2.2 General envelope equation of charged particle beam
2.2.4 The envelope equation and emittance
The envelope equation may now be derived. First, observe that Eq. (2.45) is integrable and yields the mean canonical angular momentum as a constant of integration:
P˙θ =0, (2.54)
where
Pθ =γL+γ ωc
R2
2 . (2.55)
Canonical angular momentum is not conserved for individual particles in the presence of scattering, but it is conserved in the average taken over many particles becauseδFis distributed isotropically. The virial equation (2.44) is used to eliminateV2from the energy equation (2.43) to get
γ˙ γ+1
2 d dt
γ˙ γ
d dt
R2 2 + d2
dt2 R2
2 +⟨ωβ2r2⟩ −ωcL
+
ωβ2 d dt
r2 2
+ 1
2γ d
dt(γ ωc)L= ε′
γm. (2.56) The terms involvingLmay be combined with the aid of Eqs. (2.54) and (2.55). They become γ˙
γ+1 2
d dt
(−ωcL)+ 1 2γ
d
dt(γ ωc)L=−ωc
2γ d
dt(γL) =−ωc
2γ d dt
Pθ−γ ωcR2 2
=ωc
4γ d
dt(γ ωcR2). (2.57) Substituting this expression into Eq. (2.56) along with the expressions (2.51) and (2.53) yields
γ˙ γ+1
2 d dt
γ˙ γ
d dt
R2 2 + d2
dt2 R2
2 +⟨ωβ2r2⟩
+ 1 R2
dR2
dt U+ωc
4γ d
dt(γ ωcR2) = ε′
γm. (2.58) An integrating factor (2γ2R2) has been found for the entire left-hand side of Eq. (2.58). Thus, we obtain
d dt
γ2R3R¨+γγR˙ 3R˙+γ2R2U+1
4γ2R4ωc2
=2γ2R2ε′
γm . (2.59)
Integrating once in time and dividing both sides byγ2R3gives the envelope equation, R¨+γ˙
γ R˙+U
R +ωc2R 4 − C2
γ2R3 = 1 γ2R3
Z t t0
dt′
2γR2ε′ m
t′
, (2.60)
whereC2is the constant of integration and to is the time of injection of the particular beam slice under consideration. The constant of integrationC2is immediately related to the conditions at injection by
C2=
γ2R3
R¨+γ˙ γ
R˙+U
R +ωc2R 4
t0
. (2.61)
The collection of terms on the right-hand side of Eq. (2.61), evaluated at any timet>t0, is itself constant in the absence of scattering. In the presence of scattering, it increases according to Eq. (2.59).
A more useful and physically revealing form of this expression is readily obtained. Using the virial equation (2.44) to eliminate the term proportional to ¨R, we find
γ2R3
R¨+γ˙ γ
R˙+U
R +ωc2R 4
=γ2R2
V2−(R)˙ 2+ωcL+ωc2R2 4
=γ2R2
V2−(R)˙ 2+ L
R 2
+Pθ2. (2.62)
The emittance (squared) is now defined as E2≡γ2R2
V2−(R)˙ 2− L
R
. (2.63)
E2is clearly a constant in the absence of scattering. SincePθ is always constant, Eq. (2.59) gives dE2
dt =2γR2ε′
m . (2.64)
The constantCis related toEby
C2=E2(t0) +Pθ2. (2.65)
Some insight into the nature of the emittance is gained by writing it in terms of the residual particle velocitiesδv. EliminatingV2from Eq. (2.63) with expression Eq. (2.51), we get
E2=γ2R2⟨|δv|2⟩. (2.66)
Thus,E2is proportional to the area of beam transverse phase space. This is in the area in (r⊥,P⊥) space occupied by the beam particles, with small-scale gaps and peaks of density smoothed over. As might be expected, the mean flow velocity, ˙Rr/R+Lrθ/Rˆ 2, makes no contribution to phase space area. Scattering increasesE2by adding the residual velocities without a compensating change in the particle positions.
