Direct Measurement of Focusing Fields in Active Plasma Lenses

Active plasma lenses have the potential to enable broad-ranging applications of plasma-based accelerators owing to their compact design and radially symmetric kT/m-level focusing fields, facilitating beam-quality preservation and compact beam transport. We report on the direct measurement of magnetic field gradients in active plasma lenses and demonstrate their impact on the emittance of a charged particle beam. This is made possible by the use of a well-characterized electron beam with 1.4 mm mrad normalized emittance from a conventional accelerator. Field gradients of up to 823 T/m are investigated. The observed emittance evolution is supported by numerical simulations, which suggest the potential for conservation of the core beam emittance in such a plasma lens setup.


INTRODUCTION
Laser wakefield accelerators (LWFAs) allow for the generation of extreme electric fields on the order of 100 GV/m for charged particle acceleration and can deliver beams of sub-µm normalized emittance [1,2], multi-kA peak currents [3], and femtosecond pulse duration [4][5][6]. LWFAs have shown the capability to produce multi-GeV electron beams in cm-scale structures [7][8][9]. Their application to drive compact sources of coherent X-ray beams [10,11] and incoherent MeV photons [12], ultra-fast electron diffraction experiments [13,14], and high-energy particle colliders [15] has been proposed and studied [16,17]. For all these applications small beam emittances are critical. Indeed, beams from plasma accelerators are susceptible to chromatic emittance growth in the drift following the acceleration section [18,19]. Thus, beam capturing within a few centimeters after the plasma exit is crucial for emittance preservation.
In this context, conventional focusing optics face problems: Solenoids suffer from large chromaticity and weak focusing for relativistic beams owing to their 1/γ 2 -scaling of the focusing strength, with the relativistic Lorentz factor γ. The more favorable 1/γscaling in combination with high field gradients (∼500 T/m for permanent magnets) of quadrupoles is put into perspective when considering that two quadrupoles need to be combined to achieve focusing in both transverse planes. Hence, quadrupoles, which are inherently defocusing in one plane, increase chromatic emittance growth in this plane dramatically [20].
In this work we report on a first direct measurement of the magnetic field distribution inside an APL and complement these results by experimentally detecting its influence on the emittance of a stable, well-characterized electron beam from a conventional 3 accelerator. These studies are supported by simulations and show the potential for emittance preservation.
Active plasma lenses for electron beams typically consist of a gas-filled capillary with a circular cross-section of mm-scale diameter and cm-scale length machined into glass or sapphire. A multi-kV discharge voltage is applied to the capillary ends, leading to breakdown of the gas. Subsequently, a current is driven along the generated plasma column forming an azimuthal magnetic field. In the following, we assume a azimuthally symmetric current distribution J(r), with r denoting the radial position.
Ampere's law provides the cylindrically symmetric magnetic field for r < R, with R being the capillary radius and µ 0 the vacuum permeability. The magnetic field distribution becomes B φ,ideal (r) = µ 0 I 0 r/(2πR 2 ) in case of a uniform current distribution J = I 0 /(πR 2 ), with I 0 being the total current. Differentiating this expression yields the ideal magnetic field gradient

NONLINEAR MODEL OF ACTIVE PLASMA LENSES
In general, Eq. (2) does not hold since the current distribution homogeneity J(r) is generally not uniform. A transverse temperature gradient forms due to cooling of the plasma at the capillary wall leading to a radially changing Ohmic resistance, a nonuniform current distribution, and a nonlinear magnetic field gradient. Fig. 1 shows the result of a one-dimensional magnetohydrodynamic (MHD) simulation of a capillary of R = 0.5 mm radius filled with hydrogen of n 0 = 10 17 cm −3 molecular density traversed by a current of I 0 = 364 A assuming a fixed electron temperature at the wall interface of T * = 0.5 eV. The radial position is normalized to R, the magnetic field to B ideal . Cases with I 0 = 188 A, and 740 A have also been simulated. The MHD modeling shows that for the currents used, the fraction of ionized hydrogen was well above 80%.
An analytic model for the current distribution in a plasma lens was introduced in [28]. It is based on the Spitzer collisional model of plasma, in which the conductivity σ depends on the plasma density n e and electron temperature T e via with λ D = 0 k B T e /n e e 2 , Λ = n e λ 3 D , k B the Boltzmann constant, 0 the vacuum permittivity, e the electron charge, and m e the electron mass. The scaling of σ is dominated by T e since n e appears only in the logarithm of Λ. Thus, the current density is dominated by the temperature J(r) = σE ∼ T 3/2 e . Following the work of [23,28], the temperature distribution satisfies the heat flow equation in which with m I = 1 0 u 3/7 xdx. The central region x < 1 can be written as and An important figure of merit for the linearity of an APL is its core linear magnetic field slope in comparison to the ideal magnetic field slope, defined as ∆ g = g core /g ideal = Fig. 1 is ∆ g = 1.48. This corresponds to a cold wall boundary condition. The corresponding gradients are given in Tab. I.

