Characterization of ionization injection in gas mixtures irradiated by sub-petawatt class laser pulses
A. Zhidkov, N. Pathak, J. Koga, K. Huang, M. Kando, T. Hosokai
CCharacterization of ionization injection in gasmixtures irradiated by sub-petawatt class laserpulses A . Zhidkov , , , N . Pathak , ∗ , J . K . Koga , K . Huang , , M . Kando , andT . Hosokai , Institute of Scientific and Industrial Research (ISIR), Osaka University, Mihogaoka 8-1, Ibaraki,Osaka, 567-0047, Japan. Laser Accelerator R & D , Innovative Light Sources Division, RIKEN SPring-8 Center, 1-1-1,Kouto, Sayo-cho, Sayo-gun, Hyogo, 679-5148, Japan Kansai Photon Science Institute, National Institutes for Quantum and Radiological Science andTechnology, Kizugawa, Kyoto 619-0215, Japan
Abstract
Effects of ionization injection in low and high Z gas mixtures for the laser wake fieldacceleration of electrons are analyzed with the use of balance equations and particle-in-cell simulations via test probe particle trajectories in realistic plasma fields and directsimulations of charge loading during the ionization process. It is shown that electronsappearing at the maximum of laser pulse field after optical ionization are trapped inthe first bucket of the laser pulse wake. Electrons, which are produced by optical fieldionization at the front of laser pulse, propagate backwards; some of them are trapped inthe second bucket, third bucket and so on. The efficiency of ionization injection is not ∗ [email protected] a r X i v : . [ phy s i c s . p l a s m - ph ] N ov igh, several pC/mm/bucket . This injection becomes competitive with wave breakinginjection at lower plasma density and over a rather narrow range of laser pulse intensity. ntroduction Interest in laser wake field acceleration (LWFA) of electrons [1] has rapidly grown overrecent decades owing to notable results achieved by several groups [2–7]. Production of8 GeV electron bunches, in few tens of centimeters long plasma channel, with the chargearound several pC [7] demonstrates that LWFA may become a valuable branch in theelectron accelerator family. Electron self-injection in the acceleration part of the laser wakefield is a fundamental process for single-stage electron laser wake field acceleration [8–10] aswell as for a plasma cathode in multi-stage LWFA [11, 12]. Presently, electron self-injectionis considered to be the only way to generate electron bunches with characteristics suitablefor their further acceleration in laser pulse wakes [8, 12]. Therefore, investigation of thevarious mechanisms of electron self-injection is particularly important in the developmentof staging in LWFA.So far plasma wave breaking mechanisms have been considered as the primary sources ofelectron self-injection. Several mechanisms such as wave breaking at density ramps [8, 10],parametric resonances [13], frequency chirp [14, 15], relativistic wave breaking [16, 17], andwave breaking provoked by external sources [18] have been proposed and studied boththeoretically and experimentally. Recently, another mechanism of electron self-injectionhas been proposed; it is ionization injection [19–32], which essentially differs from thecommon wave breaking process.Ionization injection should occur in a low Z gas with a high Z dope; for example He − N , He − N e and so on. Optical field ionization of inner shell electrons of the high Z dope in the vicinity of the maximum of the laser pulse field produces a number of lowenergy electrons moving with a phase different from that of the laser pulse wake [19]. Theseelectrons can be trapped and further accelerated.Separation of ionization injection from the wave breaking injection can be done inrather low density plasma, which requires high power laser pulses in order to reach theself-focusing regime for essential electron acceleration. A laser pulse should have powerequal to 2-3 times the critical power for self-focusing, P cr = 1 . × − N cr /N e [ T W ] [33],where N e is the plasma density and N cr is the critical density corresponding to the laser3avelength. This implies that for N e < cm − and λ = 0 . µm , laser pulses with powerof the order of petawatt class is required.The principle of ionization injection and acceleration in a running wake wave can beillustrated by the simplest field structure as follows. A constant electric field with a negativestrength − E , lasting from V g t to L + V g t , where t is the time, L is the length of the wavemoving with velocity V g . If the initial position of an electron in this field at t = 0 is x = 0(rear side of the wave), the electron, in order to be further accelerated or to move with thewave, must have a velocity, v e , such that v e > V g . This is the typical condition for thewave breaking process. In contrast, if an electron at t = 0 has position x = L (front of thewave), it can be trapped even if its velocity is initially zero. For this to happen the time t ,over which the electron velocity becomes equal to the group velocity of the wave, should besuch that the electron position must still exceed the position of the wave rear, x = V g t : v g = p + eEt mc (cid:114) (cid:16) p + eEt mc (cid:17) v g t (cid:54) L + mc eE (cid:115) (cid:18) p + eEt mc (cid:19) − (1)where p is the initial electron momentum. The solution of Eq.