This chapter looks at numerical simulations of photon acceleration occurring during the modulation of a long laser pulse in plasm

Bingham, Robert

Mendonça, J.T.

Murphy, C.D.

Norreys, P.A.

Silva, L.O.

Trines, Raoul M. G. M.

English

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryTrines, Raoul M. G. M. and Murphy, C.D. and Norreys, P.A. and Mendonça, J.T. and Silva, L.O.and Bingham, Robert (2007) Numerical simulations of photon acceleration occurring during themodulation of a long laser pulse in plasma. In: STFC Central Laser Facility Annual Report 2007/08.STFC, Swindon, pp. 138-141.Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/3HIGH POWER LASER SCIENCE | THEORY AND COMPUTATIONCENTRAL LASER FACILITY ANNUAL REPORT | 2007/2008138IntroductionIn laser-wakefield acceleration, the strong plasmawakefields are not only capable of accelerating electrons,but also of changing the frequency (energy) of the photonsin the driving laser pulse [1–6]. For this concept to work, thelaser frequency commonly needs to be much higher thanthe plasma frequency, so the space and time scales of theplasma perturbations are much larger than the photonwavelength and period. In this case, geometric optics canbe used to describe the motion of the electromagneticwave-packets as well as the influence of the plasma on thismotion [6]. The action of the photons on the plasma can bedescribed through the action of the ponderomotive force [4].Experimental confirmation of photon acceleration bylaser-induced wakefields was first reported by Murphy,Trines et al. [7].In this paper, we will investigate the role of photonacceleration in the modulational instability of a laser beamin a plasma. The modulational instability is an effect thattends to break up an initially uniform electromagnetic(EM) wave propagating through a plasma, and can beunderstood in terms of the ponderomotive force. If theEM wave amplitude has a maximum at some point, thenthe ponderomotive force tends to push plasma away fromthat point and to produce a region of reduced density. Thevariation with density of the plasma refractive index issuch that the EM wave tends to be refracted towardsregions of lower density, with the result that the originalamplitude perturbation is enhanced. If the break-up is inthe direction of propagation, it is called the modulationalinstability, while a transverse break-up is called thefilamentation instability. Detailed information on themodulational instability can be found in Refs. [13–15].From the work by Silva et al. [3, 4] and Mendonça [6], itfollows that the back action of the laser’s ponderomotiveforce will change the momentum (wave number) andenergy (frequency) of the laser’s photons themselves. Thus,the modulational instability will not only change theenvelope of the laser beam in real space, but also itsFourier spectrum. In particular, the spectrum will broadenand a series of peaks will emerge on that are separated bymuch less than the plasma frequency (so they can bedistinguished from mere stimulated Raman scattering),while ionization blueshift would shift the spectrum as awhole (cf. Koga et al. [12]) rather than modulating it. Theextent of the spectral broadening and the separation of thepeaks are a measure of the amplitude of the laser-drivenwakefield, and can be used to diagnose this wakefield. Inthis paper we will discuss the spectral modulations of thedriving laser pulse itself; as an alternative (e.g. when thedriving pulse is too short to be properly modulated) along, low-intensity ‘witness’ pulse can be launchedsimultaneously with the driving pulse, and the wakefieldcan be studied through modulations of this witness pulse.TheoryThe modulational instability causes the fundamentalspectral peak of the laser to split into several narrowerpeaks, having a mutual separation much smaller than ωp.It will have the same effect on any (anti-)Stokes peaks inthe spectrum, although we did not study this in ourexperiment. This distinguishes the modulational instabilityfrom e.g. stimulated Raman scattering, which onlyproduces discrete peaks separated by the plasma frequencyωp. This can be understood as follows.Consider a laser EM field E = E0 exp i(k0x – ω0t)propagating through a plasma slab of thickness s and densityn0, accompanied by a co-propagating plasma densityperturbation δn = δn0 cos(kpx – ωpt) with kp = k0ωp/ω0. Thedispersion relation of the laser wave is given by:(1)where γ denotes the relativistic Lorentz factor due toplasma electron motion and other symbols have their usualmeaning. After traversing the plasma slab, the laserfrequency will have been modulated periodically:ω = ω0 + δω, with (using ray tracing theory):D ω2– e2k2 – (e2/ε0me)(n/γ) = 0,δω ≈ E ≈ E0 exp i(k0x – [ω0 – δω0 sin(kpx – ωpt)]t)= E0 exp i(k0x – ω0t) ∑ Jl(tδω0) exp il(kpx – ωpt).