Figure 3.9: Phases of seeded self-modulation at the longitudinal centroid of long proton bunch by scan- ning seed beam initial energy (the second row), charge (the third row), and length (the fourth row). The reference seed bunch lengthσz,re f/c=2 ps. The seed electron beam is initially centered atkpeξ≈ −4.7.
Plasma density: n0=2×1014cm−3. Figures show the phase of the long proton bunch modulation is mostly affected by the seed bunch energy.
"No seed" case shows that electron bunch seeding does occur though, since its phase is different from those with electron bunch. The bottom figure shows that the phase is quite insensitive to the seed bunch length. Even though the length of seed bunch changes with propagation distance with different initial lengths, the evolution of the phase of the wakefields caused by the evolution of the bunch length is not significant. Here, we note that transition between the seeded self-modulation (SSM) and the self- modulation instability (SMI) was not observed in low energy (5 MeV) and charge (50 pC) seed beam cases, which could be different in the experiment because the noise levels are different (much lower in simulations).
3.4 Beam-driven plasma wakefields and envelope modulation seeded by
Figure 3.10: Colinearly propagating charged particle bunches in over-dense plasma. A low energy electron bunch precedes the long proton bunch. The wakefield driven by the preceding electron bunch generates wakefield for seeding the modulation of the long proton bunch.
equation for the radial plasma wakefield driven by a charged particle bunch is [11]
Er−cBθ =∓k3peE0 Z ζ
∞
dζ′sin[kpe(ζ−ζ′)]
K1(kper) Z r
0
r′dr′I0(kper′)nˆbm(kper′,kpeζ′)/npe,0 +I1(kper)
Z ∞
r
r′dr′K0(kper′)nˆbm(kper′,kpeζ′)/npe,0
.
(3.1) where the∓is the sign of charge of the bunch particle,E0=kpemec2/ethe non-relativistic wavebreaking field in cold plasma, f∥(ζ′) the longitudinal profile, and f⊥(r′) the transverse profile of the bunch, respectively. The seed from a preceding, radially-matched proton bunch which has the same energy as the following long proton bunch has important analogies with that from a RIF, such as radial profile of the wakefield and time scale of the evolution. In order to analytically estimate the transverse wakefield driven by two Gaussian bunches, we use the approximations of Bessel functionsI0(kper′)≈1 atkper′<1 andK1(kper)≈1/kperatkper<1 for the first term and ignore the second term as in Refs. [10, 11]. This approximation is valid while most of bunch particles are withinkper<1. Since the rms length of the preceding bunch is considered to be short (Ls∼k−1pe), the entire bunch is within the focusing phase of its driven plasma wakefield. The transverse wakefield behind a short Gaussian bunch is
(Er−cBθ)s,G=∓ ns
npe,0
E0Z(ζ)kper2s r
1−exp
− r2 2r2s
, (3.2)
wherensis the number density andrsis the rms radius of the seed bunch. FunctionZ(ζ)is given by Z(ζ) =
Z ζ
d(kpeζ′)sin[kpe(ζ−ζ′)]exp
−(ζ′−ζs) 2L2
, (3.3)
whereζsis the position of seed bunch longitudinal center in the beam comoving frame. The equation is simplified by setting the normalized variables ˆζ′=kpeζ′, ˆζ=kpeζ, ˆζs=kpeζs, and ˆLs=kpeLsas below.
Z= Z ∞
ζˆ
dζˆ′sin(ζˆ′−ζˆs+ζˆs−ζˆ)exp
−(ζˆ′−ζˆs)2 2 ˆL2s
. (3.4)
We assume that ˆLs≪ζˆs−ζˆ. Introducing new notations ˆζ′1=ζˆ′−ζˆsand ˆζ1=ζˆ−ζˆs, we get Z=−
Z ∞
ζˆ1
dζˆ′1sin(ζˆ−ζˆ′1)exp
− ζˆ′21 2 ˆL2s
. (3.5)
Splitting the equation into two parts using the angle difference identity of sine function, Z=−sin ˆζ1
Z ∞
ζˆ1
dζˆ′1cos ˆζ′1 exp
−ζˆ′21 2 ˆL2s
+cos ˆζ1 Z ∞
ζˆ1
dζˆ′1sin ˆζ′1 exp
−ζˆ′21 2 ˆL2s
. (3.6) The lower limits in the integrals can be replace by−∞with small errors. In the case, the second integral vanishes, and we obtain
Z(ζ) =
√
2πkpeLsexp
−k2peL2s 2
cos
kpe(ζ−ζs) +π 2
. (3.7)
Here we define the seed bunch phase−kpeζs≡φs. Since the following proton bunch is long (Lp≫k−1pe), the radial wakefield alongkpeζ on the proton bunch alternatively focuses and defocuses the envelope.
