Plasma Modulation of Harmonic Emission Spectra from Laser-Dense Plasma Interactions

Transcription

1 Plasa Modulation of Haronic Eission Spectra fro Laser-Dense Plasa Interactions R. Ondarza-Rovira and T.J.M. Boyd 2 ININ, A.P , México 80, Distrito Federal, Mexico; ondarza@nuclear.inin.x 2 Centre for Physics, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, UK We report results fro PIC siulations of the interaction of intense laser light with overdense plasa designed to exaine the effects of plasa waves generated by pulses of fast electrons on high haronic eission fro the plasa. We show that the eission spectru is odulated at the plasa frequency and identify cobinations of paraeters and circustances favourable for odulation. In particular the observed odulation is shown to depend not only on the chosen plasa electron density and intensity of the incident light but on the density profile and pulse shape.. Introduction. The generation of ultiple haronics of the incident light in laser-plasa interactions is a topic that has proved to be of enduring interest, not only because of its intrinsic iportance but for its potential as a diagnostic of conditions at the target surface and as a eans of generating intense coherent pulses at short wavelengths. Distinct echaniss are responsible for haronic generation across different intensity ranges and target characteristics. For gas-jet targets haronics up to several hundred are generated through the nonlinear response of atoic electrons. However their intensities are liited by field ionization. No such constraint applies to haronic generation when intense laser pulses are focussed on the surface of solid targets. This was first observed in experients with nanosecond pulses [] and subsequently detected for picosecond pulse lengths [2]. Conditions in short-pulse experients ensured two prerequisites essential for high haronic generation, naely an electron density uch greater than the critical density, n c, cobined with high brightness of the source. Using light of intensity I and wavelength λ L, at values of I λ 2 L in the range W c 2 µ 2 Norreys et al. [2] detected haronics up to the 75th (with conversion efficiency of 0 6 ) for.053 µ wavelength light interacting with a solid target. Subsequently haronic generation using pulse lengths of 00 fs has been observed [3]. Such short pulses with no significant pre-pulse present, create plasa free fro extensive underdense coronal regions and with extreely steep density gradients at the plasa surface. In the absence of any convincing theoretical odel for high haronic generation in this paraeter regie, particle-in-cell (PIC) siulations have proved invaluable in understanding and characterizing the eission. Gibbon [4] used PIC siulations to identify the physical echanis underlying haronic generation, showing that the current sources generating the haronics are localized just

2 within the overdense plasa boundary. Gibbon found an epirical expression for the efficiency of high-order haronic generation η (I 8 λ 2 L) 2 (/0) 5 where I 8 denotes light intensity in units of 0 8 W c 2, with λ L in icrons and is the haronic nuber. Useful as this scaling is, it akes no allowance for any dependence of haronic power levels on the plasa electron density [5]. Alternative scaling laws have been proposed by Baeva et al. [6]. on the basis of ultrarelativistic siilarity theory, independent of details of the interaction. These authors report a scaling in which the spectru decays with haronic nuber as 8/3 with a cut-off at = 8 α γax 3 where α is of order and γ is the axiu value of the relativistic factor associated with the otion of the plasa surface. Boyd and Ondarza-Rovira [7] showed that the haronic spectru is affected by plasa line eission which has been seen in a nuber of experients [8]. Moreover they reported for the first tie an unexpected feature in the spectru which they attributed to the effect of the plasa wave on neighbouring haronic lines. This so-called cobination line is in effect a footprint of the odulation subsequently observed experientally [3,9]. In this Letter we investigate these odulation effects using a odel in which the source of the odulation is identified as plasa waves generated in the supra-dense target plasa by pulses of re-entrant electrons. 2. Nuerical siulations. It is iportant at the outset to underline a liitation of our, or for that atter ost other, PIC siulations. We do not attept to odel the ionization of target aterial but treat instead the interaction of a laser pulse of length τ p with a plasa slab of initially unifor density, apart fro an interface at the front end. This vacuu-plasa interface is characterised by a density profile with scale length that is soe prescribed fraction of a wavelength across the rap. The -/2D PIC code we use ebeds the Bourdier technique [0] to allow for oblique incidence. The siulation plasa can be up to ten wavelengths in extent. Vacuu gaps extend fro both the front of the rap and the planar rear surface of the plasa to the walls of the siulation box to allow for particle and wave propagation. The initial electron teperature was chosen to be 00 ev and we used gaussian pulses of variable length and profile. The noralized quiver oentu a (I 8 λ 2 L) /2 lay in the range with electron densities between 0 n c and 64 n c. The reflected eission spectra showing the relative strength of the different oscillation odes is deterined fro the haronic contents of the reflected electric field at the vacuu gap, noralised to the fundaental. Our earlier work [5,7] established that plasa line eission was present alongside the haronics of the incident light. The source of the plasa eission was identified as plasa waves excited in the supra-dense plasa by jets of Brunel-accelerated electrons fored when p-polarized light is obliquely incident on the plasa surface []. The re-entrant electrons bunch to generate plasa oscillations at the surface before penetrating the overdense plasa. Plasa oscillations at the target surface couple efficiently to the radiation field. In Fig. we reproduce the effect for a plasa with slab electron density n e = 20 n c. Fig. a plots electron density contours across a 0.λ L section of the plasa target (3λ L wide in this siulation). The striations correspond to the pulses of fast electrons that excite plasa oscillations. A coparable density bunching effect and corresponding excitation of plasa oscillations has been reported by Quéré et al. [2] but in this case for a foil only λ L /5 thick. 2

