Magetostatic amplifier with tunable maximum by twisted-light plasma interactions D. Wu1,2,∗ and J. W. Wang2,1,† 1State Key Laboratory of High Field Laser Physics, Shanghai Institute of Optics and Fine Mechanics, 201800 Shanghai, China 2Helmholtz Institut Jena, D-07743 Jena, Germany (Dated: January 4, 2017) Laser beams with Laguerre-Gaussian (LG) mode carry orbital angular momentum (OAM),how- everwheninteracting with plasmas, thenet OAMacquiredbyplasmas isbasically zeroafter inter- 7 action. Here, we find when there exist a small magetostatic seed along laser propagation direction, 1 the barrier would be broken, giving rise to dramatic OAM transfer from LG lasers to plasmas. 0 Hence, the net OAM remained in plasmas system would continuously enhance the magetostatic 2 field, until the corresponding Larmor frequency of electrons is comparable to the laser frequency n in vacuum. Three-dimensional particle-in-cell (3D-PIC) simulations are performed to confirm our a theory,producingspace-uniform, time-stable and extremely intense magetostatic fields. J 3 PACSnumbers: 52.38.Kd,41.75.Jv,52.35.Mw,52.59.-f ] h Introduction–Generation of space-uniform, time- p stable and extremely-strong magnetostatic fields will be - m of great importance to plasma, beam physics and astro- physics [1–9]. Among many of great applications, such s a as the magnetically-assisted fast ignition approach [8], l the axial magnetic field would collimate the relativistic p electronbeamtoincreasethecouplingefficiencybetween . s the laser and the core. Furthermore, the magnetically- c i assisted proton acceleration scheme was also proposed s recently [9], the axial magnetic field could not only col- y h limate the relativistic electron beam but also enhance p the electron heating efficiency in front of the target. Al- [ thoughseveralkilotesla field has been measuredin a rel- ativistically intense laser-plasma interaction experiment 1 [10, 11], it is in small spatial and temporal scales, which v FIG. 1. (color online) Schematic of twisted light (l = 1 and 1 isdifficultinapplications. Thankstothe landmarkwork p=0) propagating in z direction. The twisted “bulk” is the 5 performed recently [12], extremely strong magetostatic 6 fieldupto1500Tisobtainedbyusinganovelcapacitor- surface ofconstant Ex. Thex-y,x-zand y-zcross-sections of 0 coil target design making use of the hot electrons gener- Ex are also presented at thecorresponding walls. 0 ated during the laser-target interactions. However even . 1 so,thisstate-of-artstrongestmagetostaticfieldisstillfar order approximation. To first order, plasma is assumed 0 smaller when comparing with laser field. to be of uniform, the longitudinal component of laser 7 Importantly, the schemes to produce relativistic fieldisignoredwhencomparedwiththetransverseparts. 1 twisted-light(withintensityover1018W/cm2)havebeen : Furthermore,instabilities[20,21],suchasfilamentinsta- v recently proposed [13, 14] and initiated new investiga- bilities,areignoredinthe consideredsituations. We find i tions of twisted-laser at relativistic intensities [14–19]. X that, when twisted light interacts with plasmas, the net Prompted by the inspiring advances of the twisted light r OAM acquired by plasmas is basically zero after inter- a intensity, the behaviour of twisted-laser-plasma interac- action. However, we also find when imposing a small tionsatrelativisticintensitiesshouldbestudiedindetail. magetostatic seed along laser propagation direction, the Relativistic intense twisted-light carries huge orbital an- barrier would