magenta), and the square of the amplitude of the laser vector potential |A|2, (blue).

Self-focusing in our case is a result of the nonlinear changes in the refractive index, nref=[1 -n /(nc 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) nref 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.

Fig. 4. The contour plots of the laser amplitude | A|, (a, e), and of the frequency shift δω (normalized to the central laser frequency) (b, f), are shown for two propagation distances. Note the large red shifted regions at the front of the pulse as well as the modulated frequency shifts at the trailing edge. (c) and (g) show the corresponding axial profiles of δω (blue) along with the laser intensity profiles (magenta). (d) and (h) show the radial variations of the frequency shifts, which typically develop at the back of the pulse.

The variations of the effective refractive index along the pulse are responsible for axial modulations of the phase and, respectively, frequency shifts accumulated due to self-phase modulation (SPM) in the course of the pulse propagation. The simulation results presented in Figs. 4(a)-(h) show the evolution of the amplitude | A | and phase ϕ (the latter is presented through the corresponding frequency shift, δω=-∂ϕ/∂τ) of the complex envelope A =| A | exp(iϕ) 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]

], elucidates the role of dispersion as well as of coupling of temporal and spatial evolution of the ultrashort pulse. These figures show contour maps of the frequency shift (“chirp”) and their axial and radial profiles together with the corresponding amplitude maps for two propagation distances. The large areas with the red-shifted frequency (δω< 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 pz2 and | A |2. The origin of the asymmetric longitudinal profile of γ, hence that of nref, 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

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]

]. The process of the downshift of the “average” frequency and of the frequency spectral broadening during the propagation of a one-dimensional laser pulse in an underdense plasma have been pointed out and analyzed in [14

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]

]. The modulations of the frequency chirp at the back of the pulse appear at a later stage and are steadily enhanced with the accompanying envelope modulations. This enhancement is well seen from Figs. 4(c) and 4(g). Initially small-amplitude undulations start to develop near the peak of the pulse. Then they propagate downhill the back slope. The accompanying phase shifts are affected in such a way that within the regions with higher amplitude the frequency is downshifted, whereas it is upshifted in the regions with lower amplitudes. Arranged in this way, the amplitude and frequency modulations enhance each other and the process turns out to be unstable and it proceeds until the back of the pulse is split into several successive pulses. This development is a result of the FOD effect. The FOD modulates phase shifts, i.e., the frequency spectrum, and through it the envelope time profile. Indeed, direct numerical simulations have shown that without the second term in Eq. (3) accounting for FOD, there is no modulation of frequency shifts and, consequently, there is no splitting of the pulse [16

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 characteristic concave (in the (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.

Fig. 5. Modulations of the pulse’s temporal profile for two subsequent propagation distances (a, b). The pulse’s head self-focuses with a conventional bell-shaped radial profile (c), while its trailing half develops a radial pattern with minimum intensity at the center surrounded by a ring in which the intensity is concentrated (d).

Figures 5 (a) and (b) show the longitudinal profiles of the pulse amplitude for two different propagation distances. The developing modulation of the time envelope and its successive splitting into sequence of pulses are evident. Figures 5 (c) and (d) show the radial profiles of the pulse at its leading and trailing edges, respectively. As it is evident from Fig. 5(d) at the trailing half of the pulse its radial profile has developed a pattern with a radial symmetry of higher order than that of the initial Gaussian profile. Note that, within the conventional paraxial wave equation, a radial transport of energy connected to SF cannot initiate changes in the radial pattern of a laser beam like one observed here [9

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

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

].

After traveling through the plasma a distance much larger than its initial length the pulse as a whole develops a longitudinal scale comparable to the transverse one. This is the evidence that we enter a parameter regime beyond the validity of our model for ultrashort pulses (ℓL ≪ ℓ). Therefore the further development is only of mathematical value.