3 noralised Fourier density In Fig. b we present a Fourier analysis of the nubers of electrons at a point just inside the siulation plasa for the paraeters used in Fig. a. Envelopes are sketched over the Fourier coponents of the density showing approxiately 3 odulation cycles across a span of 4 haronics, suggestive of odulation at the plasa frequency ω p = 4.5 ω L where ω L is the frequency of the incident light. This odulation in turn is reflected in the power spectra. <5 5-7 e-0 e-02 n e/nc 8-20 >20 e-03 e ' Figure : Plasa electron density contours for a target plasa density n e /n c = 20, a 0 = 0.5, with angle of incidence θ = 23. The tie-scale corresponds to 3 laser periods and the section of plasa shown extends over 0. λ L. Fourier decoposition of the electron density at a point x = 0. λ L inside the plasa. To explore the odulation of the eission spectru we carried out PIC siulations across a wide paraeter range that included not only the intensity of the incident light and the density of the target plasa but both pulse shape and length and the steepness of the density gradient at the front edge of the plasa box. Insofar as the behaviour of the odulation as a function of laser intensity is concerned, our conclusions are broadly in line with those of Watts et al. [9], naely the higher the intensity, the ore pronounced the odulation, albeit for different reasons. In our case at values of a 0 significantly below the range of interest ( ), the nuber of haronics with intensities above threshold (in our case 0 6 ) is correspondingly reduced. This in turn is reflected in the nuber of odulation cycles observed, or indeed by their disappearance altogether. For the odel we have adopted there is a ore fundaental reason why odulation should disappear at lower laser intensity. Since the flux of electrons injected into the target plasa depends critically on intensity, a diinished flux in turn eans that the level of plasa waves excited in the overdense plasa is correspondingly weaker. A selection of spectra is shown in Fig. 2. Figs. 2a and 2b use a 0 = 0.5, n e = 0 n c and a 0 = 0.6, n e = 9 n c, respectively, where θ = 23 o is the angle between the wave propagation vector and the noral to the target surface. For Fig. 2c, a 0 =.0, n e = 40 n c and in the highest density run shown in Fig. 2d, a 0 = 0.5, n e = 64 n c. In all four cases the pulse length used was 7 fs, chosen to iniise noise fro plasa line eission. To highlight the effect we show envelopes across the bands of odulation for P /P > 0 6 and for values of >.5 (ω p /ω L ). For the paraeters used in Figs. 2a,b and c the PIC results show 3 or 4 clear odulation cycles with average odulation frequencies 3.3 ω L, 4.3 ω L and 6.3 ω L respectively, values that correspond 3