Therefore, here we do not treat the longitudinal and transverse integrals separable [10, 30]. The trans- verse wakefield from the modulated proton bunch on any slice ofζ is
(Er−cBθ)m=− np
npe,0
E0k2pe Z ∞
ζ
dζ′sin[kpe(ζ−ζ′)]f∥(ζ′)r2p r
1−exp
− r2 2r2p
, (3.8)
wherenp is the number density andrp is the rms radius of the proton bunch. We assume that bunch energy change and coulomb scattering are negligible in the system, and envelope of the following proton bunch is affected mainly by its transverse momentum and beam-driven plasma wakefields [10, 33] as below.
d2rp
dz2 − εn,p2
γp2r3p = e γpmpc2rp
[⟨r(Er−cBθ)s,G⟩+⟨r(Er−cBθ)m⟩]
= e
γpmpc2rp
"
R∞
0 r2e−
r2 2r2
p(Er−cBθ)s,Gdr R∞
0 re−
r2 2r2
pdr
+ R∞
0 r2e−
r2 2r2
p(Er−cBθ)mdr R∞
0 re−
r2 2r2
pdr
# ,
(3.9)
wherempis the proton mass andεn,p is the normalized emittance of, andγp the relativistic gamma of the long Gaussian proton bunch. Therefore, with thin bunch approximation, the seeded modulation of the long Gaussian proton bunch in over-dense plasma is described by
d2rp dz2 − εn,2p
γp2r3p =∓2 ns
np
k2βZ(ζ) rprs2
r2p+r2s −kβ2kpe Z ∞
ζ
dζ′sin[kpe(ζ−ζ′)]f∥(ζ′)rp, (3.10)
Figure 3.11: The long proton bunch (top) envelope and (bottom) near axis current modulations seeded by the radially-matched high-energy proton seed. (top) Envelope of the long proton bunch at the Gaussian profile longitudinal center (left),σz,pahead from the center (middle), and 2σz,p ahead from the center (right). The envelope is normalized by the initial rms radiusσr,p=200µm. The modulation amplitude increases from the bunch front to the back due to the modulation feedback. (bottom) However, the cur- rent modulation near axis shows a complex behavior, which is from the difference of the seed wakefield dominance over the long proton bunch self-field. The seed wakefield is dominant at the bunch front.
The long proton bunch self-field is dominant at the bunch longitudinal center. In this case, the mode of the long proton bunch modulation is the function of the beam comoving frameζ=z−ct. The mode of the growth rate is polarized along the long proton bunch longitudinal profile.
wherekβ= (npe2/2γpmpε0c2)1/2is the proton bunch particle betatron wave number. At the early propa- gation distance in the plasma, we assume that the seed is dominant over the emittance and the modulation feedback. In particular, here we consider the situation which before the envelope modulation becomes significant, the magnetic field at the peak bunch-plasma current approximately balances the emittance, in our regime, εn,p∼0.4γpkβr2p0. If we consider a radially-matched high-energy proton seed beam (rp=rs), in the regime which the seed wakefield dominates over the following proton bunch self-filed, the equation for the purely seeded envelope modulation is
d2rp
dz2 ≈ −ns np
k2βZ(ζ)rp. (3.11)
At the phase of the long proton bunch defocusing, i.e., cos(kpeζ+φs+π/2)<0, the solution of the purely seeded modulation is
rpD=rp0cosh(As,Gkβz), (3.12)
whereAs,G≡[(ns/np)|Z(ζ)|]1/2. Once the modulation process starts, the amplitude of envelope modula- tion self-consistently grows with its driven wakefield. The contributions of emittance and seed wakefield on the envelope quickly becomes negligible during the modulation process withink z<1. Assuming
rp−rp0=rˆexp(ikpeζ)/2+c.c.and|∂kpeζr| ≪ |ˆ r|, we apply the plasma operatorˆ (∂k2
peζ+1) into Eq.