be broken, giving rise to dramatic OAM gularmomentum(OAM),oncetransferredtoplasmasin transferfromLGlaserstoplasmas. Hence,thenetOAM the interaction, one can imagine, extremely strong axial remainedinplasmassystemwouldcontinuouslyenhance magetostatic field could be generated and sustained by the magetostatic field, until the corresponding Larmor the cyclotron motion of plasmas. frequency (ω eB/m ) of electrons is comparable to c e In this paper, we have investigated relativistic intense ≡ the laser frequency (ω ) in vacuum. Three-dimensional 0 twisted-light plasma interactions basically in the first- particle-in-cell (3D-PIC) simulations are performed to confirmourtheory,producingspace-uniform,time-stable and extremely intense magetostatic fields. ∗ Email:[email protected] Theoretical model–We consider a laser pulse carry- † Email:[email protected] ing OAM, whose spatial profile will be expressed as an 2 For a relativistic intense twisted-laser, the carried total B=+0.1 OAM is huge. When interacting with plasmas, the total B=0 OAM absorbed by plasmas can be calculated by sum- B=-0.1 marizing all individual-electrons. Here, as shown in Fig. 1, a linearly polarized LG laser with E0 = E0ex irradi- ates into uniform plasmas. The laser electric field expe- riencedby anindividual electronis E /E =(l !)−1/2 x 0 (r/b)|l| exp( r2/b2) cos(lθ+ω′t), with p|=| 0. Th×e × − × x-direction velocity of that electron can be found as u /u =(l !)−1/2 (r/b)|l| exp( r2/b2) sin(lθ+ω′t), x 0 | | × × − × withu =eE /m ω′c,ω′ =ω (1 v /v ),v istheelec- 0 0 e 0 z ph z − tron longitudinal velocity and v is the phase velocity ph oflaser. Assume that the plasma is ofuniform, the total FIG. 2. (color online) Total orbital angular momentum of OAM of the plasma system can be calculated as plasmas as functions of time in 3D PIC simulations. Simula- tion parameters: normalized amplitude of twisted laser is of t a=5, normalized plasma density is of ne =0.5 and mageto- Pz(t)= rsin(θ) Lnemeux rdrdθdt, (4) static seed of B =0.1 [red line (a)], B =0 [black line (b)] or Z0 × × B=−0.1 [blueline (c)]is loaded along zdirection. whereListhethicknessofplasmas,n isplasmadensity. e After integrating over r and θ, we find that LG mode. Considering the wave propagating along z- t direction, these LG modes reads, P (t) cos(ω′t)dt; l= 1 z = ± ± (5) LG (r, θ, z)= 2p! b0 L|l|(2r2) (√2r)|l| P0  Z00; l=0, 2, 3, ... l,p sπ(l +p)! × b × p b2 × b ± ± | | r2 k r2 From Eq. (4) and(5), we can see in the considered 0 exp( ) exp(ilθ+i +iΦ), (1) uniform plasmas cases, the twisted-light could transfer × −b2 × 2R the carried OAM to plasmas only when the topological where r = x2+y2, θ = arctan(y/x), b b(z) = charge is of l = 1. However even if l = 1, the ac- ≡ ± ± b 1+(z/z )2 is the beam width, b is the beam waist quired total OAM is still zero after averagingwith time, 0 0p 0 atz =0,z =k b2/2istheRayleighrange,k isthecar- whichmeanswhenlaserpenetratesthroughplasmas,the rieprwaven0umbe0r,0R R(z)=z[1+(z /z)2]is0thephase- remained OAM in plasmas is zero. 0 ≡ front radius, Φ Φ(z) = (2p+ l +1)arctan(z/z ) is Here, we realize that it is possible for plasmas to ac- 0 ≡ − | | the Gouy phase and L|l|(ξ) are the generalized Laguerre quire a net OAM, when there exist an external longitu- p polynomials, dinalmagetostaticfield. Althoughthe magetostaticfield is small, the barrier of transferring OAM from laser to p (l +p)! plasmas can be broken. L|l|(ξ)= ( 1)m | | ξm. (2) p − (p m)! (l +m)! m! The motion of electron by considering an external mX=0 − | | magetostatic field (B = B0ez) can be written as, The two independent indices l = 0, 1, 2, ... and ux/u0 =(l !)