Nonuniform discharge currents in active plasma lenses

Active plasma lenses have attracted interest in novel accelerator applications due to their ability to provide large-field-gradient (short focal length), tunable, and radially symmetric focusing for charged particle beams. However, if the discharge current is not flowing uniformly as a function of radius, one can expect a radially varying field gradient as well as potential emittance degradation. We have investigated this experimentally for a 1-mm-diameter active plasma lens. The measured near-axis field gradient is approximately 35% larger than expected for a uniform current distribution, and at overfocusing currents ring-shaped electron beams are observed. These observations are explained by simulations.

When considering traditional transport elements for cm-scale focusing, challenges will be encountered: (i) Due to the 1=γ 2 scaling of the focusing strength, with γ the electron relativistic Lorentz factor, solenoids have weak focusing (long 10-m scale focal lengths) for relativistic electrons and have hence only been applied to energies of a few MeV or less [21]. (ii) The strong field gradients of miniature quadrupoles (of order 500 T=m [22]) are promising, as is the more favorable 1=γ scaling of the focusing strength, but the effective focal length is strongly increased when one considers that three lenses of varying and opposite strengths need to be combined to achieve equal focusing in both planes [23]. Thus, a triplet configuration still only enables > 10-100 cm focal lengths for >100 MeV class beams. Based on concepts revived from the 1950s ion accelerator community [24][25][26][27], recently Ref. [28] demonstrated that discharged plasma lenses [29][30][31][32] have the potential to be of great value to compact accelerator applications, due to tunability, high-field gradients (>3000 T=m), and radial focusing symmetry. For example, I 0 ¼ 600 A for a R ¼ 0.25 mm capillary yields a field gradient of 1800 T=m, corresponding to a focal length of 5 cm for a L ¼ 30 mm long structure and 1 GeV electrons. Here the focal length was calculated using the thin-lens approximation f ¼ 1=ðkLÞ, with k ¼ eð∂B=∂rÞ=m e γc, e the electronic charge, m e the electron rest mass, ð∂B=∂rÞ the magnetic field gradient, and c the vacuum speed of light.
Discharge capillaries are typically described as a gasfilled elongated volume of square or circular cross section inside a glass or sapphire block, with cm-scale length and mm-scale diameter. By flowing a discharge current I 0 through a circular capillary of radius R, the current density is J 0 ¼ I 0 =πR 2 for a uniform current distribution. Using Ampère's law 2πBðrÞr ¼ μ 0 R r 0 Jðr 0 Þ2πr 0 dr 0 , the radial gradient ð∂B=∂rÞ of the azimuthal component of the magnetic field is with μ 0 the vacuum permeability, and assuming cylindrical asymmetry. Although Eq. (1) is useful for estimating the focal length of an active plasma lens, in practice the transverse current distribution is a function of radial coordinate r. For a collisional plasma, the plasma conductivity σ depends on the plasma density n e and the plasma temperature T e through [33] σ ¼ , ϵ 0 the vacuum permittivity, and k B the Boltzmann constant. Due to the weak scaling of σ with density (n e only appears in the logarithmic function Λ), the current distribution is dominated by the plasma temperature. Thus, with the plasma near the wall typically colder than the plasma near the axis in a discharged capillary, a radial temperature gradient exists that is expected to drive a nonuniform current distribution.
In this paper the nonuniform transverse current distribution in the plasma lens is investigated. Section II describes magneto-hydrodynamics (MHD) simulations of the temperature and current distributions in the discharged capillary. Section III presents experimental results using a 1-mm-diameter active plasma lens. Conclusions are presented in Sec. IV.