4 closely to the plasa frequencies for each of these densities. Fig. 2c shows soe fine structure in the odulation across the range = 0 to = 8. In this case there is soe contribution fro second haronic plasa eission which perturbs the eission spectra slightly in the neighbourhood of = 2. The choice of paraeters in Fig. 2d cobines relatively high density and oderate light intensity, ensuring that the plasa wave is not too strongly driven. For n e /n c = 64 the plasa line is centred at = 8 with the blue half of the odulation structure clearly defined (haronic lines 9 to 2). The red wing is cut off at = 6, asked by the relative strength of the fourth and fifth haronic lines. Higher laser intensities are needed to extend the range of odulation at higher densities. This in turn results in the Languir waves being ore strongly driven and, at the sae tie, significantly broadened. A siple linear estiate of the bandwidth of Languir waves excited by a bea of fast electrons is given by ω ( V b /v b ) (n e /n c ) /2 ω L for a bea velocity v b with velocity spread V b. Broadened plasa line eission in turn affects neighbouring haronic lines P/P e-05 P/P e-05 e-07 e-08 e-07 e e e-0 e-02 ( c ) e-0 e-02 (d) e-03 e-03 P/P e-04 e-05 P/P e-04 e-05 e-07 e-07 e-08 e-08 e e Figure 2: Haronic odulation for a laser pulse with τ p = 7 fs and λ L = µ: a 0 = 0.5, n e /n c = 0, a 0 = 0.6, n e /n c = 9, (c) a 0 =.0, n e /n c = 40 and (d) a 0 = 0.5, n e /n c = 64. Envelopes over the haronics have been added to indicate the respective odulation frequencies. Fig. 3 illustrates the susceptibility of odulation to pulse shape and steepness of the density gradient at the front edge of the plasa box. Essentially steepening the pulse over the sin 2 profile used to obtain the data in Fig. 2 iproves the definition of the odulation (Fig. 3a). Varying 4

5 the pulse length τ p on the other hand has relatively little effect on the odulation as such, always provided the pulse is sufficiently longer than the growth tie, t g of the electron-generated plasa waves. If we ake use of the estiate given by Boyd and Ondarza-Rovira [7], this eans τ p > 0 (n c /n b )(n e /n c ) /2 ( V b /v b ) 2 τ L where n b is the density of the electron pulse exciting the plasa waves. For the paraeters in these siulations this iplies τ p > 0fs. e-0 e-02 e-02 P/P e-03 e-04 e-05 P/P e-04 e-08 e-07 e e Figure 3: Haronic odulation for a laser pulse with a 0 =.0, τ p = 30 fs, and n e /n c = 30, for a steepened pulse envelope; effectiveness of a density rap in suppressing odulation for a siulation with a 0 = 0.6, τ p = 30 fs and n e /n c = 6 for a density rap of width = 0.6 λ L. As one ight expect relaxing the steepness of the density gradient has the effect of weakening the odulation. This supports the observation by Watts et al. that the introduction of a density rap suppressed the odulation in their spectra. With a density rap present, electrons injected into the plasa excite plasa oscillations across the rap. Thus, for exaple for the paraeters used in Fig. 3b, n e = 6 n c, a 0 = 0.6, the siulations now show an enhanced = 3 line consistent with additional plasa excitation at n e = 9 n c in contrast to the sharp boundary case where only plasa oscillations with ω p = 4 ω L are present. A further distinction to be drawn is that the coupling of plasa waves to the radiation field is proportionately weaker in the case of an extensive rap. Note that the disappearance of odulation is not the only difference between the two spectra. Whereas the axiu over threshold (P /P 0 6 ) in Fig. 2b is 24, the result in Fig. 3b for a rap 0.6 λ L in width shows a range alost twice this value. Precisely why a rap should have this effect is not clear. The odulation first reported by Watts et al. was recorded over a range of about 5 haronics. Their spectra showed a distinct if irregular odulation with a frequency of between 2 and 4 ties that of the incident light. A less pronounced effect was subsequently found by Teubner et al. in experients using shorter pulses (typically 00 fs) and reduced intensities. Over a haronic range of about 0 lines they noted anoalies in eission though it is not possible to identify a odulation frequency over such a liited range. PIC siulations were carried out by both groups with a view to interpreting their data. The siulated spectru in [9] showed an average odulation frequency of 5-6, as against the 2-4 observed. Both sets of authors interpreted the odulation in ters of the so-called oscillating irror odel proposed by Bulanov et al. [3]. The ain stubling block to a fuller understanding of these anoalies is the uncertainty over 5