−1/2 (r/b)|l| exp( r2/b2) sin(lθ+ω′t) p=0, 1, 2, ...,correspondtothetopolo±gical±chargeand cos(ωct)an|d| uy/u×0 =(l !)−×1/2 (−r/b)|l| ×exp( r2/b2)× | | × × − × the number of non-axial radial nodes of the mode. Note sin(lθ+ω′t) sin(ωct),whereωc =eB/me isLarmorfre- × for l = p = 0, we recover the standard Gaussian beam. quency, which can be negative or positive depending on Hence, the vector potential can be cast in the form, the direction of B. Similarly, the total OAM of plasmas can be written as 1 E(r, z, t)= E0 LGl,p(r, θ, z)eiw0t−ik0z +c.c., (3) t 2 cos[(ω′ ω )t]dt; l = 1, ω >0, c c ω0 = ck0 is the frequency of laser, and E0 is the elec- Pz(t) =±Z0 − ± (6) tric amplitude of the laser. Note that the wave-front is P0  tcos[(ω′+ω )t]dt; l = 1, ω <0. helical, in the Coulomb gauge, the electric can have a ± c ± c Z0 preoallairsitzicatsioonlutcioomnpsohnoeunldt iinnctlhuedeprtohpisaglaotnigointuddiirneactliopno.laAr- FromEq.(6), wecanseewhen ω′ =ω ,plasma willcon- c ization component, which becomes particularly relevant tinuously acquire OAM from lasers. This condition can near the vortex axis. However, to first order, both com- be easily satisfied if the twisted-light is of relativistic in- ponents will give independent dynamics, allowing us to tensity,whoseponderomotiveforcewillalwaysaccelerate consider only the transverse part. electronsforward,approachinglightspeed. Hencetheex- In the paraxial regime, LG modes carry a discrete perienced laser frequency by electrons will decrease with OAM of l¯h per unit along their propagation directions. longitudinalvelocityuntiltheconditionω (1 v /v )= 0 z ph − 3 ω is satisfied. When ω (1 v /v ) = ω , we have c 0 z ph c cos[(ω′ ω )t] = 1, plasma−s will absorb OAM from c − twistedlightwiththevaluelinearlyincreasingwithtime. Once plasmas acquire net OAM, the imposed mageto- static field would be further enhanced and hence elec- tronswithsmallerv wouldalsoinvolvein. Themageto- z static field andOAMinplasmaswill repeatedly enhance with each other until ω =ω . c 0 Indetail,totriggerthemagetostaticamplifier,weneed to ensure that ω (1 v /v ) = ω . A higher v would 0 z ph c z − result in a smaller demanding of magetostatic seed. Ba- sically, by considering only the ponderomotive force of laser pulse, the maximal v of electrons is increasing z with laser amplitude a, i.e., v = a2/2γ , with z,max max γ =1+a2/2. Considering the phase velocity of laser max beaminplasmasisofv =1/ 1 ω2 /ω2γ,inthe low ph − pe 0 density limit, the smallest demqanding of laser amplitude FIG. 3. (color online) Isosurface visualisation of Bz with is amin = 2(1 B)/B, with B =ωc/ω0. data from 3D PIC simulations when the laser fully penetrat- − Simulation results–In order confirm the proposed ing through plasmas. The x-y, x-z and y-z cross-sections of p scheme, we have performed 3D PIC simulations by us- Ex are also presentedat thecorresponding walls. Simulation parameters are the same as shown in Fig. 1 but only with ing LAPINE code. The simulation box is 20 µm (x) × magetostatic seed of B=0.1 loaded along zdirection. 