Theory of the ultra-intense short-pulse laser interaction with underdense

Transcription

1 Theory of the ultra-intense short-ulse laser interaction with underdense lasma J. Yazdananah 1*, E. Yazdani 2, A. Chakhmachi 1 and E. Khalilzadeh 1, 3 1 The Plasma Physics and Fusion Research School, Tehran, Iran 2 Laser and otics research school, P.O. Box , Tehran, Iran 3 Deartment of Physics, Kharazmi University, 49 Mofateh Ave, Tehran, Iran Abstract A comrehensive theory is roosed to describe the roagation and absortion of ultra-intense, short laser ulse through the under-dense lasma. The kinetic asects of lasma are fully incororated using extensive article-in-cell (PIC) simulations. It is turned out that the lasma behavior is characterized by both its density and the ratio of the ulse length to the lasma wavelength. According to exact analyses and direct simulation evidences, at ultra-low densities the laser ulse is adiabatically deleted (absorbed) by the wake excitation. And the deletion is accomanied by the overall radiation red-shift. At these densities, for ulse lengths larger than the lasma wavelength the Raman tye scatterings also occur without causing instability. When the lasma density grows toward the critical density, a comletely new regime aears with the main character of highly unsteady light roagation. Here, based on analyses and simulations, the radiation ressure 1

2 induced wave breaking (RPIWB) heavily destroys the oscillatory structure of the electron wave behind the first lasma eriod. On the other hand, the electron density rofile induced by the onderomotive force ahead of the ulse begins to steeen and eventually acts as a flying mirror. As a result, the ulse becomes bake scattered and its enetration becomes limited. The radiation ressure of reflecting ulse sustains a longitudinal electric field which accelerates electrons roduced via RPIWB causing the lasma to undergo volumetric heating. Mostly imortant, based on a newly roosed model it is shown that the interaction becomes saturated at a definite time and that the overall absortion and lasma heating are directly related to this time. Also it is found that the saturation time decreases by a factor of ne / nc( n e, n c and are initial electron density, critical density and initial laser gamma factor). Other details, mainly related to different active electron acceleration mechanisms, have minor effects in absortion which are also discussed. Key words: Nonlinear relativistic laser lasma interactions, Under-dense lasma, PIC simulations, Laser absortion, Anomalous light scattering, Wave breaking, DLA. * jamyazdan@gmail.com 2

3 I. INTRODUCTION The current ability of roduction of ultra-short, ultra-intense laser ulses [1-2] have stimulated extensive researches in the field of laser-matter interactions. Attemts are mainly conducted toward the construction of comact article accelerators [3-18] and bright X-ray sources [19-2], and realization of conditions for the Fast Ignition (FI) scheme of Inertial Confinement Fusion (ICF) [21-23]. The roagation of intense laser ulse through under-dense lasma is an imortant common art of all these scenarios. This subject has recently received a vast number of attentions [14-18, 24-37]. The sub-critical densities are either roduced due to the unavoidable reulse effects in the laser-solid interaction [13] or initially exist in the form of gas jets [28] or the state-of-the-art foam targets [25]. From the alication oint of view, it is mostly desired to know how the intense laser light is absorbed and scattered during its roagation through the under-dense lasma. In the laser-solid exeriments, it is intended to know what amount of laser light is absorbed in the relasma rofile induced by the reulse in the front of target before it arrives at the critical surface where a new interaction mechanism is set. In the FI scheme of ICF the crucial question is the amount of laser enetration toward the fuel core. This issue is closely related to the light absortion and scattering in the sub-critical fuel corona. In the recent exeriments utilizing foam 3

4 target for generation of intense article currents, the absortion efficiency is the most imortant quantity. This is while; desite to these extensive quests, the theory of light roagation through the under-dense lasma has not yet been adequately develoed to answer these questions. Only few theoretical works exist which are directly related to the subject [38, 39]. And to these we must add few recent exerimental studies that have used foam targets to characterize the laser roagation through the transarent lasma [16, 17]. The traditional weakly nonlinear theories of light evolutions in lasma are based on the Raman and Brillouin scatterings [4-43]. Though elucidating, they are not directly alicable to relativistic intensities. This roblem is mainly due to the excitation of strong sace charge mode, the so called wake-field [3-6], by the intense laser ulse. This behavior is contrary to the Raman and Brillouin scatterings which suose the sace charge to grow from the noise level and to remain always small. The strong disersion modification of lasma by the wakefield and its strength itself interlay to cause the lasma to reveal notable kinetic behaviors which eventually change the nature of light scattering into a comletely new tye. The urose of this aer is to give a comrehensive descrition of the laser light evolution in the under-dense lasma. This study is based on analyses 4

5 erformed in one satial and three velocity dimensions (1D3V) assisted by the extensive fully kinetic article-in-cell (PIC) simulations. The key issues of this theory are the light scattering and absortion. Trivially, the absortion is related to the global lasma heating. The absortion of ultra-short intense laser ulse in the ultra-low density lasma is investigated in Refs. [38, 39]. Note that for resent laser arameters, ultra-low density n /.1 n corresonds to e c L P NL. Here NL n m e is the critical density and F c / is the nonlinear lasma c 2 / 2 e wavelength., m e, e, c, and ( n e / m ) are vacuum ermittivity, 2 e e electron mass, electron charge, light seed, laser carrier frequency and lasma frequency resectively. F is a relativistic factor deends on the dimensionless laser amlitude a,i.e. F 2 for a 1 and 2 F for a a. Ref. [38] indicates that the ulse deletion in this regime is adiabatic with being characterized by the radiation red-shift. And the ulse edge evolutions is investigated in Ref. [39]. In this study, we firstly give a more comlete descrition of the adiabatic ulse evolutions in the ultra-low density regime along with direct verifying evidences from PIC simulations. We also show that for ulse lengths larger than the lasma wavelength the Raman tye scatterings (light scattering off electron lasma wave) occur but without instability. In the other words, this scattering does not lay remarkable role in ulse deletion. The limitations of the 5