6 the ionization state of the target and hence the electron density. We illustrate in Fig. 4 data points fro our siulated spectra alongside the haronic efficiency easured in [3] for carbon and glass targets using frequency-doubled light fro a Ti:sapphire laser (395 n) at an intensity W c 2. Though this generated a relatively restricted range of soe 4 haronics, these experients were characterised not only by clean experiental conditions but by the choice of target aterials, both dielectric and etallic. In Fig. 4a we show the results of a siulation in which n e /n c = 30 for which the plasa line appears at 5.5 ω L. Fro the pattern found in Fig. 2 we should expect to see the first iniu in the odulated spectru at = 9 and so it appears. We have added data points fro the haronic spectru easured by Teubner et al. for a carbon target for coparison. In Fig. 4b we show results across the sae spectral range for a siulation with n e /n c = 45 which is characterised by a iniu at = and, by way of coparison, the easured spectru in this range for a glass target. In each case the haronics, siulated and observed, show a broadly coparable dependence on haronic nuber. However, in aking these coparisons we stress that the electron densities corresponding to the easured spectra were not deterined experientally. e-04 e-03 e-05 e-04 P/P P/P e-05 e e Figure 4: Haronic reflected power fro PIC siulations and fro experient: solid circles ( ) correspond to: a 0 =.0, λ L = µ, and n e /n c = 30, open circles ( ) correspond to eission easured fro a carbon target [3], : a 0 =.0, λ L = µ, n e /n c = 45 and τ p = 7 fs; : correspond to eission found fro a glass target [3]. 3. Conclusions. Our siulations lead us to conclude that the odulations we see in the haronic spectru ste fro the effect of plasa waves excited in the plasa slab. We have shown that the odulation frequency varies with the plasa density as n e /2 over a wide range. The odulation is evident for choices of intensity and pulse length for which plasa line eission does not distort the spectru to any arked degree. On this interpretation the odulation frequency affords a easure of plasa density. The presence of a density rap a fraction of a wavelength thick appears to suppress the odulation in agreeent with the observation by Watts et al. This finding is consistent with our interpretation, in that, in the presence of a rap there is no longer a unique plasa density at 6

7 the surface. The question ay properly be asked as to whether results obtained fro a PIC code such as that used in this work, liited in its diensionality, can reliably describe the physics of this interaction. In particular, should one expect the odulation effect to persist in three diensions? While it would be incautious to press this clai too strongly, we are encouraged, not least by the coparisons drawn fro Fig. 4 to believe that it ight. Acknowledgents. One of us (ROR) acknowledges support fro CONACyT under Contract No F. T.J.M. Boyd is grateful to G.J. Tallents for a helpful discussion. [] R.L. Caran, D.W. Forslund and J.M. Kindel, Phys. Rev. Lett. 46 (), 29 (98). [2] P.A. Norreys et al., Phys. Rev. Lett. 76 (), 832 (996). [3] U. Teubner et al., Phys. Rev. A 67, 0386 (2003). [4] P. Gibbon, Phys. Rev. Lett. 76 (), 50 (996). [5] T.J.M. Boyd and R. Ondarza-Rovira, in Proceedings of the International Conference on Plasa Physics, Nagoya, Septeber 996, edited by H. Sugai and T. Hayashi (Japan Society of Plasa Science, Nagoya, 997). Vol. 2, p. 78; T.J.M. Boyd and R. Ondarza-Rovira, Rev. Mex. Fís. 52 (2), 43 (2006). [6] T. Baeva, S. Gordienko, and A. Pukhov, Phys. Rev. E 74, (2006); S. Gordienko et al. Phys. Rev. Lett. 93 (), 5002 (2004). [7] T.J.M. Boyd and R. Ondarza-Rovira, Phys. Rev. Lett. 85 (7), 440 (2000). [8] U. Teubner et al., Optics Co. 44, 27 (997); D. von der Linde, Appl. Phys. B 68, 35 (999). [9] I. Watts et al., Phys. Rev. Lett. 88 (5), 5500 (2002). [0] A. Bourdier, Phys. Fluids 26, 804 (983). [] F. Brunel, Phys. Rev. Lett. 59 (), 52 (987). [2] F. Quéré et al., Phys. Rev. Lett. 96, (2006). [3] S.V. Bulanov and N.M. Nauova, and F. Pegoraro, Phys. Plasas (3), 745 (994). 7