20µm(y) 20µm(z),whichisdividedinto 200 200 × × × 400grids. AlaserbeamofLGmode (l =1andp=0)is launchedintosimulationboxatleftboundary,withwave- (a1) (a2) length of λ = 1 µm, beam radius of b = 3 µm, beam B=+0.1 B=+0.1 0 0 duration of 5 20 5T (T = λ /c) and normalized 0 0 0 − − amplitude a = 5. Here, magetostatic seed (B = +0.1 or B = 0.1) is loaded along laser propagation direc- − tion, therefore the corresponding smallest demanding of laser amplitude is a = 2(1 B)/B = 4.24. Uni- min − form plasmas with normalized density n = 0.5n (the e c correspondingcriticaldensitpyisn =1.1 1021 /cm3 for c × laser of wavelength 1 µm) is placed between z = 5 µm (b1) (b2) and z = 15 µm, with each cell containing 8 electrons. B=0 B=0 The movement of ions is deliberately turned off as its contribution is ignitable for the considered situations. Fig.2showsthetotalOAMofplasmasbysummarizing all individual electrons as function of time. Here, black line is the case without magetostatic seed, we can see that within the laser-plasma interaction window (5T < 0 t < 45T ), total OAM of plasmas just simply oscillates 0 with time. After the penetration of laser beam, the re- (c1) (c2) mainedOAMisclosetozero. Thiskindofbehaviourcan B=-0.1 B=-0.1 be well described by Eq. (5), i.e. P (t)/P = tcos(ω′t). z 0 0 Red line and blue line are the cases with initial non-zero R magetostatic seed, B = +0.1 for red line and B = 0.1 − for blue line. We can see the total OAM of plasmas is continuously increasing/decreasing with time for mage- tostatic seed of B = +0.1/B = 0.1. Even after the − penetration, the remained OAM in plasmas is still very large and kept constant. The huge OAM remained in plasmas would generate FIG. 4. (color online) The y-z and x-y cross-sections of Bz strongstaticmagneticfields. Herewehavepickedupthe are presented in first and second columns. (a) (b) and (c) specific case with initial magetostatic seed of B = +0.1. correspond to cases with different magetostatic seed of B = 0.1 [red line (a)], B = 0 [black line (b)] and B = −0.1 [blue As shown in Fig. 3, the isosurface of constant B = 0.5 z line (c)] loaded long z direction. Simulation parameters are is plotted, and the x-y, x-z and y-z cross-sections of B z thesame as shown in Fig. 2. arealsopresentedatthe correspondingwalls. The static 4 magneticfieldsproducedinplasmasareofspace-uniform tion,thereforeitisarobustandwell-controllablescheme. and extremely intense. Detailed static magnetic fields In summary, we have investigated relativistic intense profile are presented in Fig. 4. The first column corre- twisted-light plasma interaction basically in the first- sponds to y-z cross-sections and the second column x-y order approximation. To first order, plasma is assumed cross-sections. We can see, for initial magetostatic seed of uniform, the longitudinal component of laser field is of B = +0.1 [Fig. 4 (a)], the finial magnitude of B can ignored when compared with the transverse parts. Fur- z be as high as B = 1, i.e., 10000 T considering the laser thermore, instabilities, such as filament instabilities, are wavelengthisof1µm. Whileforinitialmagetostaticseed ignoredin the consideredsituations. We find that, when of B = 0.1 [Fig. 4 (c)], the final magnitude is B = 1. twisted light interacts with plasmas, the net OAM ac- − − When no magetostatic seed is loaded initially, the finial quired by plasmas is basically zero after interaction. B is week and disturbed, as shown in Fig. 4 (b). However, we also find when imposing a small mageto- z Theupperlimitoftheproposedmagetostaticamplifier static seed along laser propagationdirection, the barrier is determined whenthe correspondingLarmorfrequency would be broken, giving rise to dramatic OAM trans- iscomparabletolaserfrequencyinvacuum,i.e.,ω =ω . fer from LG lasers to plasmas. Hence, the net OAM c 0 For laser of wavelength 1 µm, after the amplification, remainedinplasmassystemwouldcontinuouslyenhance the finial static magnetic field can reach 10000 T. To themagetostaticfield,untiltheLarmorfrequencyofelec- obtain even higher static magnetic field, such as 20000 trons(ωc =eB/me)iscomparabletothelaserfrequency T,30000T,...,second-,third-orhigherorderharmonics (ω0) in vacuum. Therefore, the upper limit of static of twisted-light shall be used. magnetic field can be controlled by using higher order Discussion and Summary–We also noticed that harmonics of twisted-light. 