# Recovery time of a plasma-wakefield accelerator

### Plasma technology and characterization

A high-voltage discharge was used to create the plasma, ignited by a thyratron change working at a breakdown voltage of 25 kV, supplying roughly 500 A for a length of 400 ns. The plasma was contained inside a 1.5-mm-diameter, 50-mm-long capillary milled from two slabs of sapphire, mounted in a PEEK plastic holder, all mounted on a hexapod platform for high-precision alignment. A steady movement of argon was equipped by way of two inner fuel inlets from a buffer quantity at a ten mbar backing stress. The fuel escaped the open-ended capillary by way of holed copper electrodes (cathode upstream, anode downstream) into a big 500-mm-diameter vacuum chamber pumped to an ambient stress of 4.3 × 10−3 mbar. Broadening of spectral traces34 enabled the density on the longitudinal centre of the plasma cell to be resolved35. The profile and evolution of the plasma density had been recorded from the beginning of the discharge (0 μs) and to only after the arrival time of the electron beam (2.6 μs after discharge). The argon was doped with 3% hydrogen (outlined by atomic density) to spectrally broaden the H-alpha line. The density measurements had been fitted to acquire the plasma density, (1.75 ± 0.27) × 1016 cm−3, on the arrival time of the electron beam20.

### Electron imaging spectrometer

A dipole magnet was used to carry out vitality dispersion of the beam vertically onto a LANEX (nice) display screen mounted simply exterior the 1-mm-thick stainless-steel vacuum chamber wall, roughly 3 m downstream of the plasma cell. 5 quadrupoles (appearing as a triplet) positioned simply upstream of the dipole had been used to point-to-point picture the beam from the plasma-cell exit (the article aircraft) to the display screen (the picture aircraft) with a magnification of R11 = −5 (horizontally) and R33 = −0.97 (vertically), the place R is the object-to-image-plane switch matrix. The spatial decision of the optical system was roughly 50 μm (that’s, roughly 2 pixels), akin to an vitality decision of 0.05% for particles near the imaged vitality. Away from this imaged vitality, the vitality decision degrades relying on the vertical divergence of the bunch. The recorded two-dimensional pictures within the (x, E) aircraft could also be collapsed onto a single axis to supply a spectral density map in both x or E. The stacking of those maps—on this case a perform of bunch separation—is displayed as a waterfall plot in Figs. 2a,b, 3a and 4b .

### Spectrometer picture subtraction

In these measurements, a number of bunches work together with the electron-spectrometer scintillating display screen in its scintillation lifetime (measured to be roughly 380 μs), resulting in overlapping indicators in each area and time. A subtraction method was developed39 to allow reconstruction of the spectra of the probe bunch. This method makes use of O(100) measurements of solely the main bunch to foretell its scintillation sign (based mostly on its cost) and take away this from the spectrometer pictures within the case of the perturbed plasma. This subtraction course of contributes to the systematic uncertainty included in calculations of the vitality and transverse distributions of the probe bunch and is of the order of 10%; the magnitude of the systematic uncertainty is calculated by pixel-by-pixel comparisons of the measured scintillation sign of the main bunch solely and its corresponding predicted sign for every of the O(100) occasions. Imperfections on this subtraction process result in small variations within the driving-probe-bunch vitality spectra (Fig. 2a versus Fig. 2b). The three-bunch setup used right here (a single main bunch adopted by two probe bunches) means the scintillation sign from the trailing probe bunch is unaffected by the subtraction process (as there isn’t a overlap of the trailing probe bunch with another bunch on the scintillation display screen) and therefore its properties present the cleanest sign, motivating its use to outline the comfort of the perturbation. All properties of the trailing bunches are in contrast with optics set to picture an vitality of 1,100 MeV to enhance the decision of the trailing probe bunch. Nevertheless, comparisons between the imply energies of the driving probe bunch within the perturbed and unperturbed instances (orange knowledge factors in Fig. 2c) are carried out with a spectrometer imaging vitality of 1,050 MeV. On this case, the subtraction method is extra correct (with just a few per cent systematic uncertainty) because the change in imaging vitality minimizes imaging errors within the driving-probe-bunch spectra.