EXPERIMENTAL RESULTS
Direct measurements of the APL magnetic field gradients were performed by introducing a transverse offset of the APL with respect to the electron beam position, thus introducing a dipole kick to the beam. The centroid shifts and beam parameters of the MaMi-B beam were measured d = 25.3cm downstream of the APL at screen 7 S1 and averaged over 100 shots per offset position. The beam was focused into the APL in order to probe over the largest portion of the radius possible without beam clipping. Its dimensions at the capillary entrance were determined by backtracking the beam parameters based on the measurements at S1 without the plasma in its path. The beam size was calculated to be 80 µm rms in both planes. The offset was increased until clipping and charge loss of the beam became evident which resulted in a maximum offset of 350 µm. The resulting centroid shifts can be seen in Fig. 3 and are found to be linearly depending on the offset. The formation of fringe fields in APLswas discussed in [31]. Their influence on the emittance of a passing MaMi-type beam was simulated in ASTRA [32] and found to be negligible on the sub-percent level. The longitudinal current ramp in the fringe fields was modeled after I(z edge ) = I 0 /(1 + exp(4z edge /σ ramp )), where z edge is the distance from the capillary end and σ ramp is the ramp taper parameter, as commonly used in conventional magnet optics. Owing to the fringe fields, the effective magnetic length L = L capillary +2·L fringe of the APL extends beyond the sapphire capillary itself. So the beam offset ∆ x is dependent on the lens offset r and effective length L through the magnetic field in which p is the particle momentum, q its charge. To account for the additional uncertainty owing to the fringe field, the data in Fig. 3 was fitted for the range of L fringe ≤ 0.5 mm (which is well above the length found in [31]). The derived core gradients g core for L fringe = 0.25 mm including the systematic uncertainty for L fringe ≤ 0.5 mm can be found in Tab. I. The obtained magnetic field gradients are higher than Eq. (2) would predict from the measured discharge currents. They are, however, in good agreement with a J ∼ T 3/2 -model assuming a cold wall boundary condition with ∆ g ∼ 1.48. It is noteworthy that the relative center-of-mass jitter of the MaMi-B beam was not affected by the APL even for the extreme case of 350 µm offset (cf. Fig. 4). This means the magnetic field in the APL was highly reproducible, which may also be seen in the small error bars of the measured beam position in is measuring the emittance change of an electron beam after passage through the APL. Quadrupole scans were performed for different plasma lens settings in order to detect emittance change due to nonlinear field gradients. The currents used in the experiment were 188, 364, and 740 A. The current amplitude had a jitter of 1.5 A rms in each case. This measurement technique requires the beamline upstream of the quadrupoles used for the scan to be stable. The here reported APL stability greatly facilitated these emittance measurements and is reflected in the relatively low rms beam size variation during the scans of < 5% (100 shots were averaged per setting).
In order to probe for nonlinearities over a large fraction of the capillary diameter an rms beam size of σ y = 154 +5 −15 µm vertically and σ x = 151 +2 −12 µm horizontally and a  Other mechanisms for the emittance degradation such as self-wakefields and collisions fail to describe the observed dependence on total current which can readily be explained by a non linear field model. The driving of a self-wake can be neglected because of the low peak current used in the beam [33]. The emittance growth due to collisions can be estimated for: a) multiple scattering in neutral background gas [34], and b) transport in plasma [35,36] including the stopping power effects of collisions with free, bound and screened electrons, and Bremsstrahlung [36,37]. For the parameters relevant to this work, the normalized emittance growth due to scattering is estimated to be < 0.05 mm mrad. Owing to the small energy spread of MaMi (∼ 10 −5 ), chromatic effects were not relevant. The chromaticity introduced by the beam-plasma interaction was measured in the emittance measurements due to the dispersion introduced by the dipole in between the APL and the QMs used for the scans (cf. Fig. 2). No broadening of the energy spread was observed confirming the non-existence of self-wakefields.

SIMULATION RESULTS
The emittance growth in an L = 7.5 mm long APL was simulated with the particle tracking code ASTRA. The field was modeled to be of the form given by the J ∼

CONCLUSION
In summary, magnetic field gradients of a 1-mm diameter active plasma lens and the emittance change of a beam passing such a lens have been measured directly using the conventional accelerator Mainz Microtron. We observed excellent gradient stability. The measured gradient increase of ∆g 1.5 showed a behavior predicted for a cold wall boundary condition J ∼ T 3/2 -model. The measured emittance change of a passing electron beam agrees with predictions made by magnetohydrodynamics simulations and particle tracking simulations using the measured gradient enhancement as input parameter. Furthermore, simulations suggest that using beams of an rms size smaller than 20% of the APL radius leads to emittance preservation on the 14 mm mrad-level. Future studies will focus on mitigating emittance degradation further by manipulating the current density behavior in the APL by using different gas species and optimizing radii and current profiles.