Figure 5.5: [(a)-(f)] The longitudinal phase spaces of the witness bunch with different ionization phases for ∂kpξEz,i/E0 ∼0.34 andct =10 mm. (g: blue) The characteristic distance from the ionization to trapping positions and (g: red) the rms length of the witness bunch, according to the different phases of the ionization laser pulsekpξion the quasistatic wake potential. Here, the peak position of the wake potential iskpξ0=−2.8, which corresponds to∼160 fs behind the driver beam centroid (ξ=0).
Figure 5.6: Normalized quantities: driver (orange solid curve) and witness (dot and dashed orange curve) bunch currents, longitudinal wakefield (red dot and dashed curve), laser electric field (red solid curve), and normalized wakepotential (green solid curve).
ξ3−ξ4≡σz is the trapped rms length of the witness bunch. Reminding that ∆φ=1 is the trapping condition and neglecting any other effects,
φ4−φ2=φ3−φ1 (5.11)
Using the relationship between the wake potential and the phase shift in the bubble,
(ξ4−ξ3)(ξ4+ξ3)−2(ξ4−ξ3)ξ0= (ξ2−ξ1)(ξ2+ξ1)−2(ξ2−ξ1)ξ0 (5.12)
σz=2ξ0−(ξ2+ξ1)
2ξ0−(ξ4+ξ3)σψLi, (5.13)
whereσz≡ξ3−ξ4andσψLi≡ξ1−ξ2. All the comoving variablesξ here have negative values. When
⟨ξi⟩<ξ0, assumingξ1=⟨ξi⟩,ξ2=⟨ξi⟩ −σψLi,ξ3=⟨ξt⟩, andξ4=⟨ξt⟩ −σz, σz=⟨ξi⟩ −ξ0−σψLi/2
⟨ξt⟩ −ξ0−σz/2 σψLi≈ ⟨ξi⟩ −ξ0−σψLi/2
⟨ξi⟩ −ξ0−σz/2−∆ξσψLi
≈ −[⟨ξi⟩ −ξ0−σψLi/2]σψLi/∆ξ.
(5.14)
When⟨ξi⟩>ξ0, assumingξ1=⟨ξi⟩,ξ2=⟨ξi⟩+σψLi,ξ3=⟨ξt⟩, andξ4=⟨ξt⟩+σz, σz=⟨ξi⟩ −ξ0+σψLi/2
⟨ξt⟩ −ξ0+σz/2 σψLi≈ ⟨ξi⟩ −ξ0+σψLi/2
⟨ξi⟩ −ξ0+σz/2−∆ξσψLi
≈ −[⟨ξi⟩ −ξ0+σψLi/2]σψLi/∆ξ.
(5.15)
For Eqs. (5.18) and (5.19), ⟨ξt⟩=⟨ξi⟩ − |∆ξ|is used. Combining two cases, the rms length of the witness bunch is given in the form,
σz≈
|⟨ξi⟩ −ξ0|+σψLi
1+⟨ξi⟩ −ξ0 σψLi
, (5.16)
where Li is the rms length of the ionization laser pulse. It is assumed that the intrinsic transverse momentum induced in the laser polarization direction is negligible in Eq. (5.16), because its effect in the trapping process is inversely proportional to the wake phase velocity gammaγpwhich is close to the driver beam gammaγd [69]. Equation (5.16) confirms that the rms length of the witness bunch is minimized when the ionization laser pulse is set to the peak of the wake potential (see Fig. 5.5). We note, however, that Eq. (5.16) is not perfectly matched to the simulation results quantitatively. It is because for the majority of cases in the quasistatic approximation, the witness electrons reach the nonlinear curve ofEz at the rear part of the ion cavity, particularly when the ionization laser is set far fromkpξ0. This shortens the rms length of the witness bunch from the analytical expectation of Eq. (5.16).