– δω0 sin(kpx – ωpt),=dωdtkgωϑD/∂t k8ϑD/∂ω ω≈ =12ω0e2ε0γmeϑnϑtk08ω0∞l≈–∞D ω2 – e2k2 – (e2/ε0me)(n/γ) = 0,δω ≈ E ≈ E0 exp i(k0x – [ω0 – δω0 sin(kpx – ωpt)]t)= E0 exp i(k0x – ω0t) ∑ Jl(tδω0) exp il(kpx – ωpt).– δω0 sin(kpx – ωpt),=dωdtkgωϑD/∂t k8ϑD/∂ω ω≈ =12ω0e2ε0γmeϑnϑtk08ω0∞l≈–∞Numerical simulations of photon accelerationoccurring during the modulation of a long laserpulse in plasmaR. M. G. Trines, C. Murphy, P. A. Norreys Central Laser Facility, STFC, Rutherford AppletonLaboratory, HSIC, Didcot, Oxon OX11 0QX, UKJ. T. Mendonça and L. O. SilvaGOLP/Instituto de Plasmas e Fusão Nuclear, InstitutoSuperior Técnico, Lisbon, PortugalR. BinghamSpace Science and Technology Department, STFC,Rutherford Appleton Laboratory, HSIC, Didcot, Oxon OX11 0QX, UKContact | raoul.trines@stfc.ac.uk 3HIGH POWER LASER SCIENCE | THEORY AND COMPUTATIONCENTRAL LASER FACILITY ANNUAL REPORT | 2007/2008 139with δω0/ωp = k0sωp2/(2ω02)δn0/n0. Using standard seriesexpansions [20], we find that, after passage through theplasma slab, the modulated laser field takes the form:For fixed x, we find that the modulated frequencyspectrum of E now contains peaks at ω0 ± lωp ± δω0,l = 0, 1, 2, … Note that the frequency shifts of ±δω0 are adirect consequence of the fact that the Bessel functions Jlare approximately periodic with period 2 pi. While the(anti-)Stokes peaks at ω0±lωp will also occur for ‘classical’Raman scattering of a monochromatic beam, theadditional contribution of ±δω0 is typical for a beam thatis frequency-modulated by photon acceleration. Thesemodulations will be most visible around the fundamentalpeak of the laser spectrum, and our experiments will alsoconcentrate on the study of this peak.In realistic situations, the density perturbation will nothave constant amplitude along the length of the laserpulse; instead, the perturbation will be driven resonantlyby the modulated pulse and its amplitude will increasefrom the front to the back of the pulse. Therefore, eachperiod of the density perturbation covered by the pulsewill add contributions at ±δω0 for a different value of δω0.This will cause the fundamental laser peak to separate intoa series of narrow peaks separated by δω0 << ωp, Thisseparation increases with increasing plasma density, as δω0 ~ ωp ~ n01/2. By determining the background densityn0 and the thickness of the plasma slab s, it can be workedout what the wakefield amplitude must have been.Therefore, study of the spectral modulations induced byphoton acceleration of a long laser pulse may eventuallyyield a single-shot wakefield diagnostic.SimulationsWe have performed simulations to match the experiments,using a photon kinetic numerical code. In this code, theindividual electromagnetic wave modes that make up thefinite-length, finite-bandwidth laser pulse are treated asparticles, while the electron plasma is treated as a fluid.A detailed description of the model can be foundelsewhere [7,23]. Simulation parameters were as follows:a 45 fs, bandwidth limited pulse with λ = 800 nm waschirped (lowest frequencies first) and stretched to aFWHM of 180 fs and a peak amplitude of eA/(mec) = 0.4.The plasma density n0 was 2, 4, 8 or 16 × 1018 cm-3 givingω0/ωp = 29, 4, 20.9, 14.9 or 10.4 respectively. For n0 = 1.6 × 1019 cm-3, a snapshot was taken after ωpt = 7200, or about 9 mm propagation length.Simulation results are shown in figures 1 (n0 = 1.6 × 1019 cm-3only) and 2 (all densities). The left frame of Figure 1 showsthe frequency spectrum of the transverse EM field (laserfield) after 9 mm of propagation. It can clearly be seen thatthe fundamental peak of the spectrum has separated intoseveral