Witness electron beam injection using an active plasma lens for a proton beam-driven plasma wakefield accelerator

An active plasma lens (APL) focuses the beam in both the horizontal and vertical planes simultaneously using a magnetic field generated by a discharge current through the plasma. A beam size of $5–10\mu \mathrm{m}$ can be achieved within a short distance using a focusing gradient on the order of $100\mathrm{T}/\mathrm{m}$. The APL is therefore an attractive element for plasma wakefield acceleration, because an ultrasmall size of the witness electron beam is required for injection into the plasma wakefield to minimize emittance growth and to enhance the capturing efficiency. When the drive beam and witness electron beam copropagate through the APL, interactions between the drive and witness beams, and the plasma must be considered. In this paper, through particle-in-cell simulations, we discuss the possibility of using an APL for the final focusing of the electron beam for the AWAKE RUN 2 experiments. It is confirmed that the amplitude of the plasma wakefield excited by proton bunches remains the same even after propagation through the APL. The emittance of the witness electron beam increases rapidly in the plasma density ramp regions of the lens. Nevertheless, when the witness electron beam has a charge of 100 pC, emittance of 10 mm mrad, and bunch length of $60\mu \mathrm{m}$, its emittance growth is not significant along the active plasma lens. For small emittance, such as 2 mm mrad, the emittance growth is found to be strongly dependent on the rms beam size, plasma density, and multiple Coulomb scattering.

Figure 1

Schematic view of the AWAKE RUN 2 experiment using two plasma sources (not to scale). The electron beam is generated by combining S- and X-band RF accelerators [18] and transported through the electron beamline. Normal quad indicates the normal electromagnetic quadrupole magnet. Note that the beamline in this schematic view, including the final focusing element, is not the baseline of the AWAKE experiment (i.e., the baseline does not consider using an APL).

Figure 2

Normalized plasma density distribution along the APL. The black solid line represents the active focusing gradient, which is assumed to be constant within the lens. We note that in the time scale of proton and electron beam passages, discharge current in the APL can be regarded as stationary [7].

Figure 3

Self-modulated proton bunches (a) and longitudinal wakefield (b) inside the APL plasma column with a density of $1.4583\times {10}^{17}/{\mathrm{cm}}^3$. The propagation direction of the bunches is to the right. Subfigure (c) is the longitudinal wakefield inside the red box indicated in (b). The longitudinal coordinate $\xi =z-ct$.

Figure 4

(a) Eleven self-modulated proton bunches used for the APL simulation. (b) An electron bunch with an rms beam size of $100\mu \mathrm{m}$ was placed between the 10th and 11th proton bunches from the right. The propagation direction of the beam is toward the right. (c) Transverse density distribution of a single proton bunch.

Figure 5

Maximum normalized $E_z$ field generated by drive proton (green) and witness electron (blue) beams along the entrance region of the APL. The rms beam size and bunch length of the proton (electron) beam used for this calculation are 221(100) and $107(60)\mu \mathrm{m}$, respectively. Black dotted line indicates the APL plasma density distribution in arbitrary units.

Figure 6

Plasma wakefield obtained by the PIC simulation. (a) Longitudinal plasma wakefield $E_z$. The horizontal axis $z$ indicates the longitudinal coordinate of the APL, whereas the vertical axis $\mathrm{\Delta }\xi$ represents the longitudinal coordinates of the electron beam slice. The electron beam head is placed at positive $\mathrm{\Delta }\xi$, whereas the tail is located at negative $\mathrm{\Delta }\xi$. Negative $z$ indicates the drift space before the APL. The solid and dashed white lines are the longitudinal electron beam distribution and plasma density distribution along the APL, respectively. Both distributions are given in arbitrary units. The inset in the figure shows the longitudinal wakefield at the black dotted line between 5 and 15 mm. (b–c) Horizontal plasma wakefield $W_x$ captured at the center of the electron beam longitudinal slice where the vertical axis $x$ represents the horizontal electron beam coordinates. The solid white line is the horizontal electron beam distribution. The insets in the figure show the horizontal wakefield at the black dotted lines. The initial electron beam sizes are (a,b) 100 and (c) $250\mu \mathrm{m}$.

Figure 7

(a,b) Evolutions of the normalized emittances along the APL with different initial electron beam sizes. The APL plasma (gray-dashed line in arbitrary units) is present between $z=0$ and 20 mm, whereas the negative $z$ indicates the drift space through which the electron beam propagates. Solid lines indicate the emittance evolutions, including the wakefield effect from both the proton bunches and electrons, whereas the dashed lines represent the cases without proton bunches. (c–d) Slice emittances after propagation through the APL. Black line: Longitudinal electron beam distribution (arbitrary units), where the head is placed at positive $\mathrm{\Delta }\xi$ and the tail is located at negative $\mathrm{\Delta }\xi$. The initial emittances of the electron beams are (a,c) 10 and (b,d) 2 mm mrad, respectively.

Figure 8

Longitudinal wakefields at the exit plasma ramp region. Gray dashed line (arb. units) shows the longitudinal electron beam slice. The solid colored lines indicate the wakefields from the electron beam only, whereas the dotted lines represent cases with proton bunches. The initial electron beam sizes are 100 and $250\mu \mathrm{m}$.

Figure 9

Electron beam distributions (first column) and horizontal phase spaces (second column) at the focal point. (a–b) Initial beam size of $100\mu \mathrm{m}$. (c–d) Initial beam size of $250\mu \mathrm{m}$. (a–d) Initial emittance is 10 mm mrad. (e–f) Initial beam size of $100\mu \mathrm{m}$. (g–h) Initial beam size of $250\mu \mathrm{m}$. (e–h) Initial emittance is 2 mm mrad.

Figure 10

(a) Focusing fields generated by the discharge current. (b–c) Evolutions of the emittance along the APL. Dashed lines indicate the cases with linear focusing field only whereas the solid lines represent the cases with nonlinear field. (d–e) Horizontal phase spaces right after the APL with the initial beam size of $250\mu \mathrm{m}$. The initial emittances are (b,d) 2 mm mrad and (c,e) 10 mm mrad.

Figure 11

(a) Final emittance and its growth ratio as a function of electron beam charge. The bunch length was varied to maintain the peak current same for all cases. Solid and dashed lines indicate the cases with and without the plasma density ramp, respectively. (b) Maximum $|W_r|$ value (blue curve) at the center of the electron beam slice obtained by Eq. (8) with plasma density $n_p=1\times {10}^{16}/{\mathrm{cm}}^3$. Inset shows the transverse wakefield along $r$ in the case of 100 pC charge (red box). In this theoretical calculation, only the electron beam is considered. Orange curve shows the resonant condition $k_p{\sigma }_z$. (c) Transverse wakefields along the electron slice distribution with different charges obtained by the PIC simulation. Dashed Gaussian histograms are the longitudinal beam distributions. Here, the electron beam is moving through the flat region of the APL.

Figure 12

Evolution of emittance along the APL with several different peak APL plasma densities. The grey-dashed line represents the APL plasma distribution in arbitrary units. The initial electron beam sizes used for this simulation were 100 and $250\mu \mathrm{m}$.