6 adiabatic descrition are line out, resulting in the identification of a new interaction regime at high densities with different characteristic behaviors. Thus, in order to comlete the under-dense lasma descrition, we extend our investigation to lasma densities u to near critical values. It is turned out that the Radiation Pressure Induced Wave Breaking (RPIWB) heavily destroys the oscillatory structure of the electron wave beyond the first lasma eriod. On the other hand, the electron density rofile induced by the onderomotive force ahead of the ulse begins to steeen and eventually acts as a flying mirror. As a result, the ulse roagation becomes highly unsteady and its enetration becomes limited. The lasma undergoes volumetric heating due to hase mixing by RPIWB. The wavebreak electrons are accelerated further in the longitudinal electric field sustained by the radiation ressure of the reflecting ulse. Due to the radiation reflection, the interaction becomes saturated at a definite time. This time is shown to be the most imortant factor in the overall absortion and lasma heating with being directly related to these quantities. It decreases by a factor of ne / nc. Other details, mainly related to different active electron acceleration mechanisms, have minor effects in absortion which are also discussed. Esecially, a new mechanism for direct laser acceleration is ointed out. The alicability of results in multidimension is commented at the end of aer. 6

7 The organization of this aer is as follow; PIC simulation arameters are summarized in section II. In section III, the basic governing equations are indicated. In section IV, the ultra-low density regime is discussed. Here, firstly the adiabatic ulse deletion for L P via exact analysis and direct evidences from NL PIC simulations is described in subsection (A). Moreover, the limitations of this theory which motivate us to investigate a different behavior in high density lasmas are deliberated. Then, the long ulse behavior L NL is discussed in subsection (B). In section V. the high density regime is described using model analyses and extensive PIC simulations. This section involves three subsections. In section VI, the conclusion of the resent study and the discussion of alicability of results in multi-dimensions are rovided. P II. PARAMETERS OF PIC SIMULATIONS The simulation arameters are summarized here in order to avoid their reetition through the aer. Using PIC code which was develoed by J. Yazdananah [44, 45], we have carried out extensive 1D3V PIC simulations over wide range of system arameters; including lasma densities from.1n c to n c, laser intensities between 1 18 and 1 2 Wcm -2 ( 2 a 1 ) and laser ulse durations and ulse shaes. For most of the simulations, the laser ulse duration 7

8 has been set to L 1 fs, as the ulse envelo rises (with sinus functionality) within 3 fs to its maximum, remains constant over 4 fs and falling symmetrically within 3 fs. The laser wavelength has invariantly been considered 1m in the all cases. Other simulations with different ulse duration ( 2 fs, 14 fs and 8 fs ) and different ulse shaes (having L L L different rise and fall times) have been erformed. An imortant new asect of our simulations is that in many cases, simulation time has been chosen in such a way that achievement of the absortion saturation be insured. This deends on lasma density and laser intensity. For all run-instances, we have used hydrogen lasma with its initial lasma rofile has been invariantly ste like with initial electron and ion temeratures resectively k T 2 4 / m c 1 ( 5eV ) and kt B i ( k B is B e e Boltzmann constant). The lasma length has been chosen such that fully cover the interaction time. High satial resolution of 2 cells er laser wavelength with at least 64 articles er cell is used in the simulations. The satial resolution guaranties the lasma against the un-hysical heating roduced by the finite-grid instability, i.e. the Debye length is aroriately resolved, DX /.3 [44, D D 46]. Reflecting and oen boundary conditions have been alied for articles and fields resectively. III. THE FUNDAMENTAL EQUATIONS 8

9 General insights into the nature of mutual light-lasma interaction can be gained using general conservation laws which are currently well-known in the context of relativistic fluid mechanics and electrodynamics. In relativistic fluid mechanics, the fluid energy and momentum densities are comonents of a secondrank tensor named the energy-momentum tensor 2 Tq hquq uq / c q. Here h q is the fluid internal energy-density and u q is the velocity four-vector defined as u ( c, v ) in terms of the fluid velocity v q and relativistic gamma factor q q q q q q q c 2 1/2 (1 v. v / ). q is a second-rank tensor which includes nonzero temerature effects [47-5]. The subscrit q indicates that quantities corresond to the comonent q th (electron or ion) of lasma, note that its role is different from indices which are used to describe different comonents of vectors and tensors. At zero temerature, we have 2 hq nqmqc / q and q. Using T q, the fluid energy and momentum equations are comacted into the following single covariant equation, T F J (1) q q where is the 1 ( c / t, ) is the covariant four-derivative, F A A electromagnetic tensor with A ( / c, A ) being the otential four-vector and n is the lasma four-current with Q q and n q being elementary charge and J Q u q q q q 9

10 secial density of comonent q, resectively. If no secial assumtions has been made on q, the q th lasma comonent at arbitrary conditions is exactly described by Eq. (1). The electromagnetic tensor F aeared in the right hand side of Eq. (1) is obtained via the solution of inhomogeneous Maxwell equations given in the covariant form by, F J (2) Here J is the total four-current, i.e. J Jq. The governing equation of the light energy-momentum can be obtained from (2) [51] and takes the same form as equation (1). The so called electromagnetic energy-stress tensor K (analogous of T ) is defined as, q K u cg cg em M T, (3) where u E c B and g EB are the electromagnetic energy and em ( ) / 2 momentum densities, resectively. E and B are electric and magnetic fields resectively. M T is the Maxwell stress tensor, with its comonents being given by T ( is the Kronecker delta). K is governed by M [ 2 ] ij EiE j c Bi Bj uemij [62], K F J. (4) 1

11 Substituting equation (1) in the right hand of (4), we obtain the general form of local energy-momentum conservation being customarily written in terms of q th comonent, ( K T ) F J. (5) q qq q The article number for comonent q remains constant giving the continuity equation, J. (6) q Here we restrict ourselves to zero temerature for each lasma comonent and one satial dimension. Through the substitution of and 1,2,3, in Eq. (5) we resectively obtain the energy equation and different comonents of momentum equation for q comonent. Considering hydrogen lasma, we obtain for energy and x momentum of electrons, uem 2 ene 2 g x 2 enevex mec c mec eni vi E, (7a) t t x x 2 g x enevex enevex me [ Ey c Bz Ex ] me eni[ Ex viybz ] (7b) t t x x where subscrits e and i are used for electrons and ions resectively. These equations remain covariant among different reference frames. We can neglect the contribution of vb iy z comared to E x in the right hand side of (7b). This is because 11

12 due to their heavy mass ions remain aroximately immobile in the direction of laser olarization when intensities considered here are alied. The continuity equation (6) can also be rewritten in the form, n t n v x e e ex. (8) IV. THE ULTRA-LOW DENSITY REGIME A. ULTRA SHORT PULSE LENGTHS: THE ADIABATIC PULSE DEPLETION AND ITS LIMITATIONS Let consider the nonlinear ulse deletion due to the wake excitation in the wakefield regime. Here, the ulse length is taken to be ultra-short, i.e. L P NL and lasma is taken to be very rarefied (tyically having densities n /.1 n ) having small otical resonse. This subject has been loosely considered in some revious studies [38, 39]. Here a more comlete descrition is given along with direct evidences from PIC simulations for the laser ulse evolutions. In addition, our formalism reveals the limitations of the adiabatic interretation and hels the understanding of essential differences between the ultra-low and near critical density behaviors. To this end, let aly equations (7a), (7b) and (8) in the Pulse Co-Moving (PCM) frame which moves with velocity equal to the ulse grou velocity ( v g ) with resect to the laboratory. And assume the so-called Quasi- e c Static-Aroximation (QSA) [52-55]. Based on this aroximation, lasma 12

