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Containing intense laser light in circular cavity with magnetic trap door X. H. Yang,1,∗ W. Yu,2 M. Y. Yu,3,4,† H. Xu,5 Y. Y. Ma,1 Z. M. Sheng,6,7 H. B. Zhuo,1 Z. Y. Ge,1 and F. Q. Shao1 1College of Science, National University of Defense Technology, Changsha 410073, China 2Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China 3Institute for Fusion Theory and Simulation, Zhejiang University, Hangzhou 310027, China 4Institut fu¨r Theoretische Physik I, Ruhr-Universita¨t Bochum, D-44780 Bochum, Germany 5School of Computer Science, National University of Defense Technology, Changsha 410073, China 6Key Laboratory for Laser Plasmas (Ministry of Education), Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China 7 7SUPA, Department of Physics, University of Strathclyde, Glasgow G4 0NG, UK 1 (Dated: January 12, 2017) 0 It is shown by particle-in-cell simulation that intense circularly polarized (CP) laser light can be 2 containedinthecavityofasolid-densitycircularAl-plasmashellforhundredsoflight-waveperiods n beforeitisdissipatedbylaser-plasmainteraction. Aright-handCPlaserpulsecanpropagatealmost a without reflection into the cavity through a highly magnetized overdense H-plasma slab filling the J entrance hole. The entrapped laser light is then multiply reflected at the inner surfaces of the slab 1 andshellplasmas,graduallylosingenergytothelatter. Comparedtothatoftheincidentlaser,the 1 frequency is only slightly broadened and the wave vector slightly modified by appearance of weak nearly isotropic and homogeneous higher harmonics. ] h p Recently, intense laser interaction with plasma has InthisLetter,weproposeaschemefortrappingintense - m beenwidelystudiedinconnectionwithapplicationssuch light in a hollow shell of high density plasma by sending s asinertialconfinementfusion[1]andlaser-drivenparticle a RHCP laser light pulse through a strongly magnetized a acceleration [2–4]. Particle-in-cell (PIC) simulations and slab that serves as a transparent trap door. The laser l p theoreticalstudies[5,6]haveshownthatisolatedintense pulse can propagate through the slab with almost no re- . light pulses on the scale of the laser wavelength can be flection. Once the entire laser pulse (which can be suit- s c self-consistently trapped in plasmas as solitons. Trap- ably long) is inside the shell cavity, the magnetic field in i s ping of electromagnetic (EM) waves in plasma can also theslabisturnedoff,sothatthelightbecomescontained. y be from excitation of surface plasma waves [7–9], beam- It is then multiply reflected by the unmagnetized shell h plasmaand/orparametricinstabilities[10,11]. Recently, plasma or propagates along its surface. As a result, the p it is found that an intense laser pulse that has entered light can survive for hundreds of light-wave periods until [ into a hollow shell through a hole can also be trapped all its energy is absorbed by the slab and shell plasma. 1 when a near-critical density plasma layer is applied or The optical properties of a linear medium can be de- v self-generated to seal the entrance hole [12]. In all these scribed by its refractive index N = ck /ω = ε1/2 [25], 8 L L 1 cases,thereismuchlossofthelaserenergytotheplasma. where kL = 2π/λL, ωL, and c are the wave number, 9 On the other hand, there has also been considerable re- frequency, and vacuum speed of the light and ε is the 2 search on trapping and localizing light in atomic media dielectric constant of the medium. For simplicity, in the 0 or photonic structures [13–17]. However in these works followingweshallnormalizetheplasmadensityn bythe . e 1 nointenselaser-plasmainteractionisinvolvedbecauseof criticaldensityn =m ω2/4πe2,whereeandm arethe c e L e 0 the relatively weak laser intensities used. electron charge and mass, respectively. Accordingly, for 7 Intense magnetic fields of 103 −105T and above are unmagnetized plasma we have ε = 1−ω2/ω2 = 1−n . 