replaceRc,Gin Eq. (3.28) withRc,eq≡1.2As,eq(kβz/kpeΦ(ζ))1/3. The resultant modulation amplitude and the phase seeded by the tightly-focused low-energy electron bunch is
ˆ r≈
√ 3 8π
1/2"
δr+rp0
∑
ℓ=1
exp
−iπ 6
1.2As,eq
kβz kpeΦ(ζ)
1/32ℓ#
×exp
N+i N
√
3+i5π 12
N−1/2.
(3.41)
The dominance of seed over the long proton bunch modulation depends on the seed bunch current, not on its density. We note that the low energy electron seed at the radial equilibrium leads to largerRcthan the one from the high energy proton seed for the same bunch charge and current. The real part of the asymptotic solution is
rp−rp0≈ √
3 8π
1/2
eN
√ N
"
δrcos
−5π
12−kpeζ+φs− N
√ 3
+rp0
∑
ℓ=1
As,eq
kβz kpeΦ(ζ)
1/3
cos
−5π
12 −kpeζ+φs− N
√ 3+πℓ
3
2ℓ# .
(3.42)
For the simulation of a long proton bunch envelope modulation seeded by a tightly-focused low-energy electron bunch at seed bunch radial equilibrium in an over-dense plasma, we simply replace the massms, relativistic gammaγs, and the sign of charge of the seed bunch from the previous parameter set withme, 35.2, and−. The normalized asymptotic solution of the modulation amplitude is redefiend by ˆr/rp0≡ αrˆ0+βrˆs,eq. The recalculatedαandβ, which are mode coefficients relevant for the onset timings of two modes of long proton bunch modulation, in Figs. 3.15(a) and 3.15(b) show thatα (onset timing of self- modulation) is negligible (late) andβ (onset timing of externally seeded modulation) is approximately constant (uniform) between the bunch front [(a)kpeζ ∼ −226] and the bunch longitudinal center [(b) kpeζ ∼ −452], i.e., the long proton bunch modulation is seeded as a single mode simultaneous along ζ. Now the phase behavior of the long proton bunch modulation is simply explained by ∆arg[βrˆs,eq].
The PIC simulation result [dotted curve in Fig. 3.15(c)] of the modulation phase shift shows agreement with analytical expectation [solid curve in Fig. 3.15(c)] better than the case of the high energy proton seed, without the anomalous phase shift shown in Fig. 3.15(c). Figure 3.15(d) shows that when the long proton bunch modulation is seeded by the low energy electron bunch atIs0/Ip0≥0.75, only the externally seeded modulation survives with the simultaneous onset at the front and the center of the longitudinal Gaussian profile [β(ζ)≈constant]. The preceding low energy electron bunch seeds the long proton bunch modulation with the uniform onset alongζ in over-dense plasma.
Figure 3.15: Amplitudes of the proton bunch envelope modulation seeded by the tightly-focused low- energy electron bunch forIso/Ipo=0.75 inkβzat (a) the long proton bunch longitudinal center (kpeζ ∼
−452) and at (b) 2Lp ahead of the longitudinal center (kpeζ ∼ −226), respectively. (c) Phase shift of the proton bunch modulation for Is0/Ip0=0.75 at kpeζ ∼ −452. ∆arg(.) is the phase shift of the normalized asymptotic solution in the complex plane. (d) Onset coefficientsα andβ atkpeζ ∼ −452 andkpeζ∼ −226 inIs0/Ip0. Red dotted vertical line indicatesIs0/Ip0=0.75. Onset coefficientsα(self- modulation) andβ(externally seeded modulation) are found by fitting the curve of modulation amplitude from PIC simulation to the analytical expectation of Eq. (3.41). Onset coefficients are relevant for the mode of the growth rate and the onset timing of the bunch modulation. WhenIs0/Ip0≥0.75, the growth rate is described by a single modeβ at bothkpeζ ∼ −452 and 2Lpkpeζ∼ −226 and the onset happens simultaneously, i.e.,β(kpeζ ∼ −452)≈β(kpeζ ∼ −226). The anomalous phase shift in Fig. 3.13(c) disappears and the simulation result of the phase behavior shows nice agreement with the analytical expectation of Eq. (3.41).
linear wakefields in over-dense plasma. We found that with this low energy bunch (<20 MeV), seed wakefields are driven over less than 3 m. The phase evolution of seed wakefields in the proton bunch frame is mostly affected by the initial seed beam energy. The phase of SSM observed near the peak of the proton bunch, where we expect wakefields to reach their maximum amplitude, are also mostly affected by the initial seed beam energy. The other parameter variations, that could reflect variations in experiments, such as charge and length, did not significantly contribute to variations in phase of the SSM.
The energy spectra of the electron bunches after passing 10 m plasma were experimentally measured and compared with PIC simulations results. However, the energy losses of the electron bunches from cylindrical PIC simulations were found larger than those from the experimental results. In addition to the nature of axisymmetric PIC simulation, in which particles only move radially and longitudinally, also the unknown experimental parameters made the analysis difficult.
Also, we obtained the asymptotic solutions of the long Gaussian proton bunch modulation seeded by a preceding charged particle bunch in over-dense plasma and classified the regimes of the modulation by introducing two types of initial conditions such as initial modulation and external seed. The external seed wakefield contributes the phase shift of the modulation in the opposite direction of the modulation feedback. When the regimes of the modulation are polarized at the front and the center of the Gaussian longitudinal profile, the accumulated wakefield from the bunch front to the center interferes with the early grown wakefield at the bunch longitudinal center, leading to the anomalous behavior of the phase shift. The tightly focused electron seed easily drives a single mode modulation, suppressing the mode polarization, which can be critical when introducing plasma density step or injecting witness bunch into accelerating wakefield region.