13 quantities are time-indeendent in the PCM frame, i.e. we can set resectively in equations (7a), (7b), and (8), n / t, n v / t and e e PCM e e ex PCM ne / t. Hereafter we use the resentation X to indicate that X is PCM PCM measured in the PCM frame. For laboratory quantities no secial indication will be made. From these arguments we can immediately write equation (8) as, n v v n, (9) e ex PCM g g e with g 2 2 1/2 (1 vg / c ) being the relativistic gamma factor attributed to g v. In the right hand of Eq. (9) we have used the fact that undisturbed lasma flows with velocity vg in the PCM frame. We have also used Lorentz transformations for density of undisturbed region to write ne gne. In the same way, under the PCM QSA aroximation equations (7a, 7b) take the form, uem 2 2 { [ c gx nemec gvg e nee gvg] } t x PCM, (1a) g x { [ Ey c Bz Ex nemev g g evex ene g] } t x PCM (1b) where we have also made use of Eq. (9), E and the facts that vix vg, PCM n i PCM g e n due to the heavy ion mass. The second term in the right hand side of Eq. (7b) is neglected in arriving at Eq. (1b) according to arguments given below equation (7b). Equation (1a) can be further simlified by noting the slow 13

14 variations of the ulse in the PCM frame. Since we have E A / t and B A / x, these quantities become diminishingly small in the PCM frame due z y to the strong Doler red shift. This causes the wave momentum density to become negligible comared to the article momentum density in Eq. (1a). Also the same holds about the time derivative of the wave energy density against the sace derivative of electron momentum density. We will return to this roblem at the end of this section and for now we take them to be valid. By using above arguments Eq. (1a) is reduce into a constant, y y [ m c e ] m c. (11a) 2 2 e e PCM g e This is exactly the energy conservation law for articles in the PCM frame [45]. Eliminating in Eq. (1b) by using Eq. (11a) and alying Lorentz transformations Ex Ex and PCM in the obtained result, 2 e e g (1 vgvex / c ) PCM equation (1b) reads as, g x { [ U CW]} t x PCM (11b) Where U E c B and [ ] / 2 PCM y z PCM C E /2n m c are 2 2 W PCM x e e e resectively the ulse electromagnetic energy and the well-known wake-field energy [53]. As it will be shown at the end of this section, we could not have ignored the electromagnetic energy and momentum in comarison to the wake- 14

15 field energy here, desite to the case of momentum equation. For arbitrary ulse shaes, values of CW PCM can be exactly obtained at each time instant by solving the Poisson equation (equation (2) for ) numerically through the lasma using equations (9) and (11a) for determining density rofile in terms of. This is the wake excitation roblem which is extensively discussed in revious literatures [44,52-57]. C W remains constant behind the laser ulse [53,54], resulting in PCM the relation Ex / 2 ne mec e Ewake / 2 nemec, Ewake Ex.max resents the wake-field amlitude. As we are about considering ultra-short laser ulses, the wake is excited when L NL with resonance condition L /2 [57]. Here we NL do not aim to go through the details of wake rofile. Imortant information can be extracted from the overall variations of wave momentum and energy which are given by integration of Eqs. (11a) and (11b) over x. In this way, let use the electromagnetic field total energy H F dxuem and momentum KF dxg. Now integrate equation (11b) at time t, over the disturbed lasma region between x v t and x2 in the PCM frame. Using the identity 1 h x 2 2 dx ( / t )[...] ( / t ){ dx[...]} vh[...] and the fact that ulse vanishes at x x 1 1 x x x x 1 and x x2, after some straightforward mathematics, we obtain: 15 1

16 dk dt F PCM 1 2 Ew (12) 2 A localized electromagnetic field as whole can be described as a article with its total energy H F and momentum K F form a four vector [48, 51]. Based on the basics of relativity theory, we can always write these four-vector in the covariant form of ( HF, ckf ) H( g, gv g) in terms of the electromagnetic grou velocity, F v g. (13) H F K H H is the electromagnetic energy in PCM frame. Since H lays the Here F PCM role of the article rest mass it remains absolutely constant, i.e. we identically have HF / t, H equals the initial ulse energy in PCM frame. In the same PCM way, the total force exerted on radiation can be written in the covariant form ( f, c ) ( g g, gc ) f v F F [48], where F is the force measured in the PCM frame, i.e. the right hand side of Eq. (12). Using the above arguments equation (12) is transformed into the laboratory frame as, 2 d[ gg ] / dt gcew / 2H, with g being defined as v / c. Using the identity g g d dt d dt and the 3 [ gg ] / g g / fact that U gh ( g g( t ) U U ( t ) ), this equation takes the form, P P 16

17 d c E (14) dt 2 g 1 g w 2 2 gu P The above equation, derived for the first time in this work, fully describes the overall evolution of an ultra short laser ulse in the rarefied lasma. It states that ulse is slowly (since 1) deaccelerated and loses its energy due to the wake g excitation. At small times, when the wake amlitude has not been yet changed noticeably; the solution of Eq. (14) is tanh[ t / ] where g d 2 U / c E and 2 d P g w g 1 tanh [ ]. The full roerties of this solution together with its alications to LWFA will be discussed elsewhere. The deacceleration is accomanied by the radiation red-shift. Since the ulse shae does not undergo essential change, its sectral density remains localized around a carrier time-deendent frequency (wave-number) () t ( k () t ) during its roagation. According to the observer in the (PCM) frame, the ulse momentum becomes zero, hence the carrier wave-number should become zero accordingly. In the same way as argued for electromagnetic total energy momentum we can rove that the same covariant four vector resentation for frequency and wave-number is obtained, i.e. we have 2 (, ) g (1, g / c ) k v where is a constant. To convince yourself, you may see that the well-known relation v 2 g c / v holds 17