(1) gives a rather softcondition: U (cid:62) mc (cid:16) − γ g (cid:17) − p v g , where U = eEL is the potential difference and γ g isthe relativistic factor for V g .In particular for a laser pulse wake, assuming E = E L ω pl ω [33], p = 0 and γ g (cid:29) E L is the strength of the laser pulse, ω pl the plasma frequency, and ω the laserfrequency, one can get the requirement for the trapping length L > λ pl πa with λ pl beingthe plasma wavelength and a = eE L mcω [33]. The t can be considered to be the injectiontime, t = γ g ω pl a . This time is not short; it is of the order of one picosecond even for thismaximum plasma field. Moreover, an electron, appearing inside a laser pulse, is trappedby the laser pulse fields and is accelerated by the pulse for a rather long time acquiringmomentum opposite to the laser propagation direction owing to the ponderomotive force.The soft condition for the length results in a stronger condition for ionization injection on4he laser pulse length and plasma density: cτ < λ pl − L , where τ is the pulse durationassuming that half of the laser pulse occupies half of the plasma wake wave. It is clear thationization injection can occur for low plasma density and rather short laser pulses. Thisexplains the similarity in the results of electron acceleration in pure Ar and He gases athigh densities [34]. However, for lower plasma density, several groups have reported a cleardependence of the results on the dope quantity [26]. Nevertheless, there is an alternativepoint of view that the dope may aid in the modulation of plasma density by low intensitylaser pre-pulses [35, 36] resulting in much better focus-ability of the laser pulses. All thesefacts require detailed analysis of the ionization injection and its efficiency via particle-in-cellsimulation.There are two main approaches in the calculation of plasma ionization in particle-in-cell methods. The first is the variable particle weight technique [37–39] in the Villasenor-Buneman method, and the second is the variable number of particles technique [40–42].The first is easier practically and less time consuming while the second allows for satisfyingof the initial conditions for the particles when they are born. Not obeying the initialconditions for newly produced particles may sometimes result in non-physical solutions,see for example Ref. [40]. Unfortunately, ionization injection also requires obeying theinitial conditions, p L = 0, where p L is the particle momentum in the direction of the laserpulse propagation, and the variable particle weight technique is not applicable in its simpleform. However, the density of ’ionization’ electrons is usually small compared to the plasmadensity, the parameter η/Z (cid:28) η = 1 [here η is the doping ratio and Z is thedope element charge]. Direct use of the method of variable number of particles [41] withthe necessary accuracy becomes impractical also. On the other hand the low density of’ionization’ electrons allows for the use of a perturbative approach in the solution of theproblem of ionization injection.According to the calculations in Ref. [39], propagation of a laser pulse in high Z gaswith optical field ionization runs similarly to that in a plasma without ionization, if theplasma density is properly determined. Again, the doping of a high Z gas in a low Z gascan be considered as a small perturbation for the plasma and laser field. It is clear that5or such conditions ionization injection can be characterized by a few parameters: plasmadensity, plasma density surplus in the vicinity of the laser pulse maximum, laser pulseintensity, and, weakly, by the spatial distribution of the laser light. In the present paperanalysis of ionization injection depending upon these parameters is performed in threeparts. In the first part the kinetics of the ion states, including inner shells, is consideredfor He − N and He − N e gas mixtures to understand the spatial distribution of the’ionization’ electron density. The second part is devoted to the investigation of ’ionization’electron trajectories in real wake fields depending on the initial position of the electrons.In the third part charge loading in the acceleration phase of laser pulse wake owing to theionization injection for different doping concentrations and plasma density is investigatedvia a self-consistent particle-in-cell simulation with separated post-processing for ’wavebreaking’ and ’ionization’ electrons. This allows for an estimation of the efficiency of thedifferent mechanisms of electron self-injection in gas mixtures.
