On the robustness of spin polarization for magnetic vortex accelerated proton bunches in density down-ramps
Lars Reichwein, Anna Hützen, Markus Büscher, Alexander Pukhov
OOn the robustness of spin polarization for magnetic vortex accelerated proton bunchesin density down-ramps
Lars Reichwein ∗ and Alexander Pukhov Institut für Theoretische Physik I, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany
Anna Hützen and Markus Büscher
Peter Grünberg Institut (PGI-6), Forschungszentrum Jülich, 52425 Jülich, Germany andInstitut für Laser- und Plasmaphysik, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany (Dated: February 9, 2021)We investigate the effect of density down-ramps on the acceleration of ions via Magnetic VortexAcceleration (MVA) in a near-critical density gas target by means of particle-in-cell simulations. Thespin-polarization of the accelerated protons is robust for a variety of ramp lengths at around 80%.Significant increase of the ramp length is accompanied by collimation of low-polarization protonsinto the final beam and large transverse spread of the highly polarized protons with respect to thedirection of laser propagation.
I. INTRODUCTION
The acceleration of spin-polarized particles is interest-ing for a variety of applications, from testing the Stan-dard Model of particle physics [1] to examining the struc-ture of subatomic particles for further insight on QCD[2]. As laser-plasma based acceleration mechanisms havegrown to be more prominent due to the high achievableenergies over a shorter distance than in conventional ac-celerators [3, 4], it is the logical next step to study theacceleration of spin-polarized particles in these regimes.The current state-of-the-art is given in the paper byBüscher et al. [5].In the case of electrons, Wu et al. [6, 7] have shownthat via both laser-driven and particle beam-driven wake-field acceleration, high degrees of polarization can beachieved, if an appropriately chosen laser pulse or drivingbeam, respectively, are used. It could be seen that thereal crux for generating high-polarization electrons lieswithin the injection: due to strong azimuthal magneticfields during injection, the spins of the electrons startto precess strongly, leading to a significant loss of polar-ization, while during the acceleration phase, changes inpolarization are mostly negligible.For the acceleration of protons in general, variousmethods like Target Normal Sheath Acceleration (TNSA)[8], Radiation Pressure Acceleration (RPA) [9] or Mag-netic Vortex Acceleration (MVA) [10–12] are feasible op-tions. Wakefield acceleration of protons is also possible,although significantly higher laser intensities are neces-sary [13, 14]. If we, however, need spin-polarized beams,we have to restrict ourselves to setups where we can pre-polarize our targets, ruling out some of the options dueto the properties of the materials that are needed. Pre-polarizing the particles to be accelerated is necessary,since at the time scales and field strengths considered ∗ [email protected] for acceleration, significant polarization build-up duringthe process is not possible [15].Jin et al. [16] recently considered the acceleration ofspin-polarized protons using a near-critical density tar-get. The process, identified as MVA, works as follows:When the laser pulse enters the target, the ponderomo-tive force pushes the electron in the direction transverseto laser propagation, leaving behind a channel of low elec-tron density [10, 17]. Electrons can be accelerated in thewake induced by the laser and form a central filament inthe channel. A strong azimuthal magnetic field is createdby a current flowing in the central filament along the axisand an opposing current along the channel wall. This cur-rent also accelerates some ions in the filament structurealong the channel center. When leaving the interactionvolume, the magnetic fields can grow in the transverseregion because of the decrease in density. Strong longi-tudinal and transverse electric fields are induced by thedisplacement of the electrons with respect to the ions.Finally, an ion beam is obtained that is further acceler-ated by the prominent fields after leaving the plasma. Jinet al. showed that while higher intensities lead to betterenergies ( E p > MeV for a laser with normalized laservector potential a = eA / ( m e c ) = 100 ), it comes at theprice of lower polarization. Here, m e denotes the electronmass and c the vacuum speed of light.In this paper, we investigate the effect of density down-ramps at the end of the interaction volume onto the ob-tained proton bunches, specifically the degree of polar-ization. We consider a gaseous HCl target similar toRef. [16] in our PIC simulations, keeping all parame-ters except the length of the down-ramp fixed through-out. It is shown that the degree of polarization is ro-bust against down-ramp length and that obtaining high-quality bunches is mainly limited by the change in spa-tial beam structure due to the prevalent electromagneticfields. Only for longer ramps the spatial structure ismodulated so strongly, that lower-polarization protonsare collimated into the beam. The results presented arediscussed in the scope of the scaling laws of Ref. [15]. a r X i v : . [ phy s i c s . p l a s m - ph ] F e b Figure 1. Distribution of particle spin and field configuration for the case of L ramp = 0 λ L at t = 320 τ . All protons in theplasma have initial polarization s y = 1 . The electromagnetic fields are normalized with E = B = mcω /e . It can be seenthat the accelerated proton bunch leaving the plasma maintains a high degree of polarization, while protons surrounding theremaining filament of the coaxial channel gain transverse polarization. II. SIMULATION SETUP