18 with v / k being the electromagnetic hase velocity. Therefore we have. Putting this equation together with the same relation obtained g reviously for electromagnetic energy HF gh and dividing these two equations we obtain, HF () t () t constant (15) which is the adiabatic equation obtained via slow envelo analysis in ref [38]. In this case, the ulse deletion can be comletely described by the slow frequency red-shift due to the gradual decrease in the grou velocity. The evidences of this theory are given in Fig. 1 which is resulted from a PIC run instance for n / n.1 and 8 fs. The wake excitation is shown in Fig. 1(a) where the e c L longitudinal hase-sace of electrons together with rofile of electric otential are deicted. In Fig. 1(b), the satial laser ulse rofile is deicted in two different times. The horizontal axis is chosen in a way that the ulse front locations coincide for both time instances. On this figure, two arrows are set to oint to different locations of a definite ulse hill at different time instances. The increase in the wave-length and the red-shift is clearly observed. In Fig. 1(c), the sectral density of the radiation is lotted for two different time-instances chosen as Fig. 1(b). The total red-shift is shown here by notifying the dislacement of central frequency 18

19 with two ointing arrows at two different times. The hysics of this henomenon can be easily understood; the electrons which are ushed forward by onderomotive force exert the back reaction to radiation by ushing the field lines backward. This evolution is not a tye of light scattering of lasma but rather a deletion. It should be noticed again that Eqs. (14) and (15) describe the average ulse evolutions. Also the defined grou velocity g (Eq. (13) is an average quantity. Trivially, the local variation of electromagnetic energy-momentum and whence the grou velocity can deviate from the average trend, but average deviation is zero. An examle of this deviation can be observed in Fig 1(b) in the ulse front (see also ref [39]). As will be discussed in section V, this deviation becomes imortant when stee density gradients are encountered in the case of high density lasma and leads to the light scattering rather than the deletion. Now we return to discussion of the validity assumtions negligibility of ulse momentum made in simlification of Eq. (1a). The statement can be argued by noting the Lorentz transformations for electromagnetic fields. Using this transformations we have E ( E cb ) and B ( B E / c). y PCM g y g z z PCM g z g y Using the lane wave aroximation Ey cb z together with these transformations we obtain g E c. The relation between amlitudes of the 2 x y (1 g ) / (1 g ) PCM 19

20 transverse electric field E y and the vector otentials A A y in terms of the ulse central frequency is given by Ey A. A can be written in terms of its normalized value a / ea mec. Using these relations, after the exansion of (1 ) for 1 ( g 2 1/2 g g g 2 1 g / 2 ), we obtain for the amlitude of g x, say g x, in the PCM frame g m a c / 4 e n m c a / 4. Now x PCM e g e e g we should comare this quantity with the electrons momentum density m n v e e e ex PCM. Using continuity equation (9) together with the Lorentz transformation for 1/2 1/2 e g e(1 gvex / c) g( c vex ) / ( c vex ) ( 1 PCM g ) this quantity reads as m n v n m c ( c v ) / ( c v ). With these 2 1/2 1/2 e e e ex PCM e e g ex ex arguments, for gx to be negligible with resect to m PCM e enevex we need to PCM have, a a 2 2 g 2 2 g 4 (16a) where is defined as ( c v ) / ( c v ). When this inequality holds, using 1/2 1/2 ex ex equation (11a) together with Lorentz transformation (1 ) and e PCM g e g e aroximation g 1, we obtain 2 1 e/ mec. At non-relativistic intensities ( a 1) the inequality is trivially hold for very rarefied lasmas. At relativistic 2

21 intensities the factor inside the arenthesis is of order of unity because in this regime the gamma factor of grou velocity is aroximately given by / a /. To simlify (16a), let evaluate minimum of which 1/2 1/2 g L corresonds to the maxima of longitudinal electron velocity. The maximum attainable longitudinal electron velocity in the electron wave is g 2 1 g / 2 which occurs at the wave break threshold [53-55]. Using this relation in we find that g 1/ 2. In this way, the inequality (16a) is assured at relativistic intensities if we have 1. Using the estimate g and 1/ 2 / g L / n / n this inequality takes the form, c e n n L e 1 c (16b) Therefore, because the inequality (16b) holds for very rarefied lasmas, the wave momentum in (1a) can be safely ignored. In the same way, we can argue that we could not have ignored the electromagnetic energy and momentum in equation (11b). To understand this it should be noted that wake-field energy CW PCM is of order of n m c and the 2 e e e electromagnetic energy U is of the order of cg PCM x which is estimated in the PCM last aragrah as g n m c a 2 2 /4 2 2 x e e PCM g. Using this estimate together with 21

22 the relation a, we find g 1/ 2 / U n m c a which has the same 2 /8 PCM e e order of magnitude as C W. PCM In Fig (1b) it is observed that ulse front undergoes variations other than adiabatic lengthening. The radiation is accumulated in this region. This behavior is due to the local violence of (16b) which causes the unsteady ulse roagation. This henomenon will be described in details in the section (V) where high density lasmas are concerned. B. DISCUSSION OF LONG PULSE BEHAVIOR In the case of long ulse lengths L NL the ulse is able to excite the high P amlitude electron lasma wave (wake-field) as well as the ultra-short ulse. In this case the resonance condition for wake excitation takes the form L ( l 1/ 2) [57]. Also, as long as the condition (16b) holds, the ulse deletion is mainly due to the wake excitation and is given by Eqs. (14) and (15). The difference is that when the ulse length exceeds the lasma wavelength the light scattering of a lasma wave, say the Raman tye scattering, can make sense. But, due to the very high amlitude of the electron lasma wave (the wake) the light scattering of this wave reveals more comlex features comared to the common Raman Scattering (RS). The common RS occurs in the weakly nonlinear (quasi-linear) regime, when 22 NL

23 density erturbations are very small. In the quasi-linear theory it is the main fundamental mechanism which described light evolutions as long as laser ulse length is short enough to avoid ion resonse [4-43]. In RS different electromagnetic (EM) modes are couled to each other via the sace charge mode (the electron lasma oscillations), causing energy exchange between the EM modes. In this way, a high amlitude EM mode can feed energy into new forward or backward roagating EM modes causing their growth from the noise level. Beating among the original and new EM modes roduces onderomotive force which resonantly derives lasma oscillations and amlifies the sace charge. This causes further amlification of new EM modes and rovides the ositive feedback to the sace charge. In this way, the light roagation through the lasma goes unstable by feeding energy into definite EM and sace charge modes and growing them from the noise level. In the case of ultra-intense laser ulses, the electron lasma wave has a much more comlex role due to its high amlitude. Note that, here, this mode does not grow from the noise level by means of mode couling but by the strong laser onderomotive force. But, nevertheless, the self-modification of the roagating light when enetrating the lasma is exected as well in this case. Because the high amlitude density oscillations greatly alter the lasma disersion, the grou velocity of light becomes a comletely local quantity and causes the radiation to undergo self-modulation. The main distinguishing feature of the fully 23