(3.10), and obtain the linearized equation of the modulation amplitude ˆr[11], i.e., (∂k2
peζ+1)∂z2rp=−k2βkpe(∂k2
peζ+1) Z ζ
∞
dζ′sin[kpe(ζ−ζ′)]f∥(ζ′)rp, (3.13) (left-hand side)
(∂k2
peζ+1)∂z2rp
= 1 2(∂k2
peζ+1)∂z2[ˆrexp(ikpeζ)]
= 1
2∂z2[exp(ikpeζ)∂k2
peζrˆ+2iexp(ikpeζ)∂kpeζr]ˆ
≈iexp(ikpeζ)∂kpeζ∂z2r,ˆ
(3.14)
(right-hand side)
−k2βkpe(∂k2
peζ+1) Z ζ
∞
dζ′sin[kpe(ζ−ζ′)]f∥(ζ′)rp
=−k2
β
2 (∂k2
peζ+1) Z ζ
∞
d(kpeζ′)sin[kpe(ζ−ζ′)]f∥(ζ′)[ˆrexp(ikpeζ)]
=−k2β
2 [−ˆrexp(ikpeζ)f∥(ζ)],
(3.15)
and
∴
∂ζ∂z2+i
2k2βkpef∥(ζ)
ˆ
r=0, (3.16)
where the Leibniz integral rule was applied for estimating the derivation of the integral in the right- hand side of the equation. A factor of 1/2 at the second term is from the estimation of the proton bunch Gaussian radial profile, which can be modified for any different radial profile. We Laplace transform Eq.
(3.16) in z to p space using ˆr(z=0,ζ) =δr,∂zr(zˆ =0,ζ) =0, andLz[∂z2r] =ˆ p2Lz[ˆr]−pr(zˆ =0,ζ)as below.
∂ζLz[ˆr(z,ζ)] + i 2
k2
βkpe
p2 f∥Lz[ˆr(z,ζ)] =0. (3.17) We integrate the equation in ζ space, introducing the initial modulation (often considered as bunch noise) and the external seed, i.e.,
ˆ
r(z,ζ=0) =Θ(z)
δr+rp0
∑
ℓ=1
(As,Gkβz)2ℓ (2ℓ)!
, (3.18)
whereΘ(z)is the heaviside step function. The equation organized about the Laplace transformed mod- ulation amplitude is
Lz[ˆr(z,ζ)] = δr
p +rp0
∑
ℓ=1
A2ℓs,Gk2ℓβ p2ℓ+1
exp
i 2
kβ2kpeΦ(ζ) p2
(3.19) withΦ(ζ) =R∞ζ f∥(ζ′)|dζ′|. Bromwich integral [34] of Eq. (3.19) is
ˆ r= 1
2πi Z Γ+i∞
Γ−i∞ exp(pz) δr
p +rp0
∞
∑
ℓ=1
A2ℓs,Gkβ2ℓ p2ℓ+1
exp
i 2
k2βkpeΦ(ζ) p2
d p. (3.20)
By the method of steepest descent [35], the integrated phase locked asymptotic solution is ˆ
r= 1
2πig(p0)exp(ω0)exp(iθ) s
2π
|ω0′′|, (3.21)
whereg(p0)is the factor of the seed contributions with p0the saddle point of the Laplace transformed modulation amplitude,ω0the exponent of the integrand at the saddle point,θ the direction of steepest descent (or ascent), andω0′′the second derivative ofω0in p space, i.e.,
p0= ikβ2kpeΦ(ζ)z−113
, (3.22)
g(p0) = δr p0+rp0
∞
∑
ℓ=1
(As,Gkβ)2ℓ
p2ℓ+10 = δr ik2
βkpeΦ(ζ)z−113 +rp0
∞
∑
ℓ=1
(As,Gkβ)2ℓ ik2βkpeΦ(ζ)z−12ℓ+13
=δri−13332 4 z
332
4 k2βz2kpeΦ(ζ)13 −1
+rp0
∞
∑
ℓ=1
(As,Gkβ)2ℓe−2ℓ+16 iπ 332
4 2ℓ+1
z2ℓ+1 332
4 k2βz2kpeΦ(ζ)13 −2ℓ−1
,
(3.23)
N≡332
4 k2βz2kpeΦ(ζ)13
, (3.24)
ω0=N+i N
√
3, (3.25)
θ= 7
12π, (3.26)
and s
2π
|ω0′′|=2 2π3−5/2Nz−212
. (3.27)
Here we assumed that the effects of the initial modulation and the external seed are small compared with the modulation feedback. The resultant modulation amplitude and the phase in(kpeζ,kβz)space is
ˆ r≈
√ 3 8π
1/2"
δr+rp0
∑
ℓ=1
exp
−iπ 6
As,G
kβz kpeΦ(ζ)
1/32ℓ#
×exp
N+i N
√
3+i5π 12
N−1/2
(3.28)
withN= (33/2/4)(k2βz2kpeΦ(ζ))1/3. Withinkβz<1, as the modulation amplitudes of the long proton bunch (or the exponent N) increase, the phase of the resultant wakefield moves back inkpeζ [9, 10].