Charge states in He − N and He − Ne mixtures The ion charge distribution under the action of strong femtosecond laser fields can becalculated with the common balance equations. In the absence of recombination the set ofequations is rather simple: ∂N ∂t = − S N ∂N z ∂t = S z − N z − (2)and, ∂N k ∂t = S k − N k − − S k N k k = 1 , ...., Z − N k is the density of ions with charge k , S k is the optical field ionization rate, Z is the nuclear charge of an element. There are several approximations for the ionizationrate [43]. We use the following form 6 k = 4 ω A g k ( I k /Ry ) / ( E A /E L ) exp (cid:18) −
23 ( I k /Ry ) / ( E A /E L ) (cid:19) [ s − ]with ionization potentials I k for He , N , and N e ions taking data from the mostcomprehensive calculations and experiments [44]. Here E A = m e / (cid:126) , ω A = me / (cid:126) , Ry = m e / (cid:126) , and g k is a factor of the order of unity. For a gas mixture Eq. (2) shouldbe calculated for each gas species. In Eq. (2) we implicitly assume that the density of ionswith charge n for which ( I n /Ry ) / E A /E L (cid:28) Z (cid:88) N k = constant is not difficult. It is clear that there is a saturation level of ion charge owing to theexponential dependency of the ionization rate on the ionization potential. In He gas sucha saturation occurs at the front of a powerful laser pulse. In the presence of a high Zdope; ionization of the outer shells and inner shell of the ions is different, because theionization potential of the inner shell is essentially higher than those for the outer shells bythe parameter ξ = [2 Z/ ( Z − ; for Z (cid:29) f s (such a pulse duration is typicalfor most of the existing high lower laser facilities) are presented for He − N and He − N e gas mixtures. To make the jump more visible the concentrations of the dopes is chosen tobe 5% for N and 10% for Ne. Figs.1 (a,b,c) show the evolution of the electron density alongthe laser axis. In the case of the nitrogen dope essentially a density jump in the vicinityof the laser pulse maximum (clear in insets) is seen already for a = 1 . a = 3 .
5. Within a distance of about 1 µm near the pulse maximum the density growth (cid:52) N is less than 1% for all sets of a . With an increase of a the density jump apparentlyshifts towards the front of the laser pulse. However, along the periphery of the laser pulsethere is a density jump for a = 3 . a necessary to have a density jump in the vicinity ofmaximum of the laser pulse as seen in Fig.2 (a,b,c) (see insets). In the case of a = 6the density growth reaches several percent and exceeds that for nitrogen. For a > a = 10as shown in Fig. 2(d).The distributions of ’ionization’ electrons which can be extracted for Fig.1 and Fig.2are very important in understanding the dynamics of ionization injection into the laserpulse wake. To clarify the physical picture of the dynamics of such electrons we performan investigation of electron trajectories in the realistic laser wake fields. Probing particles in particle-in-cell simulations
To understand how the position of ’ionization’ electron influences its further dynamicswe perform tests with probe particles during the propagation of a laser pulse in plasmausing the particle-in-cell method. The simulations are performed in 2D geometry, using themoving window technique. The window has size (100 × µm ; the spatial grid resolutionis λ/
36. Laser pulses with duration of 30 f s propagate in a uniform plasma with a densityof N e = 1 × cm − . The laser pulse intensity is varied from I = 1 × W cm − to I = 3 × W cm − and the laser spot size is w = 10 µm . Probe particles with smallcharge are placed near the maximum of the laser pulse after 140 µm of pulse propagationand start moving. There are five sets of the test probe particles, which are distributedaround the peak of the laser pulse. Each set has six particles located at different positionsin the transverse direction. Two sets of the test particles are placed in front side of thelaser pulse, and two sets are placed at the rear side. One set of particles is placed at thepeak of the pulse. These positions mimic well the appearance of ’ionization’ particles withinitially zero momenta.The resulting trajectories of the probe electrons for laser pulse intensity I = 3 × W cm − are presented in Fig. 3 (a-d) at t = 3 . ps . Particles situated at the maximumof the laser pulse have the trajectories shown in Fig.3(a). Initially particles are trappedby the laser pulse v × B force and move along the laser pulse propagation direction. Thenthe particles, having been overrun by the laser pulse, are taken up by the wake field andaccelerated. After approximately one picosecond all these particles are accelerated abovethe phase velocity of the plasma wave and, therefore, are injected. The evolution of theirmomenta can be seen in Fig.4(a): after short motion in the laser pulse the probe particlesare accelerated to over 100 M eV after several picoseconds. The relativistic factor for thephase velocity corresponding to the plasma density is γ ph = (cid:113) N cr a N e ≈