4. Conclusion

In conclusion, we have studied numerically the dynamics of an ultrashort, high peak power, relativistic laser pulse propagating in an underdense plasma. Our numerical simulations are based on a model, which incorporates (i) all relevant finite-pulse-length effects, and (ii) describes laser plasma interactions within a strongly nonlinear, fully relativistic regime. The numerical studies have revealed the importance of these two factors in the evolution of a femtosecond-scale, multiterawatt, high-intensity laser pulse in underdense plasmas. Entering the plasma the pulse initiates large nonlinear changes in the refractive index. These changes are caused by the relativistic and ponderomotive effects, which contribute equally in the regime of strong coupling between the laser and the plasma response. The enhanced SPM in the presence of the GVD effect creates favorable conditions for the pulse SC, which goes synchronously with its SF. Both effects lead to a concentration of the pulse energy with a high rate. However the conventional patterns of SF and SC become significantly modified when with the further reduction of the pulse’s longitudinal dimensions the finite-pulse-length effects start to dominate in its evolution. Analyzing the evolution of the SPM induced frequency shifts together with the pulse envelope evolution it was shown that the FOD effect modulates the frequency chirp at the trailing half of the pulse. Coupled with the amplitude modulations, this process develops as an unstable backward propagating perturbation, leading to the splitting of the back of the pulse into a sequence of pulses. The combined effect of GVD, FOD and SPM actively reshapes not only the pulse’s time profile, but also its radial profile through the axial redistribution of the pulse energy. As a result the initially Gaussian radial profile during the SF changes to a profile with a higher order of radial symmetry with a ring shaped intensity distribution in the transverse plane. All these clearly show the role of dispersion effects and SPM in the evolution of ultrashort pulses in underdense plasmas. The importance of the detailed studies of these effects for intense laser-matter interactions is evident taking into account that such pulses usually have associated amplified spontaneous emission and pre- (and post) pulses. They create a low-density plasma in front of the main pulse. Thus an ultrashort pulse does propagate through underdense plasmas before reaching the main target. On the other hand, as we have shown, underdense plasma through its large and low-inertia electron nonlinearities is a medium in which an ultrashort, intense laser pulse undergoes violent changes of its temporal, spatial and spectral characteristics. Consequently these changes can significantly alter the laser-matter interactions [5

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

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]

]. Therefore detailed characterization of the self-effects governing the ultrashort pulse evolution in underdense plasmas becomes of primary interest for controlling and optimizing the spatial and temporal shape and phase of a laser pulse to improve its delivery to the target with high efficiency.

Acknowledgments

I. G. M. gratefully acknowledges the hospitality of the Istituto di Fisica del Plasma - CNR Milano, and Rainer Exner and Bärbel Exner for their invaluable support of his work.

References

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.

6.

T. Tajima and G. Mourou, “Superstrong field science,” ibid. pp. 423–436.

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]

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]

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]

11.

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]

12.

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

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. 84, 3081–3084 (2000). [CrossRef] [PubMed]

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]

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.

17.

A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986).

18.

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

20.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (AIP, New York, 1992).

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]

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

  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, 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, T. Tajima, eds., AIP Conference Proceedings, Vol.611 (American Institute of Physics, Melville, New York, 2002), pp. 385-396.
  6. T. Tajima and G. Mourou, �??Superstrong field science,�?? ibid. pp. 423-436.
  7. 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]
  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]
  9. 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]
  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]
  11. 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]
  12. P. Spangle, B. Hafizi, and J. R. Peñano, �??Laser pulse modulation instabilities in plasma channels,�?? Phys. Rev. E 61, 4381-4393 (2000). [CrossRef]
  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. 84, 3081-3084 (2000). [CrossRef] [PubMed]
  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]
  15. 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]
  16. 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.
  17. A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986).
  18. T. Brabec and F. Krausz, �??Nonlinear Optical Pulse Propagation in the Single-Cycle Regime,�?? Phys. Rev. Lett. 78, 3282-3285 (1997). [CrossRef]
  19. 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]
  20. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (AIP, New York, 1992).
  21. 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]
  22. 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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