1 p L e ubiquitous in the cosmic environment [18]. Strong mag- For magnetized plasma with the external magnetic field : v neticfieldscanalsobeself-generatedduringultraintense- B (normalized by m cω /e) along the laser propaga- 0 e L i X laser interaction with matter by compression of seeded tion direction, the dielectric constant can be written as magnetic fields [19–21], baroclinic effect [22], high- ε =1−n /(1±B ) [25, 26], where the plus and minus r B e 0 a current electron beams [23, 24], etc. In the presence signs correspond to the so-called L and R electromag- of kilotesla magnetic fields, the electron gyrofrequency netic waves, respectively [25]. Thus, for the R wave ε B ω can be comparable to or even larger than the laser is always larger than unity if B > 1, and the plasma is c 0 frequencyω . Inthiscase, aright-handcircularlypolar- transparentatanydensity. Incontrast,theLwavehasa L ized (RHCP)lasercan propagate deepintoan overdense cutoff density at n =1+B , where it will be reflected. L 0 plasma along an intense embedded magnetic field with- The proposed trapping scheme is illustrated in Fig. out encountering cutoff or resonance [25, 26]. Moreover, 1(a). Thecircularplasmashellhasonitsleftwallasmall transmission of RHCP laser into dense plasma can be section replaced by a highly-magnetized overdense flat controlled by an intense magnetic pulse in the plasma slabthatservesasatrapdoorfortheincidentlaserpulse. [27]. In order to see the function of the slab, it is instruc- 2 tive to first consider the one-dimensional linear theory of light transmission through a thin highly-magnetized high-density plasma layer, as shown in Fig. 1(b). The totalwaveelectricfieldE (x)ineachofthethreeregions y canbewrittenas[28,29]E exp(−ik x)+E exp(ik x) i L 1 L for x < 0, E exp(−ik x)+E exp(ik x) for 0 < x < d, 2 s 3 s andE exp[−ik (x−d)]forx>d,whereE andE ,E , 4 L i 1 2 E , and E are the amplitudes of the laser light incident 3 4 onandreflectedfromthefrontsurfaceoftheslab,transmitted into the slab, reflected from the back surface of the slab, and transmitted into the vacuum, respectively, k =ε1/2k isthewavenumberintheslab. Applyingthe s B L boundary conditions that at each of the boundaries (the total) E and ∂ E should be continuous, one obtains y x y after some algebra −i2RE sin(k d) E = 0 s , 1 R2eiksd−e−iksd 2RE E = 0 , 2 R(Ns+1)−(Ns−1)e−i2ksd FIG.1. Schematicof(a)theproposedscheme. Theentrance (1) holeontheleftshellwallisfittedwithanoverdenseflatslab, 2E E = 0 , whichisstronglymagnetizeduntilthelaserpulsehasentered 3 R(Ns+1)ei2ksd−(Ns−1) the cavity, and (b) the one-dimensional model for the slab 4N E used for illustrating the R wave propagation. (c) Analytical E = s 0 . 4 (Ns+1)2eiksd−(Ns−1)2e−iksd solution for linear propagation of the R wave through the slab. Here and in the following figures, E is normalized by y where N =ε1/2 and R=(N +1)/(N −1). Additional mecωL/e (corresponding to 3.22×1012V/m). s B s s internal reflections are negligible. Fig. 1(c) shows the profileoftheelectricfieldobtainedfromEq. (1)forn = e (x=0)ofthesimulationboxandisfocusedonoftheslab 20n , d = 0.2µm, B = 5, and E = 0.27 (the same as c 0 i atx=5.86µm. ItsfrontriseswithaGaussianprofilefor thecorrespondingparametersusedinthePICsimulation 17 fs to the peak intensity and remains constant for 150 below). It can be seen that the reflected light (E = 1 fs. The transverse profile of the laser pulse is Gaussian, 0.017)isonly6%ofthat(E =0.27)oftheincident,and i with spot radius 2µm. the transmitted light (E = 0.269) is very close to the 4 latter. Figure2showstheelectricfieldEy,magneticfieldBz, The relativistic 2D3V PIC simulation code EPOCH and axial Poynting vector Sx at t = 100fs. We see that [30] is used to investigate our scheme. The shell con- thelaserpulsecaneasilyentertheoverdenseslabintothe sists of Al plasma (ion mass m = 26.98m , where cavity. It propagates through the slab almost without Al p m =1836m is the proton mass) with charge Ze=10e reflection, and the amplitude of the transmitted laser in p e and density 50n . Its inner radius and thickness are the vacuum is very close to that of the incident laser, c 14.8µm and 0.2µm, respectively. The slab consists of in good agreement with the theoretical analysis (see Fig. hydrogenplasmawithdensity20n . Itsheightandthick- 1(c)). One can see from Figs. 2 for t = 100fs that there c nessare10µmand0.2µm,respectively. Thecenterofthe islittleenergylossbythelaserasitpassedtheslab. The shell cavity is located at (x,y)=(20,20)µm, so that the axial Poynting flux is everywhere positive except at the left front of the slab is at x=5.86µm. An external mag- shell-slab boundary, where some light is scattered. neticfieldofstrengthB =5(corresponding5.36×104T Reflections from the cavity wall and wave interference 0 in dimensional units) along the laser axis is embedded lead to restructuring of the EM fields in the cavity as in the slab plasma, which is shut down after the laser well as loss of wave energy to the slab and shell plas- pulse injection is complete. To account for laser-plasma mas. Figure 3 for t=350fs shows the transverse electric interaction induced plasma expansion, the