70. The particlesreach this energy after about 1 . ps . The effect of side scattering by the ponderomotiveforce is illustrated in Fig.4(b) by the transverse momenta of the particles. The effect existsbut is small for the particles, the geometrical emittance for them is within 3 × − rad .A very different picture emerges in the case where the initial positions of the particlesare before the maximum of the laser pulse field as shown in Fig. 3(b). One can observeno trapping of these particles in the first bucket of the wake. However, one particle istrapped and accelerated in the second bucket. The fact of trapping for this particle canbe proved by particle momentum shown in Fig.4(c) where the velocity of this particleexceeds the phase velocity of the wave. The emittance for this particle is not small beingof the order of 0 . rad . Particles situated further from the pulse field maximum cannotbe trapped even in the second bucket; a portion of them move to the third bucket and canbe trapped there. This effect strongly depends on the longitudinal momentum acquired bythe particle after the laser pulse has overrun it. The simulation shows that particles in thefront of the laser pulse get a larger momentum directed counter to the laser pulse motion.Moreover, we observe no trapping in the first bucket of the probe particles situated at themaximum of the laser field in the case of lower pulse intensity I = 1 × W cm − withthe other conditions being the same. For higher intensity, some particles situated behindthe maximum of the pulse field get almost zero momentum and can be efficiently trapped,as shown in Fig.4(d). However, the number of such particles is not high as seen From Fig.1and Fig.2 where the increase in the number of the ’ionization’ electrons after the peak of9he laser pulse is small or negligible compared to the whole plasma density. In contrastmost of ’ionization’ electrons are born before the maximum of the laser field. This mayresult not only in continuous injection of ’ionization’ electrons in the first bucket but quiteefficient injection of them into consequent buckets. Since the ionization injection is moreefficient in the absence of wave breaking the number of buckets can be large, which resultsin a long length for the final electron bunches of the order of picoseconds. Ionization injection in low density plasma
Effects of ionization injection including charge loading can be calculated only with the useof the particle-in-cell method. It is apparent that ionization of plasma should be included.Upon considering a lower intensity laser pulse a <
10 we neglect ion motion and considerionization processes using this approximation. Again, there are two methods to calculateplasma ionization in the framework of particle-in-cell simulation. The first is by the variableparticle weight (VPW) method. In this method, particles emulating electrons are initiallyempty and are filled with electrons in time according with Eq. (1) [39]. This method isnot time consuming. However, it has a disadvantage: it is impossible to apply arbitraryinitial conditions for the electrons. For example, in the problem of light scattering froman ionization wave the fact that the variable particle weight method results in unphysicalsolutions is explained in detail in [40]. Another method is the variable number of particles(VNP) [41,42]. N particles per a cell have the same weights W . The value N × W representsthe total possible electron density in the plasma. Initially all particles are immobile. Withtime the number of movable particles increases in accordance with Eq. (1). A new particleis involved in the motion with the necessary initial conditions for its momentum. Thedisadvantage of this approach is that large number of particles are necessary to makea smoother spatial electron density distribution. In the case of ionization injection thedensity jump, (cid:52) N , is rather small and the use of VNP requires a resolution better than (cid:52) N/N e or the number of particles N > N e (cid:52) N .There is an alternative approach allowing essential lessening of computing resourceswith a rather accurate evaluation of the ionization injection. The alternative is in the10ombination VNP and VPW. For the case of (cid:52) N/N e (cid:28) m slices with chargesproportional to (cid:52) N/m and with a motion parameter which switches from ’false’ to ’true’when a k immovable ’ionization’ particle crosses (in the moving window) the vicinity of thelaser field with strength E Lk . With the motion parameter ’true’ the ’ionization’ particle isinvolved in self-consistent motion similar to ’plasma’ particles.The simplest case is the single slice of ’ionization’ particles which start moving whencrossing the ’maximal’ strength of the laser field. We perform such alternative particle-in-cell simulations in two dimensions for gas mixtures using N e and (cid:52) N as parameters for aplasma irradiated by a 30 f s laser pulse with λ = 0 . µm and intensity I = 3 × W cm − focused to w = 10 µm . The moving box has size (100 × µm and the spatial gridresolution λ/
