PHYSICS OF PLASMAS 23, 063102 (2016) Analysis of radial and longitudinal field of plasma wakefield generated by a Laguerre-Gauss laser pulse Ali Shekari Firouzjaei and Babak Shokri Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839-63113, Iran (Received 16 February 2016; accepted 17 May 2016; published online 3 June 2016) In the present paper, we study the wakes known as the donut wake which is generated by Laguerre- Gauss (LG) laser pulses. Effects of the special spatial profile of a LG pulse on the radial and longi- tudinal wakefields are presented via an analytical model in a weakly non-linear regime in two dimensions. Different aspects of the donut-shaped wakefields have been analyzed and compared with Gaussian-driven wakes. There is also some discussion about the accelerating-focusing phase of the donut wake. Variations of longitudinal and radial wakes with laser amplitude, pulse length, and pulse spot size have been presented and discussed. Finally, we present the optimum pulse dura- tion for such wakes. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4953052] I. INTRODUCTION paper has the following structure. In Section II, we drive the basic equations of our model. In Section III, we present Laser plasma interaction is a central problem in plasma some conclusions and discuss on the possible implications of physics. It is essential to understand its different aspects spatial profile effect of laser pulses on longitudinal accelerat- which are related to laser plasma acceleration schemes, ing and radial focusing of donut wakes. Finally in Section advanced fusion concepts, and novel radiation sources.1–8 IV, the results are summarized. When a laser pulse propagates through underdense plasma, a running plasma wave is produced by the ponderomotive force of the laser pulse. This wave oscillates at the frequency II. ANALYTICAL INVESTIGATION pﬃﬃﬃ xp= c, where xp ¼ ð4pn0e2=mÞ1=2 is the non-relativistic To investigate wakefield excitation, we adopt a com- electron plasma frequency. e, m, and, n0 denote charge, moving frame ðx ¼ x; y ¼ y; n ¼ z À ct; t ¼ tÞ where (x; y) mass, and density of electrons, respectively, and finally, c is are the transverse coordinates and t and z are the time and the electron relativistic factor. The produced accelerating propagation distance, repectively.10 We further use the fields can be three orders of magnitude greater than those in quasi-static approximation @=@t ! 0.9 When a laser pulse conventional accelerators.9,10 Many different aspects of propagates in low-density plasma in the z direction, a wake- intense laser plasma interactions have been studied at length field will be excited behind the laser pulse, such that the in the last decades. In the present work, we concentrate on wakefield satisfies the well-known wave equation9,10 the problem of laser wakefield generation when the laser pulse is not purely Gaussian. In recent years, there has been
1 þ aðr; zÞ2 an increasing interest on lasers with Laguerre-Gauss (LG) @2/ 1 kÀ2 ¼ À ; (1) shape because of their two special properties: special spatial p @2n 2ð1 þ /Þ2 2 profile and orbital angular momentum (OAM) states.11–15 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p ﬃﬃﬃﬃﬃﬃﬃﬃ The usage of LG laser pulses to control the transverse focus- where kp ¼ 4pe2n0=mc2 is the plasma wave number. This ing fields in laser wakefield acceleration was explored,16 and equation can be solved in two dimensions for the scalar poten- the influence of higher order Laguerre–Gaussian laser pulses tial / which is normalized to mc2=e, and the electric field in electron acceleration was also examined.17 In another which could be inferred through Ezðn; rÞ ¼ ÀE0@/=@n and work,18 the angular momentum of particles in the longitudi- Erðn; rÞ ¼ ÀE0@/=@r, where the parameter E0 ¼ mcxp=e nal direction produced by the LG laser was investigated, and refers to the cold non-relativistic wave breaking limit.9 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ the enhancement was compared with those produced by the We use cylindrical coordinates ðr; h; zÞ where r ¼ x2 þ y2 usual laser pulses. is the transverse distance to the axis and h is the azimuthal Recent papers have introduced a new type of wakes angle. The normalized vector potential (aL ¼ eAL) of the LG mc2 named donut-shaped wake which is produced by LG laser laser pulse is given by aLðn;rÞ ¼ a0ajjðnÞarðrÞ, where a0 is pulses.19,20 In the present paper, we model the wake gener- the peak laser vector potential; ajjðnÞ is the longitudinal ated by the LG laser pulse in a weakly non-linear regime in profile of intensity given by a 2 jjðnÞ ¼ expðÀðt À z=cÞ2=2sl Þ two dimensions. Furthermore, the excited longitudinal and where s transverse electrostatic wakefields are obtained via numeri- l, the laser pulse duration, is the root-mean-square (RMS) of length which is related to the full width at half cal study of the non-linear formula. We consider several pﬃﬃﬃﬃﬃﬃﬃﬃﬃs aspects of the donut-shaped wakes and discuss the advan- maximum (FWHM) through sFWHM ¼ 2 2ln2 l; arðrÞ is the tages and challenges of these types of wakes and finally com- transverse laser profile given by arðrÞ ¼ cl;pðr=w0Þjlj pare the results with the Gaussian-driven wake case. This expðÀr2=w 2 0 þ ilhÞLjljð2r2=w2Þ, where w p 0 0 is the laser spot 1070-664X/2016/23(6)/063102/5/$30.00 23, 063102-1 Published by AIP Publishing.