We note that in Chapter 3, Eq. (2.60) is expressed as d2rp
dz2 − εn,p2
γ2pr3p = e γpmpc2rp
⟨r(Er−cBθ)⟩, (2.67)
where energy change (second term in left-hand side), externally applied solenoidal field (fourth term in lefth-hand side), and scattering (term in right-hand side) effects are ignored.
Chapter 3
Uniform Onset of the Long Proton Bunch Modulation Seeded by a Low Energy
Electron Bunch in an Over-Dense Plasma
3.1 Introduction
Over the last ten years, envelope modulation of a long, and relativistic charged particle bunch in over-dense plasma has been experimentally and analytically studied with the promise of using the SPS proton bunch at CERN for plasma-based acceleration of an electron bunch to very high energies [7, 9, 10, 15, 17, 30]. Since the optimum bunch length for generating beam-driven plasma wakefields is on the order of the plasma skindepth [31], the length of the proton bunch currently available (∼6 cm) is too long to effectively generate even low accelerating gradient plasma wakefields. However, the envelope of the long bunch can be modulated by the low amplitude wakefields it drives as self-modulation (SM) grows [30]. Self-modulation creates a train of micro-bunches with period of the plasma wavelength. The train resonantly drives plasma wakefields along the propagation axis, increasing the amplitude of the wakefields to values close to the cold non-relativistic plasma wave-breaking limit [30]. This amplitude can be several orders of magnitude greater than those of conventional accelerators.
It was demonstrated experimentally that the self-modulation process can be seeded with a relativistic ionization front (RIF) [14, 15]. Variations in seed amplitude were obtained by scanning the RIF timing along the proton bunch with Gaussian time profile. This method allows only the back half of the proton bunch to interact with the plasma, leaving the front half un-modulated. This becomes an issue when introducing an accelerator plasma after the self-modulator (see Fig. 3.1). Seeding can be obtained with a sharp rising edge in the long bunch [32], as is commonly done in numerical simulations.
The self-modulation process can also be seeded by a preceding plasma wakefield driver. The seed driver can be a short charged particle bunch or a laser pulse [1, 3]. We investigate analytically the
Figure 3.1: Comparison of two schemes in AWAKE Run 1 and AWAKE Run 2 experiments. (left) AWAKE Run 1 experiment scheme: a single electron bunch is used for the experiment. The plasma is ionized at the middle of the SPS proton bunch. Only half-back of the long proton bunch interacts with the plasma. Half-front of the proton bunch remains unmodulated. The half-front of the proton bunch can be modulated with an arbitrary modulation phase when multiple plasma stages are used.
(right) AWAKE Run 2 experiment scheme: two electron bunches are used for the experiment. The plasma is preionized by an ionization laser pulse. The first electron bunch is injected ahead the SPS proton bunch. The wakefield driven by this preceding electron bunch initially and weakly modulates the following long proton bunch. Once the modulation process starts, the modulation amplitude self- consistently grows with its driven wakefield. In this scheme, the phase and the growth rate of the long proton bunch modulation is controlled by the preceding electron bunch and the SPS proton bunch is entirely modulated.
mechanism of seeding of self-modulation when a preceding electron bunch generates seed wakefields.
This method was also demonstrated experimentally [17].
Analytical studies for the self-modulation instability have been developed assuming a bunch with a constant density profile in transverse and longitudinal directions [9, 10, 30]. The theoretical model in, which the bunch distribution in longitudinal and the transverse directions are expressed as heavy-side step functions, is advantageous in explaining the main phenomenon while simplifying the discussion, but is not suitable for predicting the growth rate of modulation amplitude with Gaussian or parabolic distributions. On the other hand, when the modulation of the proton bunch is seeded by a preceding electron bunch, the modulation process is decoupled from the seeding method. Increasing the number of degrees of freedom would make it possible to study new aspects of SM physics. In Ref. [11], the contribution of the seed appears with different aspects at the front and back of the long bunch. However, since the analytical solution of Ref. [11] was extracted from a specific order of seed contribution with a constant density profile, the relationship between the amplitude of seed wakefields and the proton bunch self-fields did not function for determining the growth rate properly. Therefore it is worthwhile to study the seeding features of the long bunch self-modulation with non-constant density profile in more detail.