II. MODELING
To quantify the radial current distribution, onedimensional radial MHD simulations were performed. For numerical reasons, the wall and the plasma temperature T e were initiated at 0.3 eV [34]. For the simulations presented next, the parameters I 0 ¼ 93 A (800 ns after the peak of the 440 A discharge pulse), R ¼ 500 μm, and He gas at 8 × 10 16 cm −3 atomic density were used, with the current pulse shape shown (solid black curve) in the bottom right inset of Fig. 2. The density and temperature (at 800 ns after the peak of the current pulse) are shown in Fig. 1(a). The plasma temperature has increased near the axis to >3 eV, while the density is reduced on axis and increases towards the wall (the formation of a plasma channel). The current density and azimuthal magnetic field versus radius are shown as circles in Figs. 1(b) and 1(c), respectively. The MHD simulations support the previous assertion that the current density JðrÞ ¼ σE ∝ T 3=2 e is dominated by the temperature distribution T e ðrÞ; the current density is maximum on axis and falls off towards the wall. Here E is the axial component of the electric field in the comoving frame in which the plasma is locally at rest.
A simple expression for the azimuthal magnetic field BðrÞ can be derived assuming JðrÞ ¼ σðrÞE ∝ T e ðrÞ 3=2 . Following the work of Ref. [34], the temperature distribution satisfies the heat flow equation where and the thermal and electric conductivities were assumed to scale as κ ¼ κ 0 T 5=2 e and σ ¼ σ 0 T 3=2 e , respectively. The boundary conditions satisfy dT e ðx ¼ 0Þ=dx ¼ 0 and T e ðr ¼ 1Þ ¼ T Ã with T Ã the temperature at the wall. The current distribution can be expressed as with Near the axis x ¼ r=R < 1, and , based on JðrÞ ∝ T e ðrÞ 3=2 , with T e ðrÞ ∝ uðrÞ 2=7 as described in Ref. [34]. One can observe an approximately 35% enhancement in near-axis current density and B-field gradients. Equation (7) predicts that the gradient near the axis is For a cold wall boundary condition T Ã ¼ 0, the solution of Eq. (3) yields uð0Þ ≃ 0.067, uð1Þ ¼ 0, m I ≃ 0.106, and ð∂B=∂rÞ r¼0 ≃ 1.48ð∂B=∂rÞ uniform . The solid curve in Fig. 1(c) shows BðrÞ based on this simplified model in case of a nonzero wall temperature, assuming uð1Þ ¼ 0.01, and is in good agreement with the MHD simulation [one can derive that uð0Þ ≃ 0.087 and m I ≃ 0.131]. It can be observed that the on-axis magnetic field gradient is larger compared to the case of a uniform current distribution. The near-axis field gradient is enhanced by approximately 35%. In Sec. III experimental data is discussed that confirms this expected field gradient enhancement.