peaks. The corresponding photon distribution as afunction of x and k can be seen in the right frame. Thephoton distribution has been modulated by the underlyingplasma wave, and various populations of photons can beseen both above and below the original wave number k0 = 10.4*kp. These photon populations are responsible forthe peaks in the spectrum: where the original photondistribution consisted of only one population which causedonly one peak, the final distribution consists of a number ofphoton populations that have been accelerated to wavenumbers different from k0 and thus contribute individualpeaks to the laser spectrum.The main features of the laser spectrum in figure 1 are thedistinctive peaks separated by about ωp. These are thestandard (anti-)Stokes peak corresponding to classicalRaman forward scattering, whose emergence has beenD ω2 – e2k2 – (e2/ε0me)(n/γ) = 0,δω ≈ E ≈ E0 exp i(k0x – [ω0 – δω0 sin(kpx – ωpt)]t)= E0 exp i(k0x – ω0t) ∑ Jl(tδω0) exp il(kpx – ωpt).– δω0 sin(kpx – ωpt),=dωdtkgωϑD/∂t k8ϑD/∂ω ω≈ =12ω0e2ε0γmeϑnϑtk08ω0∞l≈–∞Figure 1. Simulation results for the modulational instability of a long laser pulse. Depicted are the transmission spectrum (top,intensity versus ω/ωp) and the distribution of the laser’s photons in (x, k)-space (bottom, ck/ωp versus ωpx/c) after the laserpulse propagated through the plasma for 9 mm. The transmission spectrum shows large peaks separated by ωp, indicating thatthe modulational instability is fully developed to the point of saturation. The photon distribution displays extensive phasemixing of the photons, most of which have already completed several closed orbits in (x, k)-space by this time. Theaccumulation of photons at the ‘top’ and ‘bottom’ of such closed orbits leads to the peaks in the transmission spectrum.3HIGH POWER LASER SCIENCE | THEORY AND COMPUTATIONCENTRAL LASER FACILITY ANNUAL REPORT | 2007/2008140predicted above. Their presence indicates that themodulational instability is well developed at the time thesnapshot was taken and saturation has already set in. Thisis confirmed by the right frame of figure 1, which showsthe position and wave number of the laser pulse’s photons.Almost all photons have shifted considerably with respectto their original wave number k0, and have completed atleast one turn along a closed orbit in (x, k)-space.Extensive phase mixing has occurred and the spectralbroadening has mostly saturated.Although the narrow peaks at ω0 ± δω are just aboutvisible in the simulated spectra at early times, it is easier toobtain information about δω by simply looking at the fullwidth of the laser spectrum before the emergence of the(anti-)Stokes peaks at ω0 ± l*ωp. To illustrate this, both thewidth of the frequency spectrum and the amplitude of thelongitudinal electrostatic field Ex (the wakefield) have beenplotted versus propagation distance in figure 2 forsimulated densities of n0 = 2, 4, 8 and 16 × 1018 cm-3. Acareful analysis shows that the growth time of both thespectral width and the wakefield is proportional to 1/ωp3.This is different from the usual expression of 1/ωp3/2 or1/ωp2 [22], but one should bear in mind that these resultswere calculated for laser beams of infinite length. Using apulse of fixed, finite length introduces a strongerdependence of the growth time on ωp, because the pulselength increases relative to the plasma period for anincrease in ωp. Since the spectral width grows at a rateequal to the wakefield amplitude, this proves that ananalysis of the spectral width after the interaction willyield the peak wakefield amplitude reached during theinteraction. Even better results will be obtained when themodulation of a low-intensity ‘witness’ laser pulse isstudied rather than that of the driving pulse, as there willbe much less evolution of the wakefield under the witnesspulse in that case, which renders it easier to relate thespectral broadening back to the wakefield amplitude.ConclusionsWe have studied the role of photon acceleration in themodulational instability of a long laser pulse (180 fs) withan underdense plasma. The modulational instability causesthe laser pulse envelope to be modulated on the lengthscale of the laser-driven plasma wave, and it has beenpredicted