063102-2 A. S. Firouzjaei and B. Shokri Phys. Plasmas 23, 063102 (2016) size, Ljlj is a Laguerre polynomial with radial index p and az- Regarding these figures and considering the same laser p imuthal index l, and finally, c pulse amplitude, spot size, and pulse duration, there will be a l;p are normalizing factors.18,20 As we are interested in the donut type wake, we introduce a larger wake field for a donut wake (about ten times more). laser pulse with ðl;pÞ ¼ ð1;0Þ. For convenience, we have According to the previous works on Gaussian laser produced introduced the dimensionless time s ¼ x wakes,9 the most interesting and useful situation for charged psFWHM. The spatial coordinates are normalized to 1=k particle acceleration is when particles are injected near the p. Solving the above differ- axis. This is different for a donut wake. Particles should be ential equation numerically and using the LG laser profile, injected off axis in donut wakes because the longitudinal we can present and discuss the results for the transverse fo- accelerating field vanishes or has a minimum on the axis. cusing and longitudinal accelerating donut wakefield. According to Figure 1, for the LG case, the radiation source III. NUMERICAL RESULTS should be larger than the normal pulse case. Another important fact which is considered in this paper In this section, simulation results of the LG laser pro- is the overlap region for accelerating-focusing phase (AFP). duced wake are compared to the results of the Gaussian laser A particle of charge q propagating in the positive z direction pulse. Under the baseline simulation conditions, we use a in the plasma wakefield is accelerated if qEz > 0 and is laser pulse with normalized peak vector potential a0 ¼ 0:3, focused if q@rEr < 0. These inequalities define the AFP and normalized FWHM duration sFWHM ¼ kp=2c, where kp is thus determine the volume of space which is useful for elec- plasma wavelength, and normalized spot size of w ¼ 20. tron trapping and acceleration. Because of the curvature of Before analyzing the results, it should be noticed that all pa- the phase surface in the donut wake, the regions of radial fo- rameters are considered the same for both of the mentioned cusing are shifted toward longitudinal accelerating regions, laser pulses. and consequently, the overlap between the AFP increases First, we compare Gaussian laser produced wakes with respect to the Gaussian-driven case. In addition, as seen the one generated by a LG laser pulse. Figure 1 shows the from Figure 1, the AFP of a donut wake is larger for positive simulated longitudinal accelerating field Ezðn; rÞ and the charged particles than that for the negative ones. The phase transverse focusing field Erðn; rÞ in a weakly nonlinear re- difference between the accelerating and focusing fields is gime for both types of wake. Longitudinal and transverse generally well known in the quasi-linear regime. There exists fields of the donut and Gaussian pulse generated wakes are a kp=4 region that is both focusing and accelerating for either presented in Figures 1(a), 1(b) and 1(c), 1(d), respectively. electrons or positrons. However, in the quasi-linear case, the The accelerating phase of positive and negative charged par- formation of the AFP for positrons is mainly due to the varia- ticles in a donut wake consists of two slices in both sides of tion of the transverse shape and intensity of the driver pulse. the propagation axis n ¼ 0. The radius of each slice of the It can be seen from Figure 1(b) that the transverse field accelerating phase in the donut wake is about a quarter of the phase is shifted backward about p compared to the Gaussian size of the Gaussian-driven wake. case. Due to this phase shift, as presented in Figures 1(a) and FIG. 1. Accelerating and focusing wakefield generated by, (a) and (b) LG laser pulse, (c) and (d) Gaussian laser pulse. Accessible phase regions of fo- cusing and accelerating for both elec- trons and positrons are present.