36. The number of slices for ’ionization’ was one and five. We use a uniformplasma with a fixed density for easier control of the parameters. According to Ref. [39] suchan approach does not essentially change the plasma field distributions even for pure high Z gas. The transverse distribution of the charges of the ’ionization’ particles is Gaussianwith size equal 0 . w centered around the laser propagation axis.Momentum distributions for ’plasma’ and ’ionization’ particles for a plasma with density N e = 1 × cm − are shown in Fig. 5 (a-d) at t = 4 . ps pulse propagation and for (cid:52) N = 10 − N e . Such a density jump corresponds to 5% of N according to the ionizationbalance in the laser pulse field. According to ’plasma’ particles, see Fig 5(a,b), the wavebreaking injection is rather weak and occurs in the third and fourth buckets. As forionization injection, Fig.5(c,d), it starts in the first bucket and gradually appears in theother buckets. The energy of the injected electrons in the first bucket reaches 150 M eV after 4 ps and increases. The electron energy in the second, third, and fourth bucket alsoincreases similarly with a clear time delay. The evolution of the total energy distributiontaking into account the particle weights is shown in Fig. 6(a-c). One can see two parts11eparated by a peak in the figures. The lower energy part originates from the wave breakingwhile the higher energy part is from pure ionization injection. Both parts are flat withoutcharacteristic features. However, the wave breaking injection provided a larger charge ofaccelerated electrons than the ionization injection as seen in Fig. 6(c). We also have toseparate out different sorts of electrons from the ionization injection. There are manyelectrons involved in the initial acceleration by the pulse wake. These electrons acquirecertain energies around up to tens of M eV . However, they are not finally trapped in anybucket and form a sort of clouds with very low emittance. The number of such electrons farexceeds the number of injected electrons. Another negative feature of ionization injection isthe lengthening of the total electron bunch after electron injection in consequent buckets.Since the wave breaking process is not strong, as seen in Fig. 7 (a,b), the number ofbuckets is quite large. All these buckets will be filled with injected electrons which arefurther accelerated. Such beams may have durations of several picoseconds.The results presented in Fig. 5-7 have been obtained for a rather low concentration ofhigh Z gas. Fig. 8(a-d) illustrates what happens if the density of high Z gas is increasedby factor of 10. One can see that the ionization injection in this case becomes dominant.Moreover, it suppresses the electron self-injection by the wave breaking process as seen inFig. 8(a,b). However, the maximal energy of the accelerated electrons is lower than thatfor the lower concentration of high Z gas (also see Fig. 9). The most efficient injection andacceleration occur in the second bucket, which reflects upon the increase of the effect of thepre-accelerated but not injected electrons on the acceleration process. The number of suchelectrons essentially increases. It is important to note that the practical realization of suchconditions requires much higher laser pulse power. Since now the electron density is formedby a high Z gas, the power of the laser pulse should be equal to 2-3 times the critical powernot for the maximum ionization plasma density but for the minimum, which is almost 10times smaller. This is because as the laser pulse focuses the intensity is low enough sothat initially the laser is only propagating in a singly ionized plasma. To insure relativisticfocusing this requires higher laser power. For the present parameters N gas requires adensity of N e = 1 × cm − for which the laser pulse power should be about petawatt12evels. Otherwise the pulse diffraction will shorten the length of the higher density plasma.Working with high power may be critical to the ionization injection, however, this mayrestrict the process by itself. For laser pulse intensities approaching I = 1 × W cm − relativistic wave breaking [16], which occurs for a (cid:112) γ ph − λ pl a / /w (cid:29)