### Definition of residuals

Three separate residuals are used to outline the convergence of the perturbed plasma to the unperturbed state. The primary two correspond to measurements of the change in imply vitality of the driving and trailing probe bunches. That is known as the ‘relative vitality change’ in Fig. 2c and is calculated from

$$frac{{mu }_{E,{rm{u}}}-{mu }_{E,{rm{p}}}}{Delta {mu }_{E}},$$

the place ({mu }_{E,{rm{u}}/{rm{p}}}) represents the imply vitality of the unperturbed (u) or perturbed (p) bunch and (Delta {mu }_{E}) represents the typical vitality achieve and lack of the trailing and driving probe bunch, respectively, within the unperturbed scheme relative to the vitality of that bunch with out plasma interplay. The third residual is the ‘relative transverse bunch measurement’, calculated from

$$frac{{sigma }_{x,{rm{p}}}-{sigma }_{x,{rm{u}}}}{{sigma }_{x,{rm{u}}}},$$

the place ({sigma }_{x,{rm{u}}/{rm{p}}}) represents the transverse measurement of the trailing probe bunch within the unperturbed (u) or perturbed (p) scheme measured within the aircraft of the electron spectrometer. The bunch separation past which all three residuals return to, and stay at, zero inside experimental uncertainties is outlined because the restoration time of the plasma.

### Timescale for the formation of an on-axis density spike

In a plasma-wakefield accelerator, ions contained in the plasma wake focus the electrons within the passing beam. Within the course of, the beam electrons may even exert an equal however reverse power on the ions, which varies each in time, t, and in area, r. Assuming, for simplicity, a cylindrical bunch of space (2{rm{pi }}{sigma }_{r}^{2}) and a present profile I(t), the radial power on the ions within the radius of the beam is

$${F}_{r}left(t,rright)=frac{e{Z}_{{rm{i}}}Ileft(tright)r}{4{rm{pi }}c{sigma }_{r}^{2}{varepsilon }_{0}},$$

the place Zi is the ionization state of the ions, and e, c and ε0 are the electron cost, pace of sunshine in a vacuum and permittivity, respectively. Rosenzweig et al.14 used an analogous start line to mannequin the movement of ions throughout the preliminary plasma cavity. Nevertheless, within the current examine the movement of the ions is negligible on the timescale of the plasma-electron frequency. As an alternative, the overall radial impulse,

$$varDelta {p}_{r}left(rright)=int {F}_{r}left(t,rright){rm{d}}t=frac{e{Z}_{i}{Qr}}{4{rm{pi }}c{sigma }_{r}^{2}{varepsilon }_{0}},$$

induces a (non-relativistic) radial ion velocity (varDelta {v}_{r}=varDelta {p}_{r}left(rright)/{m}_{{rm{i}}}), the place ({m}_{{rm{i}}}) is the ion mass and (Q=int Ileft(tright){rm{d}}t) is the overall bunch cost. Assuming that plasma electrons don’t considerably alter the collective ion movement, the ions are all ‘centered’ onto the axis in a time

$$varDelta {t}_{{rm{spike}}}=frac{r}{{v}_{r}}=frac{4{rm{pi }}c{varepsilon }_{0}{m}_{{rm{i}}}{sigma }_{r}^{2}}{e{Z}_{{rm{i}}}Q},$$

which represents an approximate higher certain to the timescale of the on-axis ion-peak technology. For this experiment (Fig. 2), working in singly ionized argon (({Z}_{{rm{i}}}=1), ({m}_{{rm{i}}}=6.64times {10}^{-26},{rm{kg}})) with a median leading-bunch cost of 590 pC and r.m.s. transverse beam measurement of 5 ± 1 µm, the ensuing formation time for the density spike is estimated to be roughly 0.5 ± 0.2 ns.