extent of B field E , magnetic field B , and EM field energy density 0 y z is slightly wider than the slab. The initial temperature W = 1((cid:15) |E|2+µ−1|B|2), where (cid:15) and µ are the free- 2 0 0 0 0 of the slab and shell is 100 eV. The simulation box is spacepermittivityandpermeability,respectively. Wesee 40µm×40µmwith4000×4000cells. Eachcellcontains50 that light is trapped in the cavity and its structure is macroparticlesperspecies. Att=0,aRHCPlaserpulse rather complex. Because of constructive interference of ofwavelengthλ=1µmandintensityI =2×1017W/cm2 the reflecting light in the cavity, locally the magnitudes 0 (or laser parameter a = eE /m cω = 0.27, where E of the wave electric and magnetic fields can be up to L L e L L is the peak laser electric field) enters from the left side 1.61 and 2.35, respectively, times that of the incident 3 are generated around them. One can also see in Fig. 4(a) for t = 250fs that, as expected, at this instant the flow of light energy is still mainly around the ±x directions. Figure 4(b) for t = 350fs shows that at this later time interaction of the trapped light with plasma takes places mainly in three regions: at the slab and two regions on the shell wall. That is, the Poynting flux is at first localized along the axis region, but then spreads to other regions at latter times. Focusing and defocusing of the reflected light on the cavity axis can be observed at x ∼ 28µm and x ∼ 17µm. The 20n highly mag- c netized H slab plasma at the laser entrance is strongly heatedandexpands, buttheveryhighchargedensityAl shell plasma remains almost unaffected. The reason is that for the level of light intensity here, energy absorption by the shell plasma is mainly due to vacuum heat- ing [31, 32], whose efficiency depends on the skin depth, FIG. 2. Distribution of the transverse (a) electric (E ) and y which for the 50n Z = 10 Al shell plasma is 5 times c (b) magnetic (B ) fields, and (c) the axial Poynting flux S z x smaller than that of the 20n H slab plasma, where effi- att=100fs. (d)E andS (averagedover0.5µmaroundy= c y x cient resonant absorption dominates, especially after the 20µm)alongthexdirection. Hereandinthefollowingfigures, B isnormalizedbym cω /e(correspondingto1.07×104T) expansion [32, 33]. At still later times the interaction re- z e L and S is in units of W/m2. The highly localized E and gions on the right inner shell wall are wider spread, but x y S fields at the top and bottom slab-shell boundaries can be the structure and behavior of the trapped light in the x attributedtothecomplexlaser-plasmainteractionthere. See cavityremainsimilar. Duetocontinuousenergytransfer also Figs. 3(c) and 4. totheslabandshellplasma,afterthelaserpulsehasfully entered the cavity at t ∼ 190 fs, the total light/plasma energydecreases/increasesalmostlinearlywithtime,but laser. Figure 3(d) for the evolution of the energies of as mentioned, the total energy remains constant. How- the trapped light and the plasmas shows that the former ever, no stationary or quasistationary state was found increases with time until the laser pulse has completely evenatt>∼500fs, whenthelightfieldhasbecomeweak. entered the cavity (at t ∼ 190fs, recall that the slab is 5.86µm away from the left side of the simulation box). Figure5showsthespatialEMfieldspectraE (k)and y Then it gradually decreases. One can also see that the B (k) at two instants after the laser pulse has fully en- z energy of the plasmas first increases rapidly, mainly due tered the cavity. One can see that even though the spa- to light absorption by the slab electrons, which is heated tialwavestructureintherealspaceisrathercomplex,its andexpands,asshallbediscussedlater. Figure3(d)also Fourierspaceisrelativelysimple. Asmentioned,initially shows that after the laser pulse has completely entered the±k componentsfromthejust-transmittedlaserlight x the cavity at t∼190fs, energy absorption by the plasma dominates (not shown), but k components also appear y continues at a slower rate, and the total energy of the as reflections take place since the laser is of finite spot- light in the cavity and the plasma in the slab and shell size and the cavity wall is curved. Figures 5(a)–(c) show is well conserved. that except around θ ∼ π/2 and 3π/2 (apparently due On the other hand, Fig. 3(d) also shows that even to less light reflections occurring around these angles in thoughtransferoflightenergytotheslabandshellplas- thephysicalspace), the|k|=1mode(smalldarkpartial mas starts as soon as the laser enters the slab, even at