3D-PIC simulations are per- strongmagnetostaticfieldscanbeproducedbyusingcir- formed to confirm our theory, producing space-uniform, cularly polarized Gaussian laser, through the so-called time-stable and extremely intense magetostatic fields. “self-matching mechanism” [22] in near critical den- sity plasmas. To achieve “self-matching mechanism”, laser intensity is usually of extremely strong, at 1021 ACKNOWLEDGMENTS 1022 W/cm2. Under such conditions, non-linear effect∼s, suchas laserself-focusing andhelicalelectronbunchfor- ThisworkwaspartiallysupportedbyNationalNatural mations [20, 21] would become the dominant contribu- Science Foundation of China (No. 11605269, 11674341 tion. Howeverourtheoryworksinfirstorderapproxima- and 11675245). [1] X. H. Yang, W. Yu, H. Xu, M. Y. Yu, Z. Y. Ge, B. B. [12] S. Fujioka, Z. Zhang, K. Ishihara, K. Shigemori, Y. Xu,H.B.Zhuo,Y.Y.Ma,F.Q.Shao,andM.Borghesi, Hironaka, T. Johzaki, A. Sunahara, N. Yamamoto, H. Appl.Phys.Lett. 106, 224103 (2015) Nakashima, T. Watanabe, H. Shiraga, H. Nishimura, H. [2] X. T. He, J. W. Li, Z. F. Fan, L. F. Wang, J. Liu, K. Azechi, Sci. Rep. 3, 1170 (2013). Lan,J.F. Wu,andW.H.Ye,Phys.Plasmas 23, 082706 [13] Y. Shi, B. Shen, L. Zhang, X. Zhang, W. Wang, and Z. (2016). Xu, Phys.Rev.Lett. 112, 235001 (2014). [3] D. Wu, S. I. Krasheninnikov, S. X. Luan, and W. Yu, [14] J.Vieira,R.M.G.M.Trines,E.P.Alves,R.A.Fonseca, Nucl.Fusion 57, 016007 (2017). J.T.Mendonca,R.Bingham,P.Norreys,andL.O.Silva, [4] W. Feng, J. Q. Li, Y. Kishimoto, Phys. Plasmas, 23, Nat. Commun. 7, 10371 (2016). 032102 (2016). [15] J. T. Mendona, B. Thid, and H. Then, Phys. Rev. Lett. [5] D. Wu, S. I. Krasheninnikov, S. X. Luan, and W. Yu, 102, 185005 (2009). Phys.Plasmas 23, 123116 (2016). [16] J. Vieira and J. T. Mendona, Phys. Rev. Lett. 112, [6] S.X.Luan,W.Yu,F.Y.Li,D.Wu,Z.M.Sheng,M.Y. 215001 (2014). Yu,and J. Zhang, Phys. Rev.E 94, 053207 (2016). [17] J. T. Mendona and J. Vieira, Phys. Plasmas 21, 033107 [7] G.Ma, W.Yu,M.Y.Yu,S.X.Luan,andD.Wu,Phys. (2014). Rev.E 93, 053209 (2016). [18] X.Zhang,B.Shen,Y.Shi,X.Wang,L.Zhang,W.Wang, [8] W. M. Wang, P. Gibbon, Z. M. Sheng, and Y. T. Li, J. Xu, L. Yi, and Z. Xu, Phys. Rev. Lett. 114, 173901 Phys.Rev.Lett. 114, 015001 (2015). (2015). [9] A. Arefiev, T. Toncian and G. Fiksel, New J. Phys. 18 [19] W. Wang, B. Shen, X. Zhang, L. Zhang, Y. Shi, and Z. 105011 (2016). Xu, Sci. Rep.5, 8274 (2015). [10] M. Tatarakis, I. Watts, F. N. Beg, E. L. Clark, A. E. [20] R. H. Hu, B. Liu, H. Y. Lu, M. L. Zhou, C. Lin, Z. M. Dangor,A.Gopal,M.G.Haines,P.A.Norreys,U.Wag- Sheng, C. Chen, X. T. He and X. Q. Yan, Sci. Rep. 5, ner,M.S.Wei, M.ZepfandK.KrushelnickNature415, 15499 (2015). 280 (2002). [21] B. Liu, R. H. Hu, H. Y. Wang, D. Wu, J. Liu, C. E. [11] U. Wagner, M. Tatarakis, A. Gopal, F. N. Beg, E. L. Chen,J.Meyer-ter-Vehn,X.Q.Yan,andX.T.He,Phy. Clark, A. E. Dangor, R. G. Evans, M. G. Haines, S. P. Plasmas 22, 080704 (2015). D.Mangles,P.A.Norreys,M.S.Wei,MatthewZepfand [22] B. Liu, H. Y. Wang, J. Liu, L. B. Fu, Y. J. Xu, X. Q. K.Krushelnick, Phys.Rev.E 70, 026401 (2004). Yan,andX.T.He,Phys.Rev.Lett.110, 045002 (2013).