24 nonlinear regime is that the modulation enveloe does not mimic the sace-charge attern. This henomenon is caused by both the comlex shae of relativistic electron wave rofile and the locality of grou velocity. Concerning the latter, we remind the fact that based on Eq. (14) the grou velocity decreases in the ulse front due to seeding of the wakefield. Since the hase velocity of roduced wake is equal to the ulse grou velocity in this location, the back art of radiation flows forward through the lasma wave. In this way, when a density eak is encountered, the light iss accumulated on its back and is disersed on its front. PIC simulations at very low densities confirm these statements. As an examle we have deicted simulation results for n /.1 n, a 6 and 14 fs in Fig. 2. In Fig. (2a).the e c resultant modulation is observed to have characters described above. Accordingly, the couling introduced by the sace charge between different electromagnetic modes is not of resonant tye. This causes not a few but an infinite number of EM modes in the light sectrum to become couled to each other, i.e. the frequencies of scattered EM modes does not obey the common relation but rather by L the relation NL NL with 1,2... and (see also [58]) / L [46, 59], 1 a / 2. The factor L is introduced in lasma frequency to account L 2 for relativistic mass increase. The nonlinear lasma frequency in this case is NL.48. Using this calculation, the growth of modes with frequencies 24

25 NL is clearly observed in Fig. (2b) where we have deicted the sectral density of radiation. The amlified modes are secified by arrow on the figure. Because the intercalation between the scattered light and lasma wave is of nonresonant tye, this henomenon does not result in instability but instead very slow ulse modulational. When the ulse length is not large enough, the accumulation of scattered light does not introduce significant radiation back-reaction on the lasma wave. When very long ulse lengths are considered, the radiation back reaction becomes noticeable and causes induced wave breaking, say the radiation ressure induced wave breaking (RPIWB). At these conditions, the scattering evolves from Raman tye to Thomson tye. The stimulation mechanism is related to the disintegrability of the electron motion in the resence of back-scattered light which is extensively discussed in recent literatures [33-35]. The difference between very long and long ulse behaviors has been described in detail elsewhere [6]. When the lasma density grows toward critical value the lasma otical resonse becomes strong enough to cause much faster RPIWB, causing its occurrence with short ulse lasers. This is described in the next section. V. NEAR CRITICAL PLASMA DENSITIES A. NONLINEAR ENERGY DEPLETION AND ANOMALOUS LIGHT SCATTERING 25

26 From arguments given in section IV.A, if the condition of the Eq. (16b) becomes violated, say when n / e Ln c grows toward unity, the adiabatic ulse deletion becomes doubtful. At these conditions, the ulse can undergo raid changes during its roagation. For arameters of current intense lasers, high densities corresond to the regime of L P NL. In section IV.B., it has been noticed that the light scattering off the lasma wave behaves strangely when high amlitude waves are encountered. At very low densities, the light scattering does not introduce noticeable modification in the sace charge mode due to the very slow develoment. In site, when the light scattering becomes raid due to the rise in n / e Ln c, serious modifications are encountered. Here, those beating between scattered EM modes which are non-resonant with the sace charge can result in strong wave breaking. When a density eak is encountered, the light is accumulated on its back and is disersed on its front as already we have described on Fig. (2a). As a result, a net forward radiation ressure is exerted on the density column. This ressure accelerate the lasma electrons in the direction of the lasma wave hase velocity [58]. Under these conditions, the electron lasma wave undergoes early wave breaking as observed in Ref. [61]. When the ulse length, L, exceeds noticeably the lasma wave length c /, the wave break strongly affects the lasma oscillations causing them to undergo wild modulation. 26

27 This leads to inalicability of the concet of radiation scattering off a lasma wave. Naturally, at these conditions, the hase-matching relations between the radiation and sace charge disaear. Furthermore, wave breaking results in the saturation of the lasma scattering and the lasma back reflection [62]. In the ulse front electrons are accumulated by the laser onderomotive force resulting in a ositive density gradient. This density gradient can accumulate and even back reflect the light. The distinctive feature of the front is that the wave-breaking is absent because of no electrostatic field being exist beyond the front. Thus the lasma scattering cannot be saturated by wave-breaking. In this location, the lasma can reflect the light in the same way as it does in the radiation tra inside the density valleys described in Ref. [63]. If the eak density in the frame comoving with it, reaches the critical value n c in this frame, the reflection takes lace. Since the ositive density gradients are amlified by receiving ositive feedback from radiation, the reflection eventually occurs in the absence of dissiative rocesses. However, the laser energy conversion to the electrostatic and kinetic energy of electrons exists as a dissiating rocess as discussed in IV.A. When the density scale length is short enough, the accumulation can gain over the dissiation and eventually can lead to the back reflection. The density scale length NL can be estimated by the nonlinear lasma wavelength F c /. The time e 27

28 NL needed for reflection is given by /2 v, with vg being the deviation of r g the maximum local grou velocity with resect to the average value given by Eq. (13). Note that the maximum of local grou velocity occurs at the density NL minimum dislaced by /2 with resect to the front. When the reflection time is shorter than the deletion time, d, the ulse reflection takes lace. And this occurs again when the n / e Ln c grows toward unity. Our PIC simulations mentioned in Sec. II show when the lasma density is raised toward values of order of.1n c and higher, the electron ile-u formed in the ulse front, quickly becomes reflective and then roagates as a solitary disturbance along the lasma. The longitudinal electric field roduced by this ile-u converts the laser energy into the wave break electrons continuously. This rovides an indirect mechanism for electron acceleration. The formation and roagation of the electron soliton (ileu) can be directly observed in PIC simulation results resented in Fig. 3. In anels (a) and (c) the longitudinal hase sace together with vector otential and electron density rofile are sketched for a 6, n / n.3 and 1 fs at two different e time instances. In anels (b) and (d) the longitudinal electric field is lotted together with the density rofile in the location of electron ile-u, showing the double-layer structure of the erturbation. Electron acceleration by the formed structure is also directly observed in this figure. The radiation becomes gradually 28 c L