Assuming that the phase slippage of the witness bunch on the wake potential is negligible after the trapping process is done, the energy spread of the witness bunch can be estimated by the combination of three effects: the intrinsic energy spread from the ionization time interval, work done by the wakefield slope, and longitudinal space-charge field of the witness bunch. The energy spread from the wakefield slope during the acceleration process is given byσE,a≈eσz(z−z0)∂ξEz,t, wherez0=zf+p
π/2σψZR
is the virtual point in which the witness bunch charge is accumulated up to almost N. Here, N ≈ π(σψwi)2(√
2π σψZR)nHIT is the maximum number of the ionized witness electrons [57]. The energy spreadσE,sfrom the longitudinal space charge field is approximately given by [70]
σE,s≈e Z z
z0
Ez,sdz′ (5.17)
with
Ez,s≈ N 4π ε0
e
γ2σz2logγ σz
σr
, (5.18)
whereγ=γi+eEz,t(z−z0)/mec2is the relativistic gamma of the accelerating witness bunch,γithe initial gamma of the witness bunch atz=z0, andσrthe witness bunch radius. By the initial matching condition kpwi∼2ai [57], the witness bunch radius in the laser polarization direction is approximately matched toσr=q
εr/γkβ with the betatron wave numberkβ =kp/√
2γ. Since the contribution ofσzσr−1 term becomes quickly small compared to γ, it can be assumed that the possible mismatching effect of the witness bunch radius is not significant for the relative energy spread [see Eq. (5.19)]. The average energy of the witness bunch is⟨E⟩=γimec2+eEz,t(z−z0)≈eEz,t(z−z0), where it was assumed that the initial witness bunch energy is negligible.
Reminding that the intrinsic energy spread σE,i determines the size of the semi-minor axis of the longitudinal phase space ellipse, and the forces acting on the witness bunch in the opposite directions affect the tilt angle of the semi-major axis, we suggest the following expression for the relative energy
Figure 5.7: The average energy and energy spread of the witness bunch forkpξi=−3.3. (blue dashed line) The asymptotic behavior of the relative energy spread. (blue solid curve) The analytic expression (10) for the relative energy spread. (blue circles) The PIC simulation result of the relative energy spread.
(red circles) The PIC simulation result of the average energy.
spread of the witness bunch:
σE
⟨E⟩≈ σE,i+|σE,a+σE,s|
⟨E⟩
≈ σψZR
z−z0+
1 Ez,t
∂Ez,t
∂ ξ σz− N 4π ε0
e2 σz
(eEz,tσz)−1
×
log(γ σzσr−1) +1.25
γ2 −log(γiσzσr,i−1) +1.25 γ γi
,
(5.19)
whereσE is the total rms energy spread of the witness bunch andσr,i the initial witness bunch radius.
It is assumed that any wakefield effect driven by the witness bunch itself is negligible in the present TH injection regime. For the case in which the ionization laser pulse is synchronized near the peak position of the wake potential, Ez,t and∂ξEz,t are specified by E0 andkpE0/2, respectively. We note thatσE,i and σE,s of Eq. (5.19) are comparable toσE,a atct =10 mm in Fig. 5.5. Due to the high bunch density, Figs. 5.5(a) to 5.5(f) illustrate that the energy chirp of the witness bunch is space-charge field dominated when the ionization laser is synchronized near the peak position of the wake potential atkpξ0=−2.8, whereas it is plasma-wakefield dominated when the ionization laser is synchronized far fromkpξ0. Equation (5.19) impliesσE/⟨E⟩∝n0atz→∞, where the effects ofσE,i andσE,s vanish.
We also note that there is a ultrahigh density limit of witness bunch which would not follow Eq. (5.19).
Energy modulation which is seeded during the trapping process has been found in ultrahigh density witness bunch cases. A partly similar simulation result as in Fig. 5.7 has been reported in Ref. [26]. In Ref. [26], a high-charge escort bunch was additionally injected to distort the wakefield gradient. In this work, instead, we tried to figure out the minimization condition of the rms length and considered the self-field effect [53] of the witness bunch. The simulation result of the relative energy spread in Fig. 5.7 shows a reasonable agreement with Eq. (5.19) (compare blue circles and blue dashed line in Fig. 5.7).