circles in the figures) associated with that (k = 1eˆ ) of x t = 500fs the trapped light still retains about 78.6% of the input laser remains dominant at all times, which can itsmaximumenergy,namelythatatt∼190fs. Wehave alsoberoughlyseenfromtheanglesofthePoyntingvec- not attempted to optimize the light trapping, which has tors in Fig. 4. In fact, close examination shows that the to be done by trial and error adjustment of the initial |k| < 1 region is more highly populated around the ±x parameters. For example, simulations indicate that the directions, as to be expected. Weak but distinguishable energy of the trapped light can increase with the cavity spatial harmonics |k| = 2,3,4,... can also be observed. size. It should also be possible to improve the design of The spectra continue to evolve with time as energy loss theslabsothatabsorptionofreflectedlightbyitscavity to the slab and shell plasma continues, but their over- facing side is minimized. allprofilesremainalmostunchanged. Inparticular, they Figure 4 for the plasma density and light Poynting remain peaked at |k| = 1 (except around θ ∼ π/2 and flux distributions shows that the slab-shell boundaries 3π/2) and k = 0. Figures 5(e) and (f) show the fre- aremodifiedandstronglocalelectricandmagneticfields quency spectra E (ω) and B (ω) at the center of the y z 4 FIG. 3. Distribution of (a) E , (b) B , and (c) EM field y z energy density (in J/m3) at t = 350fs. (d) Evolution of the energiesofthelightwavesintheshell(redcircles),theplasma (including the shell and slab, black squares) and the sum of theformerenergies(bluetriangles). Thegreendashedlinein (d)markstheterminationofthelaserpulseinjectingintothe shell. Recallthatthelatterhasashort(17fs)Gaussianfront, FIG.4. DistributionoftheinstantaneousPoyntingfluxS = followed by a long (150fs) flat-top tail. S +S (arrows, their lengths indicating the magnitudes), x y and the electron density (color coded) of the slab and the shell at (a) t=250fs and (b) t=350fs. cavity. At other locations in the cavity they (not shown) are very similar. We see that the frequency spectrum is ingintenselightinthecavityofasolid-densityAlplasma highly peaked at the incident laser frequency ω , which L shell. PIC simulations demonstrate that the laser light together with the the dominance of the wavelength near can enter the cavity with the help of an axial magnetic that of the incident laser, confirms that the light of the field embedded in an overdense H slab covering a hole in latteristrappedwithoutalteringitsbasiccharacteristics. the shell. Inside the cavity the light wave is reflected as Theslightfrequencybroadeningcanbeattributedtothe well as partially absorbed by the slab and shell plasma. presenceinthecavityoflow-densityelectronsdrivenout Because of the multiple reflections, the wave frequency by the laser-wall plasma interaction, so that the vacuum is slightly broadened but remains close to that of the in- light-wave dispersion relation is slightly modified by incident laser. The wave vector, originally only along the clusion of the plasma frequency ω ((cid:28) k c). However, p L laser axis, also acquires new directions and weak spatial as already pointed out, we were unable to identify the harmonicsaregenerated,whichthenbecomeazimuthally EM field or Poynting vector structures with that of an isotropicexceptaroundθ ∼π/2and3π/2. However, the eigenmode solution of the vacuum wave equation in cir- wavelength of the incident laser still remains the domi- cular geometry [34], even though the trapped light has nating spatial scale. Because of energy transfer to the quite well defined frequency and wavelength, as well as slab and shell, the trapped light continues to decay until E (ω,k) and B (ω,k) relationship. The reason could y z it vanishes. The results here may be useful for under- be that there is insufficient time to form an eigenmode, standing storage of intense light, as well as interpreta- the cavity is not really circularly symmetric because of tionofphenomenonassociatedwithhighlylocalizedlight the slab, and/or the wave-plasma interaction at the slab [35, 36]. and cavity wall cannot be ignored. Furthermore, one can see in Fig. 5(a)-(d) that the region near k = 0 is We would like thank S. H. Luan for useful discussions. also enhanced, which can be attributed to the multiple This work was supported by the NNSFC (11305264, light reflections at the slab and shell plasma boundaries, 11275269, 11374262, 11375265, and 91230205) and the wherek changessign. 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