29 reflected by the electron ile-u. In the soliton rest (SR) frame, the radiation is the suerosition of the aroximately equal-amlitude incident and reflected lights. The electron ile-u is conserved by the radiation ressure roduced by the light reflection. Let the soliton to roagate with velocity v s. The incident ulse in the SR frame exeriences the relativistic frequency down-shift with resect to the lab frame, given by ( ) / ( ) c v c v. Hereafter subscrits i and r are used to 1/2 1/2 i SR s s resent incident and reflected quantities, resectively. The notation X is used to SR indicate that X is measured in the SR frame. The ulse length in the SR frame can be obtained using the invariant normalized hase number, N /2, where is the hase difference between the start and the end of the ulse. In terms of this quantity, we have L / ( ) ( ) / ( ). N L L c v c v. 1/2 1/2 i SR i SR i SR s s In transformation from SR into the lab frame, the reflected radiation exeriences another frequency down-shifted due to the negative relative frame velocity vs, i.e. 1/2 1/2 r ( c i v s) / ( c v s). Therefore, the back reflection in the lab frame SR receives the double frequency down-shifted, ( c vs) r. (17) ( c v ) s 29

30 Corresondingly, in the same way as discussed above for the incident ulse, the reflected ulse becomes stretched with resect to the initial ulse as follow, s L. r L c v s ( c v ). (18) ( ) Since the transverse vector otential remains invariant in Lorentz transformations, the exression Ey / also should remain invariant. Hence the electromagnetic energy density of the reflected radiation, u EM. r, is reduced by a factor of ( / ) with resect to the incident radiation. Using this argument with together equations (17) and (18), the reflectivity ( R ), the fraction of the total reflected electromagnetic energy ( HF. r uem. rl. r ) over the incident energy ( HF. i uem. rl. i ) is estimated to be: r 2 R c v c v s s. (19a) The absortion coefficient, 1 R, is estimated to be, 2v s c v s. (19b) Note that stored electromagnetic energy in terms of longitudinal electric field has been ignored in above derivation. As will be shown in simulation results, this ignorance is satisfactory. 3

31 In the hysical model resented so far, the interaction ends u as soon as the ulse is being comletely reflected, i.e. an absortion saturation is attained by the reflection comletion. The saturation time, t sat, can be obtained easily by calculating the time-difference between two relativistic events (time-sace oints) in the laboratory frame which corresond to the begin and end of the ulse reflection in the SR frame. These events occur resectively at ( t1, x1) (,) and SR ( t, x ) ( L / c,) 2 2 SR i. SR. Time difference between these events in the laboratory frame is given by Lorentz time transformation, t L. / c, with s being sat s i SR s (1 v / c ). Therefore, using relation for L i. given in the last aragrah we s 2 2 1/2 s have, t sat L c 1 v / c s. (19c) An estimate of lasma temerature in terms of hysical arameters n e and a can be made using obtained relations for and tsat in the form of kbte. h HF / nevst sat, with H F being the initial electromagnetic energy. In this way, using equations (19), this estimate reads as, u c v c v k T m c a 2 2 em s s B e e ne c v s c vs 2 (2) 31

32 where 2 a dx, calculated at t, is the ulse shae factor ( 1, 1 2 / al for rectangular shae ulse). We can also made an estimate of the soliton velocity, v s. Since the electromagnetic energy remains conserved in the SR frame, according to discussions given in section IV.A above Eq. (14), this frame is the secial frame of radiation which moves with the grou velocity of Eq. (13), i.e. s g. By v v assuming local equilibrium between lasma and radiation inside the first lasma cavity behind the electron ileu, the velocity of electron soliton can be estimated using equations (1a) and (1b). The local equilibrium means n / t, e e PCM n v / t, u / t and g / t. In the other hand, since the e e ex PCM EM PCM electromagnetic field in the PCM frame is a standing wave, formed by suerosition of incident and reflected wave, its momentum is zero in average. In this way equation (1a) reduces to the wake constant (11a) again. Using this statement, the roblem of obtaining induced density rofile reduces to solving for x wake excitation by a ulse with its amlitude (the standing wave amlitude), a, being two times the original laser amlitude ( a 2a ). Hereafter the rime is used for the standing wave quantities. Now using equation (11b) and arguments given below this equation we obtain in the lasma region considered here PCM 32

33 U. Recalling arguments given above equation (16a), we have PCM 2 Ewake /2 U E (1 ) / (1 ) m c a / 4e. On the other hand we have y g g e g PCM Ewake 2( e.max 1) mec / e [56, 57] where e.max is the maximum electron gamma factor inside the wake. Using these last two equations into the former we obtain, g a e.max 1 (21) At weak laser intensities ( a 1), 1a and we recover the well known 2 e.max linear result. At ultra relativistic intensities ( a 1), e.max scales as e.max a and 1/2 Eq. (21) scales as the well known relation a. This equation shows 33 g / increase against the laser intensity and decrease against the lasma density. As will be shown this trends are confirmed by our simulations. The imortant features of the radiation dynamic are visualized in Figs. 4 and 5 using PIC simulations. In anels (a) and (b) the satiotemoral evolution of the vector otential is resented for arameters same as Fig. 3. The radiation gradual back reflection and overall de-acceleration and lengthening of the reflected ulse can be easily seen in these lots. In anel (c) the history of the vicinity of the radiation front is sketched for two values of intensity a 6,1 and two values of density n / n.3,.5, with other arameters being chosen as (a) and (b). This e c

34 lot verifies the redictions of (21) about decrease (increase) of grou velocity versus increasing the lasma density (laser intensity). In Fig. 5, the satiotemoral evolutions of the sectral density of the vector otential are resented for arameters same as Fig. 5a. The figure shows the gradual deoulation at the original laser wav-number k k and oulation at lower wave numbers. This is consistent with the model of light reflection from the moving radiation front. The broadening of the radiation sectrum bellow k k is due to comlex motion of electrons in the strong radiation which causes strong harmonic generation [57]. The saturation of reflection is also seen in this lot as the sectrum shae remains invariant after a certain time. In Fig. 6 we described different asects of the radiation absortion. In Figs. 6(a) and 6(b), we have shown time histories for electromagnetic and lasma energies and momentums for a 6, n /.3 n and 1 fs. Both the total energy-momentum conservation and absortion L saturation are seen in this figures. In Fig. 6(b), it is shown that total energy inside the box is decreased due to the back-reflection. It is also shown that the stored energy in the longitudinal art of electric field is neglectable comared to the absorbed energy, an assumtion which has been made in the roosed model. The validity of the roosed model can be best demonstrated by changing the ulse length and keeing the other arameters constant. Since v s deends only on e c 34