The phase shift from the modulation feedback rotates the position of ˆrin counterclockwise direction in the complex plane. On the other hand, in the physical system, the seed wakefield keeps the modulation phase fixed in the seed dominant region. The series summation in Eq. (3.28) draws a clockwise spiral curve in the complex plane. Since in the regime of interest, the radius of convergence with Gaussian seed bunchRc,G≡As,G(kβz/kpeΦ(ζ))1/3 in Eq. (3.28) is less than unity [see Fig. 3.12(a)], the series converges. When the seed contribution to the phase shift is larger than that of the modulation feedback,
Figure 3.12: (a) Colormap ofRc,G(radius of convergence with the Gaussian proton seed) in(kpeζ,kβz) space forns/np=3/4. Equations (3.28) and (3.29) converge only ifRc,G<1. IfRc,G≥1, the bunch modulation feedback is negligible and the purely seeded modulation exactly describes the long proton bunch modulation. (b) Proton seed and the following long proton bunch currents inkpeζ. Two black dashed-vertical lines atkpeζ =kpeζ0∼ −452 andkpeζ =kpe(ζ0+2Lp)∼ −226, respectively.
i.e., the seed contribution is dominant over the modulation feedback, Eq. (3.28) is unphysical and the amplitude of the purely seeded modulation is valid for describing the envelope. We note that by selecting a specific order of seed (ℓ), this series solution is partly reduced to the ones in Refs. [10, 11]. 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,G
kβz kpeΦ(ζ)
1/3
cos
−5π
12 −kpeζ+φs− N
√ 3+πℓ
3
2ℓ# .
(3.29)
For the simulation of a long proton bunch envelope modulation seeded by a radially-matched high- energy proton seed in an over-dense plasma, the physical and numerical parameters are set as follows.
The ambient plasma electron number density is npe =1.0×1014 cm−3. The Gaussian proton seed bunch initial parameters are: number density ns=1.8×1012cm−3 (Qs≈150 pC), mean relativistic gamma⟨γs⟩=426, bunch particle massms=mp, rms lengthLs=1.4/kpe≈744µm, rms radiusrs= 0.4/kpe≈213µm, and normalized transverse emittanceεn,s=1µm. The Gaussian long proton bunch initial parameters are: number density np=2.4×1012cm−3 (Qp≈16 nC), mean relativistic gamma
⟨γp⟩=426, rms length Lp=6 cm≈113/kpe, rms radius rp =0.4/kpe ≈213 µm, and normalized transverse emittanceεn,p=0.4γpkβr2p0≈1.8µm. The simulation is 2D axisymmetric and the resolution is set to∆z=0.02/kpe,∆r=0.005/kpe, and∆t=∆z/cin the laboratory frame. In order to suppress the effect of the hard-cut proton bunch at the front of the simulation window, the length of the simulation window is set toLw=4Lp. Only the front half of the long Gaussian proton bunch is simulated. The radius of the simulation window is set toRw=1.1 mm≈1.88/kpe, which is enough for the early stage
Figure 3.13: Amplitudes of the proton bunch envelope modulation seeded by the radially-matched high- energy proton bunch forns/np=0.75 inkβz at (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 ns/np=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 inns/np. Red dotted vertical line indicatesns/np=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. (3.28). Onset coefficients are relevant for the mode of the growth rate and the onset timing of the bunch modulation. At the long proton bunch longitudinal center (kpeζ ∼ −452), the modes are mixed, e.g., bothαandβ are not zero. With the high density proton seed bunch (ns/np=2), the self-modulation growth becomes negligible atkpeζ∼ −452.