1, may make the wave breaking injectiondominant even for low density plasma.The pure ionization injection case, which is at lower density than the previous cases,is illustrated in Fig. 10 (a-d). In a plasma with density N e = 5 × cm − and (cid:52) N =5 × cm − irradiated by laser pulses with intensity I = 3 × W cm we observeno results of the wave breaking processes. It clearly seen in Fig.10 (a,b) where thereis no visible self-injected plasma electrons. In contrast, one can see in Fig.10 (c,d) anefficient injection to the third bucket with continuous electron acceleration after 4 ps oflaser pulse propagation. In the first and second buckets there are only pre-acceleratedelectrons. At this time, t = 4 . ps , it is impossible to say whether these electrons will befurther accelerated or not. In Fig.11 spatial distributions of the total electron density andplasma electric field are shown. One can see a small channel-like structure appearing alongthe laser axis. We anticipate that this structure is formed by pre-accelerated electronsmoving through the buckets. The evolution of the total distribution function is shown inFig. 12. Even for t = 4 . ps we observe no electrons with velocity exceeding the phasevelocity of the plasma wave, which for this condition should be about γ = 100. Conclusion
In conclusion, we have characterized the effect of ionization injection for the laser wakefield acceleration of electrons in gas mixtures using self-consistent particle-in-cell simula-tions. First, we applied the technique of test probe particles to investigate the condition ofparticle trapping in the acceleration phase of the wake field. Second, we performed two di-mensional particle-in-cell simulations splitting ’plasma’ electrons and ionization’ electrons,which allowed us to investigate the entire process of electron pre-acceleration, trapping,and further acceleration. 13irect simulations have shown the efficiency of continuous ionization injection in a lowdensity plasma with low and high Z components. Electrons, generated by optical fieldionization of the ion inner shell in the vicinity of the maximum of the laser pulse field,after their propagation inside the pulse can be pre-accelerated by the wake field to energieshigh enough for their further acceleration. These electrons can form a high energy, lowemittance beam. The total charge of the beam depends on the matching between the laserintensity and the high Z component. Electrons appearing before the maximum of laserpulse field due to ionization of inner shell cannot be trapped in the first bucket and finallyform a cloud of pre-accelerated electrons or are injected in wake buckets far behind the laserpulse. The later process results in the formation of a long electron bunch with a durationof several picoseconds. Optimal matching can provide a density jump equal ηN e /Z with η being the concentration of the high Z gas. With plasma density increase the ionizationinjection cannot compete with the wave breaking injection. Moreover, when the laser pulselength exceeds the plasma wavelength, the ionization injection vanishes.Typically in experiments the density of the high Z gas is small being of the order of afew percent. For densities of ionization electrons (cid:52) N = 10 − cm − [(5 −
10) % of high Z dope] and a diameter of the injection section of about 5 µm (see Fig.1 and Fig.2) one canestimate the total charge per mm of acceleration length to be (40 − pC/mm . However,the low efficiency for trapped electrons, which is less than 10% of the total pre-acceleratedparticles, essentially reduces the charge of accelerated electrons. The total charge maybe considerably higher after ionization injection fills many wake buckets. However, suchbeams can be quite long with their duration being picoseconds. Acknowledgements
This work is funded by the JST-MIRAI program grant no. JPMJMI17A1, and was par-tially supported by the ImPACT R&D Program of Council for Science, Technology andInnovation (Cabinet Office, Government of Japan). We are grateful to Prof. Yuji Sano forencouragement and helpful discussions. We also acknowledge the use of Mini-K computingfacility at SACLA, RIKEN, SPring-8 Center.14 eferences [1] T. Tajima, and J. M. Dawson, Phys. Rev. Lett. , 267 (1979).[2] S. P. D. Mangels, C. D. Murphy, Z. Najmudin, A. G. R. Thomas, J. L. Collier, A.E. Dangor, E. J. Divall, P. S. Foster, J. G. Gallacher, C. J. Hooker, D. A. Jaroszynski,A. J. Langley, W. B. Mori, P. A. Norreys, F. S. Tsung, R. Viskup, B. R. Walton, andK. Krushelnick, Nature (London) , 535 (2004).[3] C. G. R. Geddes, Cs. Toth, J. van Tilborg, E. Esarey, C. B. Schroeder, D. Bruhwiler,C. Nieter, J. Cary, and W.P. Leemans, Nature (London) , 538 (2004).[4] J. Faure, Y. Glinec, A. Pukhov, S. Kiselev, S. Gordienko, E. Lefebvre, J. P. Rousseau,F. 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5, and (c,d) a max = 3 .
5. Arrows in (b) shows the range of ’plasma’ electronsand ’ionization’ electrons. Insets show the electron densities around the peak of the laserpulse. 19 a) (b)(c) (d)
Figure 2: Dynamics of the electron density in the gas mixture He − N e with 10% of
N e along the laser axis (a-c), and off-axis (d) depending on the laser pulse intensity: (a) a max = 3 .