III. EXPERIMENTAL OBSERVATION
To experimentally study the role of discharge current nonuniformity in active plasma lenses, the LPA setup as depicted in Fig. 2 was operated at LBNL's BELLA Center. A 1.8 J laser was focused by a 2 m focal length parabola to a spotsize of w 0 ¼ 22 μm onto a deLaval gas jet of 840 μm diameter, tilted 30°from normal. As described in Ref. [35], a razor blade on a translation stage was attached to the body of the jet, allowing for a sharp density shock profile normal to the laser beam to be present. The longitudinal profile of the density (up-ramp to shock, sharp shock-induced density drop, followed by a gentle down-ramp to vacuum), as shown in the bottom left inset in Fig. 2, was controlled with jet pressure and blade location. By flowing hydrogen gas at electron densities of 2-6 × 10 18 cm −3 , tunable stable electron beams of energy  MeV were produced, with the energy spread on the sub-10% (rms) level. Charge was 10-50 pC at divergence 2-7 mrad, depending on the parameters of operation. Although shot-to-shot pointing fluctuations were of order 10%-20% of the beam divergence, at times minute-scale pointing drifts of <3 mrad with respect to the laser axis occurred (under investigation).
Following a vacuum drift of 30 cm after the jet, the electron beams entered the active plasma lens. The lens was a sapphire-based cylindrically symmetric structure of length L ¼ 15 mm and radius R ¼ 0.5 mm. When filled with He at 8 × 10 16 cm −3 atomic density (2.5 Torr), the plasma density n e was estimated to be ≃5 × 10 16 cm −3 based on the MHD simulations in Fig. 1(a). The current trace through the capillary is shown in the bottom right inset in Fig. 2, at 440 A peak current. The timing of the LPA laser and electron beam with respect to the discharge pulse was adjustable. A phosphor screen was positioned 140 cm downstream of the plasma lens, and a magnetic spectrometer 240 cm from the plasma lens. The relevant properties of the magnetic spectrometer (energy resolution and fringe fields) are described in Ref. [36]. The imaging resolution of the phosphor screen was estimated to be 0.3 mm (rms). Although the beam divergence was variable over the range 2-7 mrad (rms), the angular acceptance of the capillary was AE1.7 mrad.
The results of a discharge timing scan are shown in Fig. 3, taken at 8 × 10 16 cm −3 atomic He density in the capillary and peak current of 440 A. The blue circles display the peak signal level on the camera (in arbitrary units), and the red circles the beam size (in units of mm). The error bars are defined as the standard error of the mean. The timing was varied on the falling slope of the discharge pulse (see bottom right inset in Fig. 2). The electron energy was 62 AE 2 MeV with a 6% (rms) energy spread. One can observe that the smallest electron beams where recorded around 50 A. Note that a uniform-current-density capillary FIG. 2. Schematic concept of the experimental setup. Electron beams produced by a jet-blade assembly [35] were propagated through the plasma lens to either a transverse charge diagnostic (phosphor screen) or energy distribution diagnostic (magnetic spectrometer). The bottom left inset shows the (simplified) plasma profile at the LPA, while the bottom right inset displays the measured discharge current profile (red curve) and profile used for the MHD simulations (black curve). would have predicted a current of 68 A for focusing (approximately 35% larger than measured); see dashed blue line in Fig. 3. This observation supports the assertion that current nonuniformity from near-axis heating leads to a 35% near-axis gradient enhancement; hence 35% less current is needed to achieve the ∂B=∂r gradient required for focusing.
The left column in Fig. 4 shows representative images of the charge distribution at the phosphor screen (same scan as Fig. 3). For currents above 60 A in Figs. 4(c)-(e), where the lens is stronger, the electron beam has overfocused and is diverging at the location of the screen. One can observe that the beam has developed a ring-shaped profile. However, at 27 A in Fig. 4(a), where the lens is weaker and the electron beam is still converging onto the screen, the electron beam is larger but without the ring-shaped profile.
To understand these features, the observations are compared to a particle tracking simulation based on the propagation of > 10 5 particles from LPA source (5 μm source size was used) to phosphor screen, using the experimental conditions for the LPA and transport line described above. Space charge effects during transport and electron beam driven wakefields in the lens have been evaluated, and were not found to significantly impact the results. The particle simulations in Figs. 4(f)-(j) used the B-field profile of Fig. 1(c) in the lens, with gradient enhancement 35%. One can observe that the overall features of the experimental data are reproduced. For currents > 60 A the central part of the electron beam in the lens is overfocused, leading to a weak charge density. However, the electrons near the walls experience a weaker field gradient and have not expanded significantly. This leads to the ring-shaped beam profile. For weaker currents <30 A both near-axis and near-wall electrons are still converging to the phosphor screen, maintaining a filled-in profile. When the transport simulations are repeated for a uniform current density plasma lens, the ring-shaped profile is not observed and the current needed to focus the electron beam has shifted from 52 A to 68 A. Note that the mm-level electron beam size at focus, observed in experiment [ Fig. 4 Fig. 4(e)], is a result of the chromaticity of the active plasma lens combined with the electron beam energy spread, as well as the imaging resolution of the phosphor screen camera. These effects were included in the tracking simulations. The profile asymmetries in Figs. 4(c)-(e) can be qualitatively explained through comparison to transport simulations by a <3 mrad electron beam pointing offset causing a nonuniform charge transmission through the capillary. It was also verified using transport simulations that a tilted misaligned active plasma lens does not reproduce the asymmetry.

(b)] and simulation [
The transport simulation results demonstrate that the current distribution parametrization based on the MHD simulations and simplified model of JðrÞ ∝ TðrÞ 2=3 , discussed in Sec. II, provides good agreement with experimental data, highlighting the role of the current density distribution as shown in Fig. 1(b). Depending on the electron beam size at the plasma lens, this current nonuniformity can manifest itself as a degradation of emittance, since the focal strength of the lens contains a radial dependence. For an electron beam confined to < R=2 inside the lens, the field gradient [cf. Fig. 1(c)] is approximately constant and little emittance degradation is expected. Also, MHD simulations presented in Ref. [32] indicate that current nonuniformity is minimized at lower peak current and higher fill pressures. This could imply that operating a 440 A peak discharge pulse at 200 A timing (trailing edge) is more detrimental to the emittance than using a 200 A peak discharge pulse at peak timing. The benefit of higher fill pressures is that ionization takes longer to accomplish, leaving less time for (detrimental) plasma heating [32]. For a fixed nonzero wall temperature, the reduction in on-axis temperature then results in improved current uniformity. These concepts will be studied in future experiments.

IV. CONCLUSIONS
In summary, experiments were performed on 1-mmdiameter active plasma lenses using LPA-produced electron beams. Through comparison to transport simulations and MHD-simulated plasma lens properties, several key experimental observations were explained by the role of nonuniform radial current density (more current near the lens axis than near the wall): (1) The near-axis field gradient ∂B=∂r was enhanced, meaning less current was needed to focus the electron beam than expected for a uniformcurrent density lens. (2) For overfocusing currents, the beam developed a ring-shaped profile, while (3) at underfocused currents the beam remained with a filled-in profile. Future studies will focus on characterization, and mitigation, of emittance degradation from an active plasma lens with radially varying focusing strength.