Low energy electron bunch as a seed

Figure 3.14: (a) Longitudinal profiles of seed wakefields atk_βz≈2.5×10⁻²(z≈10.6 cm). (b) Radial profiles of the plasma electron number densities and (c) seed wakefields at [k_βz≈2.5×10⁻², (proton seed) kpeζ ≈ −11.6 and (electron seed) kpeζ ≈ −8.5]. (d) Radial seed wakefields averaged on the proton bunch Gaussian radial profile weighted with r ink_βz. In our analytical approach, the curve of proton seed (orange line) in Fig. (d) is handled as it is, which the averaged wakefield increases as cosin hyperbolic function. The curve of electron seed (blue line) in Fig. (d) is assumed to be constant within the short propagation distance. Both approaches for estimating the bunch modulation growth rates show nice agreement with PIC simulation results in Figs. 3.13 and 3.15.

work. We note that the case whichα andβ are 1 corresponds to the immediate onset of two modes.

case (orange solid curve). Instead of dealing with the unknown bunch profile at equilibrium, one can introduce a radial profile of the perturbed plasma electrons behind the low current electron driver, which is empirically found in PIC simulations, i.e.,

feq(r) =exp

−2r rB

. (3.30)

Herer_Bis known as the "blow-out radius" of plasma electrons defined by [36]

rB=2 n_s0

npe

1/2

r_s0. (3.31)

By the previous treatment of Bessel functions, the radial wakefield driven by a short and low current electron bunch at the radial equilibrium is [dashed curve in Fig. 3.14(c)]

(Er−cB_θ)_s,eq≈E₀Z(ζ)k_per²_B 4r

1−exp

−2r rB

1+2r

rB

. (3.32)

The envelope equation of the long proton bunch behind the tightly-focused low energy electron bunch is d²r_p

dz² − ε_n,²_p

γ²_pr³_p = e γpmpc²rp

[⟨r(E_r−cB_θ)_s,eq⟩+⟨r(E_r−cB_θ)m⟩]. (3.33) Since⟨r(E_r−cB_θ)s,eq⟩in Eq. (3.33) is not integrable in the integration limit 0≤r<∞with the proton bunch Gaussian radial profile, we set the integration upper limit as√

3rpand expand the Gaussian near axis up tor⁶order. Then, considering the low current seed bunch, i.e.,r_B≪r_p,

⟨r(E_r−cB_θ)_s,eq⟩= R_∞

0 r²e⁻

r2 2r2

p(Er−cB_θ)_s,eqdr R_∞

0 re⁻

r2 2r2

pdr

≈ 1 r²_p

k_per_B²

4 E₀Z(ζ) Z

√3rp

0

drr

1− r² 2r²_p+ r⁴

8r⁴_p− r⁶ 48r⁶_p

1−exp

−2r rB

1+2r

rB

≈ 1 r²_p

k_per_B²

4 E₀Z(ζ)186 256r²_p.

(3.34) Therefore, the purely seeded modulation of a long proton bunch at the seed bunch radial equilibrium is approximately described by

d²rp

dz² ≈ 93

256 npe

np

Z(ζ) k²_βr²_B rp

!

. (3.35)

At the phase of the proton bunch focusing, i.e., cos(kpeζ+φs+π/2)<0, definingA≡(93/256)(npe/np)

|Z(ζ)|k²

βr_B², we integrate the equation inzspace.

Z z^′=z z^′=0

dr_p dz^′

d²r_p

dz^′2dz^′=−A Z z^′=z

z^′=0

dr_p dz^′

1 rp

dz^′,

1 2

drp

dz^′

2z^′=z z^′=0

=−Ah lnrp

iz^′=z z^′=0.

Considering the initial conditions(drp/dz)z=0=0 andrp(z=0) =r_p0, drp

dz 2

=−2A(lnrp−lnr_p0).

Again integrating the equation inzspace with the initial conditionrp(z=0) =r_p0, drp

dz =±(2Alnr_p0−2Alnrp)^1/2,

Z dz

(2Alnr_p0−2Alnrp)^1/2 drp

dz =± Z

dz,

±z= 1 (2A)^1/2

Z dr_p

(lnr_p0−lnr_p)^1/2 =− r π

2Ar_p0erfp

lnr_p0−rp

,

lnr_p=lnr_p0−erf⁻¹ ∓ 1 r_p0

r2A π z

!2

.

Finally, organizing the equation aboutr_p, the resultant equation is rp=r_p0exp

−erf⁻¹ ∓ 1 r_p0

r2A π z

!2 .

In order to obtain a monotonically increasing modulation amplitude, we select + sign.

rpF =r_p0exp

"

−erf⁻¹

93 128π

n_pe np

|Z(ζ)|

1/2 rB

r_p0

k_βz 2#

. (3.36)

Using an approximation withinz<1 as below, exp(−erf⁻¹(z)²) =exp

− π

4z²+π²

24z⁴+13π³

1440z⁶+O(z⁸)

=1−π

4z²−π²

96z⁴− 7π³

5760z⁶−O(z⁸)

(3.37)

and

2−cosh rπ

2z

=1−π

4z²−π²

96z⁴− π³

5760z⁶−O(z⁸). (3.38) Therefore,

exp(−erf⁻¹(z)²) =2−cosh rπ

2z

+O(z⁶), (3.39)

which has the error in O(z⁶), we obtain the purely seeded solution in the form of hyperbolic cosine function as we did with the radially-matched high-energy proton seed, i.e.,

rpF ≈r_p0[2−cosh{1.2A_s,eqk_βz}], (3.40) whereA_s,eq≡[(I_s0/I_p0)|Z(ζ)|]^1/2with the seed and proton bunch peak currentsI_s0andI_p0. Since except the zeros order, the mathematical form (hyperbolic cosine) of Eq. (3.40) is same as Eq. (3.12), we simply

replaceR_c,Gin Eq. (3.28) withR_c,eq≡1.2As,eq(k_βz/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+r_p0

∑

ℓ=1

exp

−iπ 6

1.2A_s,eq

k_βz 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−r_p0≈ √

3 8π

1/2

e^N

√ N

"

δrcos

−5π

12−kpeζ+φs− N

√ 3

+r_p0

∑

ℓ=1

A_s,eq

k_βz k_peΦ(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 withm_e, 35.2, and−. The normalized asymptotic solution of the modulation amplitude is redefiend by ˆr/r_p0≡ αrˆ₀+βrˆ_s,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[βrˆ_s,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 atI_s0/I_p0≥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.