### Origin of the density-independent betatron-mismatch bands

The driving probe bunch occupies a wide variety of wakefield section, and therefore longitudinal-field amplitude, for the vary of plasma-electron densities used within the experiments. Because of this, the betatron section advance, and, subsequently, the divergence of particular person vitality slices, varies considerably throughout the bunch on the plasma exit. When fixing the focal vitality of the capturing optics, this variation manifests itself as bands of raised depth on the spectrometer display screen, separated by an nπ section advance. Within the unperturbed case, the vitality at which these bands seem is fixed over an orders-of-magnitude plasma-density vary (Prolonged Information Fig. 2). That is because of the linear wakefield response generated by the top of the driving probe bunch, which is comparatively low in density due to the coherent synchrotron radiation induced in the course of the transport of the bunch to the plasma.

On this regime, the focusing and decelerating fields at a given longitudinal slice are linked with the focusing-field power40,41, which is approximated as

$$frac{{F}_{r}}{r}=-{left(frac{8{varepsilon }_{0}L{varDelta }^{3}}{9{e}^{2}{n}_{{rm{b}}0}{w}^{2}}proper)}^{1/2},$$

the place L is the size of the beam, Δ is the magnitude of deceleration of that slice, nb0 is the height bunch density and w is the width of the beam. On this area of the driving probe bunch, the present profile may be approximated as being longitudinally triangular (L = 60 μm) and transversely Gaussian (r.m.s. w = 40 μm), with a cost of 125 pC (giving nb0 ≈ 1.2 × 1016 cm−3). To acquire this expression, one can get the analytic expression for the pseudo-potential within the beam utilizing Inexperienced features40 and extract the corresponding transverse and focusing forces. The expression can then be readily derived. With these values, the mannequin predicts three shifts of π within the closing section of betatron oscillation over the top of the bunch, all separated by roughly 10 MeV, that’s, in good settlement with the experimental outcomes of Fig. 3a.

### Quantification of the betatron-mismatch bands

The vitality of the principle betatron-mismatch band within the perturbed-plasma case (Fig. 3a) is calculated for every separation by becoming a peak to the spectrometer picture projected onto the energy-dispersed axis. The imply vitality of those bands is given by the height of the match, with error bars representing the typical full-width at half-maximum of the height. Each the imply vitality and errors are overlaid on the simulated spectra of Fig. 4b. These values are the signature used to derive the curvature of the evolving radial ion profile (see the next subsection). At bunch separations round 10 ns, a number of focal traces seem in a small vitality vary within the spectra, resulting in a scientific enhance within the common full-width at half-maximum. On the shortest timescales, a big fraction of the driving-probe-bunch cost is misplaced and therefore the identification of the peaks within the spectra carries an related increased uncertainty.

### Derivation of ion-channel-profile parameters

The primary two experimental signatures—(1) the modification of the vitality slice that’s maximally centered by the post-plasma imaging optics because of the betatron mismatch, and (2) the oscillations of the r.m.s. transverse measurement of the trailing probe bunch—are a results of the movement of electrons within the probe bunches as they propagate within the plasma:

Within the first experimental signature, an electron that propagates within the linear portion of the wakefield, that’s, on the head of the driving probe bunch, undergoes transverse oscillations because of the focusing power supplied by the wakefield. Within the presence of a parabolic transverse-plasma-density profile, n(r)  =n0(1 +αr2), the focusing power at a given longitudinal slice for a uniform density profile is modified by the issue (1 +αr2). As such, the energies of the longitudinal slices that exit the plasma having acquired the suitable betatron section to correspond to bands of raised depth noticed on the spectrometer depend upon α by way of the relation

$${varDelta }_{alpha ,i}={varDelta }_{0,i}{(1+alpha {r}^{2})}^{2/3},$$

the place ({varDelta }_{alpha ,i}) and ({varDelta }_{0,i}) are the deceleration of slice i in a plasma with a non-zero and nil curvature, respectively. This allows reconstruction of the curvature as a perform of the separation between the main and probe bunches by becoming to the distinction in vitality between the band of raised depth for the perturbed case and that within the equal unperturbed case.