that the plasma wave will also periodicallymodulate the laser frequency.We have developed an analytic model for the spectralmodulations that result from a periodic acceleration/deceleration of the laser’s photon population. This modelpredicts that the fundamental spectral peak (and theStokes satellites, which we did not study however) will splitinto two peaks with frequency ω = ω0 ± δω with δω0/ωp =k0sωp2/(2ω02)δn0/n0, s is the thickness of the plasma slaband δn0 is the plasma wave amplitude. Since the plasmawave amplitude will grow throughout the laser pulse, eachplasma period will contribute its own pair of spectralpeaks at a different value of δω, leading to the generationof a number of narrow peaks, separated by considerablyless than the plasma frequency.For a maximum separation of up to 15 nm, as seen in arecent experiment [23], we have δω/ωp = (ω0/ωp)(δλ/λ0) = 0.2,and a gas jet of about 3 mm wide will give us a densityperturbation of δn/n0 = 0.016. This value is in line withexpectations for the case of a long, low-intensity laserpulse interacting with a tenuous plasma, and confirms thevalue of photon acceleration as a wakefield diagnostic.Photon kinetic simulations show the separation of thefundamental peak into secondary peaks, as predicted bythe theory and observed in the experiment. Thesimulations don’t contain an ionisation model and don’texhibit an overall blueshift of the spectrum. This confirmsthe notion that a shift of the spectrum as a whole is causedby ionisation blueshift, while modulations of the spectrumare caused by wakefield-driven photon acceleration.Figure 2. Simulation results for the modulational instability of a long laser pulse. Depicted are the amplitude of thelongitudinal electric field eE/(meωpc) (top) and the width of the laser’s spectrum δω/ω0 (bottom) versus propagation distancein mm, for background electron densities of n0 = 2 (black), 4 (blue), 8 (green) and 16 × 1018 cm-3 (red). It is found that bothquantities, in particular the wakefield amplitude, grow exponentially once the modulational takes off properly. The growthtime has been found to scale with 1/ωp3 due to the effect of finite pulse length on the development of the modulationalinstability. For the highest density, it is found that the modulational instability saturates after about 4 mm of propagation.3HIGH POWER LASER SCIENCE | THEORY AND COMPUTATIONCENTRAL LASER FACILITY ANNUAL REPORT | 2007/2008 141AcknowledgementsOne of the authors (RT) would like to thank T. Katsouleasfor valuable comments. This work was supported by theSTFC Accelerator Science and Technology Centre.References1. S. C. Wilks, J. M. Dawson, W. B. Mori, T. Katsouleas,and M. E. Jones, Phys. Rev. Lett. 62, 2600 (1989).2. J. M. Dias, L. O. Silva and J. T. Mendonça, Phys. Rev.ST Accel. Beams 1, 031301 (1999).3. J. T. Mendonça and L. O. Silva, Phys. Rev. E 49, 3520(1994).4. L. O. Silva, R. Bingham, J. M. Dawson, and W. B. Mori, Phys. 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Bingham and C. N. Lashmore-Davies, PlasmaPhysics [Plasma Phys. Contr. Fusion] 21, 433 (1979).16. N. H. Matlis, S. Reed, S. S. Bulanov, et al., NaturePhysics 2, 749 (2006).17. J. P. Geindre et al., Opt Lett. 19, 1997 (1994).18. J. R. Marque´s, J. P. Geindre, F. Amiranoff,P. Audebert, J. C. Gauthier, A. Antonetti and G. Grillon, Phys. Rev. Lett. 76, 3566 (1996).19. C. W. Siders, S. P. Le Blanc, D. Fisher, T. Tajima,M. C. Downer, A. Babine, A. Stepanov and A. Sergeev,Phys. Rev. Lett. 76, 3570 (1996).20. M. Abramowitz and I. A. Stegun, Handbook ofMathematical Functions with Formulas, Graphs, andMathematical Tables (Dover, New York, 1964).21. C. J. Hooker et al., Journal de Physique IV 133, 673(2006).22. C. J. McKinstrie and R. Bingham, Phys. Fluids B 4,2626 (1992).23. R. Trines, C. Murphy, et al., Plasma Phys. Contr.Fusion, in press (2008).

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Numerical simulations of photon acceleration occurring during the modulation of a long laser pulse in plasma

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Numerical simulations of photon acceleration occurring during the modulation of a long laser pulse in plasma

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