However, the value ofα is still smaller than that atkpeζ∼ −226, i.e., the onset happens later. This mode polarization leads to an anomalous phase shift in Fig. (c).
of the long proton bunch modulation. Each simulation particle of the seed and the long proton bunches represents 1000 physical particles.
The amplitudes of the proton bunch envelope early (kβz<1) modulation seeded by the radially- matched, high-energy proton bunch are calculated as 0.5(rp,max−rp,min)/rp0, which is equivalent to
|ˆr/rp0|for PIC simulation result [dotted curves in Fig. 3.13(a)-(b)] late along the bunchkpeζ=kpeζ0∼
−452 and early along the bunchkpeζ=kpe(ζ0+2Lp)∼ −226 [indicated by black dashed vertical lines in Fig. 3.12(b)], where ζ0 is the position of the long proton bunch longitudinal center. For kβz<
0.2, the self-modulation process is not on the point, where the bunch noise from the finite number of beam simulation particles is responsible for initial modulation amplitude δr. In order to compare the prediction of Eq. (3.28) with PIC simulation results, we set ˆr/rp0≡αrˆ0+βrˆs,G[black solid curves in Figs. 3.13(a)-(b)]. Here ˆr0and ˆrs,Gare the normalized asymptotic solutions of the modulation amplitude from the initial modulation and the external seed of Eq. (3.28), respectively andα andβ are additional coefficients which are related to onset timing of each process and are determined while simulation data is fitted to Eq. (3.28). Atkpeζ ∼ −452 [Fig. 3.13(a)], the amplitude of the current induced focusing field is larger than that of the radial seed wakefield. In this case (ns/np=0.75),α(≈0.5)≫β(≈0.12) and the self-modulation ˆr0 grows faster than the externally seeded modulation ˆrs,G. The growth rate is the mixture of two modes ( ˆr0 and ˆrs,G). On the other hand, atkpeζ ∼ −226 [Fig. 3.13(b)], the long proton bunch self-field is negligible compared to the seed wakefield and only the externally seeded mode (ˆrs,G) survives. Therefore, we note that with the radially-matched high-energy proton seed, the onset coefficients (α andβ) are different at the front and the longitudinal center of the long proton bunch, i.e., mode of the long proton bunch modulation is polarized alongζ. Since the wakefield phase shift results from the shift of the modulated bunch density peaks [10], the phase fixed seed wakefield contributes the phase shift in opposite direction of the modulation feedback. As expected, analytical estimation of the phase shift with the external seed [black solid curve in Fig. 3.13(c)] is smaller than that without the external seed [black dashed curve in Fig. 3.13(c)]. We also note that the PIC simulation result (dotted curve) of the phase shift, which is the phase trace of the longitudinal wakefield zeros, shows an anomalous behavior. The PIC simulation result implies that when the modes of the modulation are polarized at the front and the center of the Gaussian longitudinal profile, the phase shift is not simply explained by Eq. (3.28) and the wakefield accumulated from the bunch front to the center interferes with the early grown wakefield at the bunch longitudinal center. Figure 3.13(d) showsαandβdependence on ns/npatkpeζ∼ −452 andkpeζ∼ −226. Red dotted vertical line indicates the case of Figs. 3.13[(a)-(c)].
The maximum value of the self-modulation onset coefficientα is found atns/np=1 for this radially- matched, high-energy proton seed, where focusing force of the linear plasma wakefield is maximized without defocusing force. α and β may depend on the magnitude of the long bunch noise, and the profiles and the amplitudes of the seed wakefields and the proton bunch self-fields, and therefore are functions ofζ. However, the rigorous analytical estimation of the numbers is outside the scope of this
Figure 3.14: (a) Longitudinal profiles of seed wakefields atkβz≈2.5×10−2(z≈10.6 cm). (b) Radial profiles of the plasma electron number densities and (c) seed wakefields at [kβz≈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 inkβz. 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.