35 the lasma density and ulse intensity, the absortion should remain constant as is redicted by (19b) and the saturation time should change exactly roortional to the ulse length as is redicted by (19c). In Fig. 6(c), the lasma energy (absortion amount) is lotted at the laser intensity a 6 for two lasma densities, n /.3 e nc and n /.5 e nc, and two ulse lengths L 1 fs and L 2 fs. The saturation times are ointed by arrows in this figure with their values being resented. On can see that resented values are in excellent agreement with the model formulas (19b) and (19c) in the sense described just above. At this end let further comare redictions of model equations with direct simulation results. As can be read from Fig. 3(b), the osition of the formed electron soliton (ile u) is x. 17 m at time t 3 fs. The initial osition at sf t is the left lasma boundary x. 4m. The average soliton velocity is s s. f s. i f i si v ( x x )/( t t ).74c. Using this value as an estimate in equation (19b) we get.85 which is in good agreement with the direct result.75. The origin of extra absortion calculated via (19b) is due to the ignorance of radiation back-reflection from lasma arts other than the electron ile-u. A art of the light is also reflected in the rocess of soliton formation. From equation (19c), the saturation time is estimated to be t sat 728 ( t sat fs ) which is very close to the direct result, t sat 68 (see Fig. 6(c)). The calculated temerature from equation

36 (2) is kt 9.2MeV. With.7 which is tyical of our ulse shae, we B eh obtain kt 6.4MeV. The direct result is kt 4.5MeV. The error is due to B eh B eh inaccuracy in the calculated absortion discussed above. At the end of this subsection we wish to add comments on the descrition of lasma heating in the roosed model. In this model equations (19b) only indirect laser heating mediated by the longitudinal electric field of the formed double layer (see Fig. 3) has been taken into account. Based on grou velocity behaviors described by Eq. (21) and Eq. (19b), the overall heating (total absortion) is redicted to decrease with increase in the density. This trend is already confirmed in our simulation results for absortion history given for examle in Fig. 6(c) for different lasma densities, n /.3 n and n /.5 n. Based on indirect e c electron acceleration mechanism used in the roosed model, we exect that sloe of the absortion-curve has the same behavior. However simulations result does not satisfy this exectation. Quiet reverse, it is seen in Fig. (6c) that this sloe is larger at the higher lasma density ( n /.5 n ). This increase in the sloo is in e fact due to the direct laser absortion by electrons which has not been taken into account in the Eq. (19b). The direct laser absortion occurs when freed (unbounded) electrons enter the radiation region (see Fig. 3) [29-34]. A more detailed discussion of the lasma heating is given in the next subsection. c e c 36

37 B. PLASMA HEATING AND SUPER-PONDEROMOTIVE ELECTRON GENERATION The electron acceleration by intense laser inside the under-dense lasma has been extensively studied reviously, with the large majority of attentions having been aid to the observed suer-onderomotive electrons (electrons with energies larger that 2 2 m /2 ec a ). The energy scale of these electrons is beyond the scale of attainable energy inside the lasma wave body ( of order of 2 mec a ) and their total current is very larger than what roducible by the wakefield acceleration [3-5, 52]. In this resect, a variety of direct laser acceleration (DLA) mechanisms have been roosed for descrition of these electrons, including acceleration in the ion channel [29-32], stochastic acceleration by evolving radiation [33-35] and enhanced electron acceleration by anti-dehasing effects [37, 64]. Here, our roosed radiation model suggests a new mechanism for lasma heating and sueronderomotive electron generation. The radiation induced wave breaking, described in IV.B and V.A, lays a rimary role in this mechanism. Based on arguments resented in the last subsection, the laser absortion is dominated by the indirect acceleration of wave break electrons in the sace charge field sustained by the radiation ressure. The hase mixing attained by intersection between trajectories of the wave break electrons and oscillation body electrons leas to the 37

38 volumetric lasma heating. Furthermore, a new direct laser acceleration mechanism can be identified based on described radiation model. When the ulse length exceeds the lasma wavelength electrons can be liberated inside the radiation region by means of wave breaking (see Fig. 3). In the PCM frame the radiation is described by a standing wave roduced by suerosition of equal amlitude incident and reflected lights. It has been shown reviously that the electron motion inside the standing wave is disintegrable and can become chaotic at nonlinear regime [35]. These characters of motion can lead to stochastic energy gain by electron [33-34]. Therefore, inside the PCM frame electrons are directly heated by laser via the stochastic mechanism. Inside this frame, the maximum attainable energy can be as large as the onderomotive scale m c (2 a ) / 2 2m c a. Since the PCM frame moves with velocity v g with e e resect to the laboratory, the results for forwardly directed electrons are boosted by a factor of g in the laboratory frame. For examle the Lorentz transformations for momentum imlies, ( v v ), (22) ex g e. st g ex. st with ex. st v and. e st being resectively the stochastic electron velocity and gamma factor in the PCM frame. This equation redicts that maximum electron energy increases versus g for a definite laser intensity, a result which will be confirmed 38

39 by our simulations (see V.C). Also it redicts larger absolute momentum values for forwardly directed electrons ( vex. st ) with resect to backwardly directed ones ( v ). The fact that can be already seen in Fig. 3 and is frequently observed in ex. st revious ublications. Furthermore it is inferred from Eq. (22) that electron can attain energy of several times the onderomotive value, as frequently observed reviously. The DLA mechanism roosed here may be termed as the relativistically boosted stochastic heating. In Fig. 7, the hase variables of two selected test electrons are resented for the simulation instance of a 6, n / n.5 and 1 fs. The t x and t y e c diagrams resented in anels (a) and (b) demonstrate the free (non-bounded) motion of test articles in the radiation field. Fig. 7(a) shows the test article initially located at the left lasma boundary enters the free motion after a transient bounded hase. It aroximately travels the whole disturbed region along the ulse roagation. The second article moves freely oosite to the ulse roagation until it reaches to the lasma boundary. After then, it is returned back by the sace charge (see Fig. 7(b)). This article is a art of the return current which is roduced due to the strong wave-breaking. In Figs. 7(c) and 7(d), x y hase slice is lotted for test electrons. It is clearly seen that after assing several chaotic hases, L 39