5, (b) a max = 6, and (c,d) a max = 10. Insets show the electron densitiesaround the peak of the laser pulse. 20 a) (b)(c) (d) Figure 3: Trajectories of test probe particles placed at distance (cid:52) x from the maximumof the laser pulse with 30 fs duration and intensity I = 3 × W cm − . (a) (cid:52) x = 0 µm ,(b) (cid:52) x = − µm , (c) (cid:52) x = − µm , and (d) (cid:52) x = 2 µm . Different colors correspondsto different transverse positions of the probe particles: green- (cid:52) y = − . µm , blue- (cid:52) y = − . µm , red- (cid:52) y = − . µm , cyan- (cid:52) y = 0 . µm , black- (cid:52) y = 0 . µm ,magenta- (cid:52) y =0 . µm . Inset shows magnified view of different regions in the wave buckets. Laser pulse ispropagating from right to left hand side. The colorbar shows electron density normalizedby critical density. 21 a) (b)(c) (d) Figure 4: Evolution of the longitudinal and transverse momentum of the probing particlesplaced at a distance (cid:52) x from the laser pulse with 30 fs duration and intensity I = 3 × W cm − . (a,b) (cid:52) x = 0 µm , and (c,d) (cid:52) x = − µm . Different colors correspondsto different transverse positions of the probe particles: green- (cid:52) y = − . µm , blue- (cid:52) y = − . µm , red- (cid:52) y = − . µm , cyan- (cid:52) y = 0 . µm , black- (cid:52) y = 0 . µm ,magenta- (cid:52) y =0 . µm . 22 a)(b) Figure 5: Spatial distribution of electrons momenta at t = 4 . ps in a plasma with N e = 10 cm − , and (cid:52) N = 10 cm − irradiated by a laser pulse with I = 3 × W cm − , τ = 30 f s . (a) ’plasma’ electrons, and (b) ’ionization’ electrons. Laser pulse is propagatingfrom right to left hand side. 23 a)(b)(c) Figure 6: Evolution of the total energy distribution for a plasma with N e =10 cm − , (cid:52) N = 10 cm − irradiated by a laser pulse with I = 3 × W cm − , τ = 30 f s at (a) t = 1 . ps , (b) t = 2 . ps , and (c) t = 4 . ps .24 a) (b) Figure 7: Spatial distribution of (a) electrons density and (b) x-component of the electricfield for a plasma with N e = 10 cm − , and (cid:52) N = 10 cm − irradiated by a laser pulsewith I = 3 × W cm − , τ = 30 f s at t = 4 . ps . Laser pulse is propagating from rightto left hand side. In (a) the colorbar shows electron density normalized by critical densityand in (b) the colorbar shows axial field in normalized units ( eEx/mωc ).25 a)(b) Figure 8: Spatial distribution of electrons momenta at t = 4 . ps in a plasma with N e = 10 cm − , and (cid:52) N = 10 cm − irradiated by a laser pulse with I = 3 × W cm − , τ = 30 f s . (a) ’plasma’ electrons, and (b) ’ionization’ electrons. Laser pulse is propagatingfrom right to left hand side. 26igure 9: Total energy distribution for a plasma with N e = 10 cm − , (cid:52) N = 10 cm − irradiated by a laser pulse with I = 3 × W cm − , τ = 30 f s at t = 4 . ps .27 a)(b) Figure 10: Spatial distribution of electrons momenta at t = 4 . ps in a plasma with N e =5 × cm − , and (cid:52) N = 5 × cm − irradiated by a laser pulse with I = 3 × W cm − , τ = 30 f s . (a) ’plasma’ electrons, and (b) ’ionization’ electrons. Laser pulse is propagatingfrom right to left hand side. 28 a) (b) Figure 11: Spatial distribution of (a) electrons density and (b) x-component of the electricfield for a plasma with N e = 5 × cm − , and (cid:52) N = 5 × cm − irradiated by a laserpulse with I = 3 × W cm − , τ = 30 f s at t = 4 . ps . Laser pulse is propagating fromright to left hand side. In (a) the colorbar shows electron density normalized by criticaldensity and in (b) the colorbar shows axial field in normalized units ( eEx/mωc ).29 a)(b)(c) Figure 12: Evolution of the total energy distribution for a plasma with N e = 5 × cm − , (cid:52) N = 5 × cm − irradiated by a laser pulse with I = 3 × W cm − , τ = 30 f s at (a) t = 1 . ps , (b) t = 2 . ps , and (c) t = 4 . psps