Electron Trapping

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2. The Physics of Propagation of Ultrashort Electron Bunches in Underdense Plasma

Figure 2.11: Demonstration of re-acceleration and temporal evolution of trapped particles in a co-moving frame. The input parameters are the same as Fig. 2.4.

The comparison of electric and magnetic fields are shown in Fig. 2.10. Interestingly, the electric field defocusses the particles in the beginning as well as in the end of the process while the magnetic field is always focusing, as shown in Fig. 2.10(a) and (b), and the total contribution of both fields is always focusing within the wake as shown in (c).

2.2 Propagation of Electron Bunches Driven by LWFA 43

(a)

0 500 1,000 1,500

−0.4

−0.2 0 0.2 0.4

x-position (a.u.) Electricfield(E0)

(b)

200 400 600 800 1,000 0.98

0.99 1 1.01 1.02

Time (1/ω_p)

Phasevelocity(β) 1

2 3 4

Figure 2.12: Monitoring of local phase velocity evolution of the wakefield driven by an electron bunch with parameters ofγ = 71.4≡γ_i,σ_r = 1.51µm,σ_x = 0.39µm, total charge of 40 pC. (a) illustrates the color-coded denotation of the selected peaks/troughs on a longitudinal electric field of the wakefield at an arbitrary time.

The longitudinal position of the electron driver is located around x=1300. The corresponding phase velocity as a function of time of the selected phases with the same color-coding is plotted in (b), and the four areas with different colors correspond to four steps of evolution described in section 2.2.3.

σ_x = 0.14k_p⁻¹ = 0.39 µm, total charge of 40 pC and a constant plasma density ne= 3.6×10¹⁸ cm⁻³. From now on, the results of this section are based on these parameters implicitly if not otherwise stated. To look for injection conditions, we focus on the local phase velocity of specific phases of the wakefield as indicated in Fig. 2.12. The local phase velocity β is defined as ∆xt/c∆t, where ∆xt is the difference of the position of a specific phase of the plasma wave between time step t and t+ ∆t, and ∆t is the size of one time step in the simulation. As seen in Fig. 2.12(b), the phase velocity is not uniformly distributed across the wake and goes higher or lower than c. The mechanism behind this effect is explained by the progression of the longitudinal electric field and density distribution of the elec-tron bunch as what are shown in Fig. 2.13. The phase velocity evolves in following steps:

1. From T=34 to T≈140: Initially, the bunch has medium strength to drive a slightly nonlinear plasma wave. This is accompanied by an increase in the driving strength due to bunch self-focusing, and the wakefield becomes more and more nonlinear. Similar to LWFA, this effect is attributed to the relativistic mass increase of plasma electrons and causes a downshift in the

44

2. The Physics of Propagation of Ultrashort Electron Bunches in Underdense Plasma

Figure 2.13: Lineouts of the longitudinal electric field (dashed green line, in units ofE₀) and the density profile of the electron bunch (solid blue line, a.u.) from the middle of simulation box. The plasma background is stepwise distributed, and the bunch encounters plasma atT ≈34.

2.2 Propagation of Electron Bunches Driven by LWFA 45

Figure 2.14: Tracks of selected particles which are color coded by energy. The particles are homogeneously distributed within the middle of the cross section of the bunch which are the same as what is indicated as crosses in Fig. 2.9. It is well visible that dominantly low energy electrons are scattered out.

oscillation frequency and increase in the corresponding plasma wavelength [Rosenzweig, 1987]. The phase velocity of the first trough (color coded with green in Fig. 2.12) is much slower than β_i, which is the corresponding nor-malized velocity of γ_i which is the initial energy of the driver. However, since no significant amount of particles experience strong deceleration, there are few particles trapped at this stage. On other hand, the increase in ex-pelling force pushes the first peak forward, i.e. the lead of wakefield (blue in Fig. 2.12), and results in a phase velocity greater than c. The process approaches a quasi-steady wakefield at T≈120, where the magnitude of the longitudinal electric field also reaches its maximum.

2. From T≈150 to T≈300: The bunch starts to collapse due the low energy electrons moving backwards relative to the average position of the driving bunch, hence the whole wake is slowed down, as seen from T=145 to 177 in Fig. 2.12(b). Shortly after this intermediate step, the significant flux of electrons are decelerated to reach the backside of the first plasma period and built up the second bunch, which further pushes the second period of wakefield moving backward. This effect causes a feedback to additionally reduce the velocity and trap more and more electrons.

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2. The Physics of Propagation of Ultrashort Electron Bunches in Underdense Plasma

3. From T≈300 to T≈400: The increasing amount of charge in the second bunch neutralizes the original electric field driven by the first bunch and starts to drive its own wakefield. Because of the relatively low charge inside second bunch and the interference with the field driven by first bunch, the wakefield becomes more linear. Together with the fact that the distance between first peak and the center of mass of the second bunch is shorter than the first period of the wake, the second period moves forward. This is noticed in Fig. 2.12(b) where the velocity of the last three denoted positions are all increased until T≈350. The movement of the second period, however, shifts the second bunch into the defocusing and deceleration field, and the composite are scattered away, as observed in Fig. 2.14.

4. From T≈400 to T=∞: The charge loss reduces the driving power of the second bunch, and the dominant contribution of the driving force is gradually shifted to the first bunch again, also the electron scattering rate decreases because of weakening of the defocusing field. Therefore, the second period of plasma wave experiences a reduction of the phase velocity until T≈650.

From this point, however, the wakefield becomes even weaker due to bunch elongation. Because of the resulting decrease in the electron flux from the first bunch to the second bunch, the charge in the acceleration phase is not enough to load the wakefield. The phase velocity is then determined by the remaining high energy electrons, no significant trapping happens again and considerable amount of the low energy electrons fall behind the deceleration phase which are scattered by the defocusing field and get lost eventually (see Fig. 2.14).