For our simulations we use the PIC code VLPL [18]that includes the precession of particle spin s accordingto the T-BMT equation d s i d t = − Ω × s i , (1)where Ω = qmc (cid:104) Ω B B − Ω v (cid:16) v c · B (cid:17) v c − Ω E v c × E (cid:105) (2)is the precession frequency for a particle with charge q ,mass m and velocity v . The prefactors are given as Ω B = a + 1 γ , Ω v = aγγ + 1 , Ω E = a + 11 + γ , (3)with a and γ being the particle’s anomalous magnetic mo-ment and its Lorentz factor, respectively. This equationdescribes the change in spin for a particle that traversesthrough some arbritary configuration of electric fields E and magnetic fields B .In general, more spin-related effects would have to beconsidered, like the Stern-Gerlach force that describes theeffect of spin onto a particle’s trajectory, and also theSokolov-Ternov effect, that links radiative effects withspin. It has, however, been shown by Thomas et al. [15],that these two effects can be neglected for the parameterregimes considered in the following.For our setup, we choose a circularly polarized laserwith a = 25 and wavelength λ L = 800 nm. The lengthof the pulse is τ = 10 λ L /c and it has a focal radius of w = 10 λ L (at /e of the intensity).The target consists of HCl gas with a peak density of n H = n Cl = 0 . n cr , leading to a near-critical elec-tron background. Here, n cr denotes the critical density.This specific gas is chosen because it allows for an easily Table I. Results of the simulations with different ramp lengthsin terms of average polarization and peak density of the pro-ton bunch. The average polarization of the proton bunch isobtained by selecting the particles in the high-density regionleaving the plasma channel. Note that for longer ramps ( λ L , λ L ) the shape of proton bunch is increasingly ill-defined. L ramp [ λ L ] (cid:104) s y (cid:105) n peak [ n cr ] 0.209 0.126 0.044 achievable pre-polarization of the protons (see Ref. [7]for a detailed description of the process). In our case, forall protons, we initially choose s y = 1 .The interaction volume starts with an up-ramp risingfrom vacuum to peak density over a distance of λ L , thenmaintaining peak density for λ L . The down-ramplength at the end is varied in the range of λ L up to λ L (see Table I).In our simulations, we use a box of size (100 × × λ L that is moving alongside the laser pulse until theaccelerated protons leave the plasma. The grid size ischosen as h x = 0 . λ L (direction of propagation) and h y = h z = 0 . λ L . We do, however, use a feature ofVLPL that allows for the increase of cell size the furtherwe go from the central axis in the transverse directionin order to reduce computational effort. The solver usedfor the simulations is the RIP solver [19], which requiresthat the time step is ∆ t = h x /c . III. DISCUSSION
When the laser pulse enters the target, the electronsare driven out in the direction transverse to laser prop-agation, leaving behind an ionic filament that is pushed
Figure 2. Density and spin polarization for the simulations with ramp lengths λ L , λ L and λ L (left to right) after theaccelerated proton bunch has left the plasma (end of ramp in box middle). Note that the density plots are clipped at . n cr for better visbility. The density plots show that increasing the ramp length is accompanied by a higher transverse spread ofthe resulting proton beam, which is located at X ≈ λ L for the case of L ramp = 0 λ L . out at the end of the plasma due to the electromagneticfields. Since all of our simulations have the same configu-ration at start, the created proton bunch will be identicaluntil the start of the down-ramp. We can see that thecentral filament initially keeps its polarization very wellwhile the region around it starts to depolarize due to theelectromagnetic fields (compare Fig. 1).As we enter the down-ramp region, we can start to seethe effects of the different ramp lengths L ramp . For thetarget with a hard cut-off in density, i.e. L ramp = 0 λ L ,the usual MVA fields can be observed: the magneticvortex starts to appear and a uniform longitudinal elec-tric field E x that drives the protons further out of theplasma. The proton energies that can be achieved for acomparable setup are discussed in the work by Jin et al.