Within the second experimental signature, the trailing bunch has a low O(mm mrad) emittance and is targeted to a centimetre-scale β-function on the entrance of the plasma; therefore, it has a transverse measurement of roughly 8 μm, which is way smaller than even the steepest ion channel discovered within the experiment, the place a rise of roughly 5% is predicted within the measurement of the trailing bunch and a doubling in density from the worth on axis happens at a radial place of roughly 40 μm. The trailing probe bunch subsequently experiences restricted modifications to its off-axis focusing power because of the presence of the parabolic channel (confirmed in PIC simulations) and its divergence as a perform of the on-axis plasma density follows the relation

$${{rm{sigma }}}_{x{rm{{prime} }}}propto <?RetainOpenmmlmfenced separators=”” open=”|” shut=”|”?>|{ok}_{beta }{rm{sin }}<?RetainOpenmmlmfenced separators=”” open=”(” shut=”)”?>({ok}_{beta }{L}_{{rm{p}}})|$$

for a hard and fast plasma size Lp, the place ({ok}_{beta }={omega }_{beta }/c). The plasma size is assumed to be fixed over the O(100 ns) timescale thought of right here. The divergence of the bunch on the plasma exit instantly pertains to the r.m.s. transverse measurement measured within the aircraft of the electron spectrometer. Bunches with minimal and maximal divergence on the exit of the plasma may be centered to each small and huge sizes and excessive and low intensities, respectively, on the scintillating display screen. Due to this fact, the measured oscillations within the transverse measurement of the trailing probe bunch (Fig. 3a) may be correlated to extrema of the divergence of the trailing probe bunch on the plasma exit. This allows reconstruction of the on-axis plasma density as a perform of the separation between the main and probe bunches.

As the 2 experimental signatures are decoupled, the related equations can, subsequently, be solved independently by way of numerical becoming of every equation to the pertinent experimental observable, that’s, the vitality of the betatron-mismatch bands for α and the r.m.s. transverse beam measurement for n0 (Fig. 3a). The fitted values of the parabolic channel (and the related becoming errors) are proven in Fig. 3b.

### 6D beam reconstruction

For correct modelling of the plasma acceleration course of, sturdy measurements of each the transverse and longitudinal section areas are required. A sequence of 11 quadrupoles downstream of the plasma cell was used to move the electron bunches to an X-band transverse deflection construction (X-TDS)42, the place the beam was streaked onto a cerium-doped gadolinium aluminium gallium garnet display screen for measurements of the longitudinal section area. The energy-dispersed axis of the longitudinal section area was supplied by a dipole positioned between the X-TDS and the display screen. The overall size of the bunch was roughly 195 μm with a peak present of 1.5 kA for the main bunch and 420 μm with a peak present of 1 kA for the unscraped probe bunch. The X-TDS has the function of with the ability to streak in all transverse instructions. As such, it was potential to derive slice data of the horizontal and vertical planes of the beam by streaking within the vertical and horizontal instructions, respectively. Slice emittance measurements had been carried out in each planes for the main and probe bunches, offering beam measurement and emittance data for each 8 μm slice. The X-TDS was solely operated with non-plasma-interacted bunches and relaxed beam focusing because of the complexity of transporting high-divergence bunches the complete distance (33 m) from the plasma to the X-TDS measurement display screen.

### Particle-in-cell simulations

The 3D quasi-static PIC code HiPACE++ (ref. 43) was used to simulate the complete evolution of the beam–plasma interplay. The enter beam was generated based mostly on the 6D phase-space data of the experimentally characterised beams. It was modelled with 2 × 106 constant-weight macro-particles. A 32-mm-long flat-top plasma-density profile of peak 1.75 × 1016 cm−3 was estimated based mostly on density measurements (see above). The plasma was sampled with 16 particles per cell. A simulation field of measurement 600 × 600 × 480 μm3 (in x × y × ξ, the place ξ = zct represents the co-moving body) was resolved by a grid of 512 × 512 × 512 cells, developed with a continuing time step of 4.5ωp−1, the place ωp is the plasma frequency.