063102-3 A. S. Firouzjaei and B. Shokri Phys. Plasmas 23, 063102 (2016) 1(b), the AFP for the positive charged particles is situated in focusing field amplitude. Furthermore, the nonlinear effects the first half of the bucket, so they could be injected into the of wave steepening and period lengthening are clearly evi- first bucket. This result is in contrast to Figures 1(c) and 1(d) dent. This will cause an enhancement in nonlinear plasma which show that the first bucket has no AFP for the positive wavelength.10 Thus, one can change the wakefield properties charges. (length and amplitude) by varying the radial profile of the This phase shift also leads to a shift in the AFP of the laser pulse. negative charged particles from the second quarter to the first To understand more the latest result, Figure 3 is plotted quarter in the first bucket. This finding is important in the self for varying pulse amplitude and pulse spot size simultane- injection process of a highly nonlinear regime and also other ously. It is found out that the longitudinal accelerating field methods of injection in which charged particles are injected could be greater than the cold wave breaking limit even for at the rear point of the first plasma wake. In the Gaussian pro- the quasi-linear laser amplitude case. So, it can be concluded duced wake, as the background plasma electrons are trapped that there is a transition from the quasi-linear to the non- in the first half of the wake region (self-injection), negative linear wake regime. This finding is really important. In previ- charges will experience a radial expulsion force. Therefore, ous works, it was shown that the maximum wake amplitude this will increase the radial emittance, resulting in the produc- for a Gaussian laser pulse in the linear laser amplitude re- tion of broad energy spectra of charges before being acceler- gime is Ez ¼ 0:76a2E 0 0, which is really smaller than the cold ated in the AFP region. In fact, this feature of the donut wake wave breaking limit.9,10 will be favorable for laser wakefield accelerators (LWFA). In Figure 4, the effect of the pulse duration on the donut After the above general comparison, we start to investi- wake is presented while keeping pulse amplitude fixed. At a gate parameter sensibility of donut wakes. Plots of Figure 2 fixed electron density, we examine three different pulse show details of variation of the longitudinal and radial fields lengths csFWHM ¼ kp=2 ; kp=4 ; kp=6. It is obvious that the versus laser spot size. It is obvious that laser spot size has a pulse duration influences the wakefields. One could find strong effect on both the longitudinal accelerating and radial FIG. 2. Variation of (a) longitudinal and (b) transverse, component of donut FIG. 3. Variation of (a) longitudinal (b) transverse donut wakefield as a wakefield with spot size as a function of kpn and kpr for spot size changed to function of kpn and kpr. Normalized spot size changed to w ¼ 25 and nor- w ¼ 25. malized amplitude changed to a ¼ 0:6 simultaneously.
063102-4 A. S. Firouzjaei and B. Shokri Phys. Plasmas 23, 063102 (2016) FIG. 4. Longitudinal wakefield gener- ated by different pulse durations, (a) csFWHM ¼ kp=2, (b) csFWHM ¼ kp=4, and (c) csFWHM ¼ kp=6. with those of the Gaussian produced wakes. This regime could be well characterized, and the acceleration process could be optimized for maximum electron and positron ener- gies. We showed that the donut wake has a larger focusing- accelerating phase for positive charged particles compared to the Gaussian wakes. Our results also show that unlike Gaussian-driven wakes, there is no possibility to inject charged particles on axis in donut wakes. Finally, variation of the laser spot size and laser amplitude were presented and their effects were discussed. In addition, effects of laser pulse duration were examined, and the optimum pulse duration to resonantly drive the wakefield was obtained. FIG. 5. Behaviour of normalized maximum donut wake amplitude versus pulse length. 1D. Jaroszynski, R. Bingham, E. Brunetti, B. Ersfeld, J. Gallacher, B. van der Geer, R. Issac, S. Jamison, D. Jones, and M. de Loos, Philos. Trans. R. from Figure 4 that the maximum wake amplitude will be Soc. London A 364(1840), 689–710 (2006). 2 generated by the pulse length cs S. Kneip, S. Nagel, S. Martins, S. Mangles, C. Bellei, O. Chekhlov, R. FWHM ¼ kp=2. To find out Clarke, N. Delerue, E. Divall, and G. Doucas, Phys. Rev. Lett. 103(3), the pulse length at which the laser pulse will most efficiently 035002 (2009). drives the plasma wave, we plot the maximum normalized 3W. Leemans, B. Nagler, A. Gonsalves, C. Toth, K. Nakamura, C. Geddes, wake amplitude versus the pulse length in Figure 5. It is evi- E. Esarey, C. Schroeder, and S. Hooker, Nat. Phys. 2(10), 696–699 (2006). 4 dent that the optimum pulse duration for a donut wake occurs G. A. Mourou, T. Tajima, and S. V. Bulanov, Rev. Mod. Phys. 78(2), 309 (2006). at s ¼ 2:6. This is different from the results reported for the 5K. Nakamura, B. Nagler, C. T oth, C. Geddes, C. Schroeder, E. Esarey, W. Gaussian-driven wakes.9,10 Leemans, A. Gonsalves, and S. Hooker, Phys. Plasmas 14(5), 056708 (2007). 6C. Schroeder, E. Esarey, C. Geddes, C. Benedetti, and W. Leemans, Phys. Rev. Spec. Top.: Accel. Beams 13(10), 101301 (2010). IV. SUMMARY AND CONCLUSION 7M. Tabak, J. Hammer, M. E. Glinsky, W. L. Kruer, S. C. Wilks, J. Woodworth, E. M. Campbell, M. D. Perry, and R. J. Mason, Phys. In this paper, we have studied the case of the laser gener- Plasmas 1(5), 1626–1634 (1994). ated wakefield by an intense LG pulse in a weakly non-linear 8T. Tajima and J. Dawson, Phys. Rev. Lett. 43(4), 267 (1979). 9 regime. An analytical formula was introduced and numerical E. Esarey, C. Schroeder, and W. Leemans, Rev. Mod. Phys. 81(3), 1229 (2009). results have been demonstrated. We have explained some 10E. Esarey, P. Sprangle, J. Krall, and A. Ting, IEEE Trans. Plasma Sci. general features of the donut wakes and then compared them 24(2), 252–288 (1996).
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