[16], where they reached E p ≈ MeV for a laser with a = 25 and a HCl plasma of similar density, but with L ramp = 5 λ L .Going to a longer ramp length, we can see that, dueto the lower densities in those regions, the fields startto grow transversely while the proton bunch is still inthe plasma (not shown here). An approximation for thestrength of the magnetic field in a down-ramp is givenby Nakamura et al. [10]. This change in field config-uration leads to differences clearly visible when lookingthe the density plots in Fig. 2: the accelerated proton bunch is modulated such that for longer ramps it fur-ther spreads in the transverse direction. Especially inthe case of L ramp = 75 λ L and λ L , the protons leavingthe plasma hardly form a consistent bunch structure any-more. Transverse density profile of the different beamsas well as peak density are shown in Fig. 3 and Table I.The change in bunch structure can be attributed totwo factors. Firstly, increasing the ramp lengths in afashion as we do in our simulations, also effectively leadsto a longer interaction volume, meaning that the laseris depleted of more energy. Secondly, the down-rampallows for the transverse fields to grow, leading to a widerchannel (also visible at the left boundary of the densityplots in Fig. 2) and therefore the transverse growth ofthe proton bunch.Besides the quality of the bunch in terms of transverseand longitudinal structure, the degree of polarization ob-tained at the end is of main interest. We can directlytell by looking at the precession frequency Ω in equation(2) that the change in proton spin should be significantlylower than for electrons, as | Ω | ∝ m − . To measurethe polarization of the bunch, we consider the particlesclose to the central axis. We subdivide the longitudinaldirection into several bins for which we calculate the av-erage polarization (cid:104) s y (cid:105) . Depending on the spatial beamstructure, different degrees of polarization can be located Figure 3. Transverse beam profile (at the plane with peakdensity) for a selection of different ramp lengths. Longerramps lead to a widening of the accelerated proton beam,reducing the peak density.Figure 4. Exemplary polarization data for the case of L ramp =0 λ L at time step τ . The spin for each PIC particle is as-signed to a corresponding bin in the longitudinal direction forwhich the average spin polarization (red line) and the numberof PIC particles (blue, dashed) are given. along the volume (compare Fig. 4). This is due to thefact that protons that end up in the beam front experi-ence slighty different electromagnetic fields than the onesin the beam’s stern.In total, for most of the ramp lengths considered, ahigh polarization of around 80% is maintained. Onlyin the simulation with L ramp = 100 λ L we see a signif-icant decrease to roughly 60%. It has, however, to bestrongly emphasized that high-polarization protons arestill pushed in propagation direction (see spin plot inFig. 2), only in a non-collimated form. Instead, someprotons with lower polarization (red region in the spinplots) make up a significant part of the proton bunchvisible in the density plots. These negative effects can partly be mitigated bychoosing different laser-plasma parameters, although ithas to be noted that for higher laser intensities the po-larization degree will also decrease as it was shown in[16]. This, as well as polarization decrease in our case oflong ramps, is explained by the scaling laws derived byThomas et al. [15]: A particle beam can be viewed asdepolarized, once the angle between initial polarizationdirection and the final spin vectors is in the range of π/ .The time after which this is to be expected is called theminimum depolarization time T D,p and scales as T D,p ∝ max ( | E | , | B | ) . (4)This means that stronger electromagnetic fields inducedby the laser pulse lead to a faster depolarization of theprotons. Further, the longer interaction volume due tolonger down-ramps also may decrease the polarizationonce we reach the range of the depolarization time. Whileshorter interaction volumes are desirable for high-qualityproton bunches, this may come at the cost of experi-mental realizability due to limitations of the nozzles andblades usable for the creation of a pre-polarized plasmatarget. In the case of electrons, a mechanical setup for fil-tering out unwanted spin contributions has recently beenproposed [20]. Depolarization after the initial accelera-tion of the protons out of the channel gets increasinglynegligible, as the prefactors of the precession frequency(3) get smaller for higher energies γ . IV. CONCLUSION
We have studied the effect of down-ramp length for anear-critical HCl gas target that we use to obtain highlyspin-polarized proton bunches via MVA. The interactionplasma has been pre-polarized, since polarization build-up over the course of acceleration is negligible. We ob-serve that longer down-ramps modulate the spatial bunchstructure, leading to ill-defined bunches. For most of theramp lengths examined, the yielded polarization robustlystays around 80% due to the inert proton spin. Signif-icantly longer ramps lead to the collimation of lower-polarization protons instead of the wanted ones. The de-teriorative effects of longer down-ramps can be compen-sated by adjusting the parameters of the laser and plasmaused to some extent. Generally, as-short-as-possible in-teraction volumes are preferable, since the minimum de-polarization time for the bunch is inversely proportionalto the field strength experienced by the protons.
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