We first start with the study of the low energy electron bunch dynamics. In the regime of AWAKE
Figure 3.2: (left) A schematic image and (right) the experimental result of electron bunch seeded SPS proton bunch modulation in AWAKE Run 2(a) experiment. (left) A low power laser pulse preionizes Rubidium vapor. A low energy electron bunch is injected into the preionized plasma and generates plasma wakefield for seeding the envelope modulation of the following SPS proton bunch. Optical transition radiation (OTR) is emitted when the proton bunch enters an aluminum-coated silicon wafer, positioned at 3.5 m downstream of the plasma exit and is imaged on the entrance slit of a streak camera.
(right) Time-resolved images of the modulated proton bunch at the OTR screen obtained by averaging 10 single-event images (210 ps long window with 2 ps resolution,Qp=14.7 nC). (a) No plasma. (b) preionized plasma (npe=1.02×1014cm−3) and seed electron bunch withQs=249 pC. (c) Same as (b), but seed electron bunch is relatively delayed by 6.7 ps. (d) On-axis time profiles of (b) (blue line) and (c) (red line), which shows the phase control capability of the SPS proton bunch modulation.
experiment, the ambient plasma electron number denstiy is on the order of∼1014cm−3. The electron or proton bunch number density is on the order of∼1012cm−3. Thus, the system in in linear, or over-dense plasma regime. At the beginning, we confirm that there is no radial beam-plasma matching condition for the low energy (∼18 MeV) electron bunch. Since the pinched electron bunch density is usually on the order of the ambient plasma density, the beam-plasma interaction can not be linearized and the exact electron bunch profile after the evolution is analytically unknown. For the low energy electron bunch, the efficiency for the stable wakefield generation is maximized at the bunch lengthσz,e=√
2/kpe, which is the well-known optimum bunch length condition for PWFA. The energy of the electron bunch is the most significant parameter which affects the phase of the proton bunch modulation over other parameters like the bunch charge, emittance, and length. During the wakefield gneration, the significant portion of the bunch particles are radially outward and do not participate the wakefield generation. In particular, this bunch particle escape becomes more significant when the bunch energy is small and the emittance is large. 2D RZ symmetric simulation shows its limitation for describing this regime.
In order to estimate beam-driven plasma wakefield with non-constant radial profile, we assume that bunches (e−, p+) initial rms radius is much smaller than the plasma skin depth k−1pe, with kpe= (npee2/ε0mec2)1/2, wherenpe is the ambient plasma electron number density,ethe elementary charge,
ε0 the vacuum permittivity,me the electron mass, andcthe speed of light in vacuum. The growth rate of the long Gaussian proton bunch modulation in over-dense plasma is estimated for the two cases of a radially-matched high-energy proton and of a tightly-focused low-energy electron seed bunches. The effect of seed in the asymptotic solution of the envelope modulation amplitude appears as series sum- mation, causing phase shift against the modulation feedback. By considering the modulation process as the linear combination of self-modulation and externally seeded modulation, and by introducing ad- ditional coefficients for explaining the timing of onsets of two modes, we characterize the regimes at the front and the center of the long proton bunch Gaussian longitudinal profile. In particular, we show that the electron seed not only drives an early growth rate larger than that driven by the proton seed, but also seeds a single mode modulation simultaneously along the entire long proton bunch, suppress- ing mode polarization with anomalous phase shift. The analytical approach shows nice agreement with FBPIC [28] simulations.