40 the test electron eventually exeriences the DLA and gains energies beyond the onderomotive scale. There are three tyes of motion inside the disturbed lasma which corresondingly roduce three tyes of oulation in the electron sectrum. The first tye is the motion inside the electrostatic oscillation-body. The second tye belongs to the electrons which are roduced in the wave-break and exchange energy with lasma oscillations. The last tye corresonds to the most energetic free electrons which undergo the DLA mechanism. In Fig. 8, we have given PIC simulation results for electron sectrum at four different times for a 6, n / n.3 and 1 fs. Since the disturbed region is increased in length due e c L to laser roagation, the overall hot electron oulations increase accordingly. To understand heating rocess in the course of time, we refer to the simulation results given in Fig. 9. Here, we have resented time histories of electron temerature and maximum gamma factor for different lasma densities and two laser intensities ( a 6 and a 1) at 1 L fs. It is observed that the electron temerature is aroximately saturated at a definite time, say t sat. Afterward it grows very slowly u to the end of the interaction (absortion saturation t sat ). Our additional simulations erformed with different ulse shaes (not given here) have indicated that the shae of temerature history before the saturation time is directly related to 4

41 the ulse shae in its rising art. Esecially t sat is roortional to the ulse rise time, as is exected according to arguments we have given about Eq. (19c). The maximum gamma factor shows the same behavior as the temerature, excet for fluctuations in the saturation regime. C. Simulation Trends In Fig. 1, we have deicted the total absorbed electromagnetic energy,, versus the lasma density (in anels (a) and (b) for a =6 and a =1, resectively) and laser intensity (in anel (c) for n /.5 n ). The corresonding saturation times, t sat e c, are also given in this figure. It is noticed that in agreement with our model Eqs. (19b), (19c), (2) and (21), both the absorbed energy and the saturation time decrease versus the lasma density and increase versus the laser intensity. Note that the electromagnetic grou-velocity given by Eq. (21) decreases (increases) by increasing the lasma density (laser intensity). Trends for saturated value of electron temerature and corresonding saturation time versus the lasma density and the laser intensity are given in Fig. 11. Parameters are chosen as Fig. 1. Again, both quantities decrease versus the lasma density (Fig. 11(a)) and increase versus the laser intensity (Fig. 11(b)). For the same arameters, trends for the maximum electron energy are given in Fig. 12, 41

42 in anel (a) versus the density and in anel (b) versus the intensity. Observed trends are the same as those of temerature and confirm the model equation (22). All trends given above indicate that the total absortion is mainly dominated by the saturation time. Larger saturation time means larger absortion. Equally seaking the laser absortion is dominated by the indirect mechanism described in section IV.A. VI. CONCLUSION AND DISCUSSION In conclusion, we have resented a comrehensive one dimensional theory of intense short-ulse laser interaction with under-dense lasma. The kinetic asects of lasma have been fully incororated using extensive article-in-cell (PIC) simulations. As described in the introduction, our study might be very imortant in understanding of the recent laser lasma exeriments. Also, results resented here could find alications in the fundamental theory of intense light roagation through under-dense lasma. Based on this study, it has been turned out that the lasma behavior is characterized by both its density and the ratio of the ulse length to the lasma wavelength. In this resect, different interaction regimes have been identified and described in details in sections IV-V. Accurate estimates for the overall laser absortion and lasma heating have been obtained for different regimes identified in this study. Though our study is one dimensional, the 42

43 fundamental henomena described here can occur as well in multi-dimensions. The descrition of light deletion by wake excitation and light scattering by induced lasma rofiles remains indeendent of sace dimensions. In the same way, the imortant role of these henomena in lasma heating remains unchanged when multi-dimensional sace is considered. The most imortant evidences for these statements come from very recent exeriments carried out using sub-critical foam targets [16, 25, 27]. In these exeriments, it has been demonstrated that the enetration deth is the most imortant determining factor in the total laser absortion and overall lasma heating. And it has been demonstrated that this deth increases with density decrease. These outcomes are both in agreement with our results. The agreement confirms the fact that the multi-dimensional effects cannot rule out the henomena described here but they can only cause modifications. In multi-dimensions, the laser light can undergoes self-focusing [27]. If this self focusing could be imroved by density increasing and this imrovement could eventually result in further radiation imrisonment inside the induced lasma, it might be able to modify the simulation trends observed in V.C. Moreover, the direct electron acceleration may become arametrically amlified via betatron oscillation resonance in the multi-dimensions [36-37, 65]. Because these electrons are oorly oulated, this henomenon is unlikely to introduce significant effects. An imortant asect of multi-dimension which may seem more 43

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49 FIG. 1. (Color online) Longitudinal hase-sace maximum and the carrier frequency resectively. 49 x (left axis) and otential (right axis) (a), laser ulse at two different time instances (b), and the ulse sectral density at these times (c), all for a 6, n / n.1 and 8 fs. In the anel (b) the horizontal axis shows x v t, with f e c L v being the ulse-front velocity. Arrows in anels (b) and (c) show different locations of a definite f x

50 FIG. 2. (Color online) The ulse rofile in the ositive half-lan (left axis) together with x hase sace at t 28 fs (a), and the sectral density of radiation at different times (b), for arameters same as Fig.1 but with 14 fs. In (a) the dashed line is for eye-guiding. In (b) locations of scattered modes are marked by arrows. L x 5

51 FIG. 3. (Color online) The longitudinal electron hase sace (left axis), the vector otential (right axis) and the electron density rofile (right axis) (a), and the longitudinal electric field (left axis) and electron density (right axis) in the front region (b), at t 18 fs, together with lots for the same issues at t 3 fs (c, d). Here a 6, n /.3 n and 1 fs. e c L 51

52 FIG. 4. (Color online) The satiotemoral evolutions of electromagnetic wave; satial rofile of the vector otential at different time instants (a), the full histogram of the vector otential intensity over the whole simulation time (b), and the history of radiation front location for different laser intensities and lasma densities. In (a) and (b) arameters are same as Fig. 3. In (c) 1 fs. L 52

53 FIG. 5. (Color online) Evolutions of the radiation sectral density; sectral density at different time instants (a), and its full histogram over the whole simulation time (b), arameters are same as Fig

54 FIG. 6. (Color online) The histories for field (left axis) and lasma (right axis) momentums (a) and energies (b) both for, a 6, n /.3 n and 1 fs, and histories for lasma energy at the same e c (left axis) and energy of longitudinal electric-field (right axis) are also deicted. 54 L laser intensity but two different lasma densities n / n.3 and n / n.5 and two different ulse lengths 1 fs and 2 fs (c). In anel (b), the total energy inside the simulation box L L e c e c

55 FIG. 7. (Color online) t x, t y (a, b) and corresonding x y diagrams (c, d) for two selected test articles with laser lasma arameters a 6, n /.5 n and 1 fs and initial conditions ( x, y, v, v ) (4,, 4.3e 4, 7.3e 3) and ( x, y, v, v ) (71.58,, 6.5e 3,1.6e 3). x y x y e c L 55