Radiation Reaction of Betatron Oscillation in Plasma Wakefield Accelerators

A classical model of radiation reaction for the betatron oscillation of an electron in a plasma wakefield accelerator is presented. The maximum energy of the electron due to the longitudinal radiation reaction is found, and the betatron oscillation damping due to both the longitudinal and transverse radiation reaction effects is analyzed. Both theoretical and numerical solutions are shown with good agreements. The regime that the quantum radiation takes effect is also discussed. This model is important for designing future plasma based super accelerators or colliders.

In a PWA, the transverse focusing force of the beam, due to the radial E-field and azimuthal B-field of the wake, exists along the whole accelerator. As a consequence, the normalized emittance of the beam may grow due to the betatron oscillation (BO) and finally saturate [10], with the saturation normalized emittance commonly ranging from 0.1 to 100 mm·mrad, depending on the injection mechanism, e.g. self-injection, density gradient injection, optical injection, ionization injection and so on [11][12][13][14][15][16][17][18][19][20].
Radiation reaction (RR) is the friction on an electron due to EM radiation of itself. The RR effect has been studied for laser-plasma interactions, typically on energetic electrons colliding with high-intensity laser pulses. It is predicted that more than 35% of laser energy is converted to EM radiation at a laser radiance of Iλ 2 > 10 22 Wµm 2 /cm 2 in a duration of 20 fs [21]. The quantum correction in RR has been confirmed by recent experiments of the head-on collision between energetic electron bunches and high-intensity laser pulses [22,23].
These researches focused on an extreme condition of highly intense EM-field in a short period of the fs-order to study the elementary process of a radiating electron. On the other hand, the long-time accumulation of the RR effect has also been studied, e.g. for runaway electrons in Tokamak to engineer good plasma conditions of magnetic confinement plasma [24].
For a regular PWA length of the order of 0.1 m or shorter, RR is negligible because the RR force is several orders of magnitude smaller than the acceleration and focusing forces of the wake. However, the magnitude of the RR force becomes significant at high beam energies.
Moreover, the RR force is a damping force and its effect accumulates, which may finally become noticeable at long acceleration distances. Such effects are crucial for developing future super plasma wakefield accelerators and colliders which have acceleration distances of the order of 10 to 10 3 meters [25].
In this work, we show a classical model of the RR effects for the single electron BO in a PWA. The theory is reinforced by numerical results obtained by the code PTracker [26], which uses 4-th order Runge-Kutta Method to solve the simplified equations of motion of an electron in a PWA. This paper is organized as follows. Section II gives the general form of BO of an electron in a PWA. Section III shows the BO amplitude scaling during acceleration or deceleration processes. Section IV presents a classical model of RR during BO, and Section V discusses the limit between classical and quantum RRs.

II. BETATRON OSCILLATIONS WITHOUT RADIATION REACTION
For a single electron trapped (with relativistic longitudinal velocities) in a blowout plasma wakefield and moving in the z-x plane, the focusing force (in x direction) and accelerating force (in z direction) near the center of the blowout bubble are [27] f ext where ζ = z − β w t is the wake co-moving frame with ζ = 0 and x = 0 being the center of the blowout bubble, β w = 1 − 1/γ 2 w is the wake phase velocity normalized to speed of light in vacuum c, and γ w 1 is the Lorentz factor of the wake. Note we have adopted the normalized units where length is normalized to k −1 p , time is normalized to (ck p ) −1 , momentum is normalized to m e c, energy and work are normalized to m e c 2 , force is normalized to m e c 2 k p , with k p = µ 0 e 2 n p /m e being the wavenumber of plasma wake, µ 0 being the vacuum permeability, e being the elementary charge, n p being the plasma density and m e being the electron mass. Without RR, the equation of motion iṡ with where a dot on top means time derivative, p is the normalized momentum and γ is the Lorentz factor of the electron.
Assume |x| is a small quantity. Then the trajectory in the wake co-moving frame features a figure "8" with the frequencies in x-axis being ω β and in z-axis being 2ω β where ω β = 1/ √ 2γ is the normalized BO frequency [28] x = x 1 sin ω β t, where subscriptions 0 / 1 indicate "slow" / "fast" components with characteristic time scale much longer than / comparable to ω −1 β . The normalized velocities are where β z0 = β w +ζ 0 is β z averaged in the ω −1 β time scale. Thus where high order terms are neglected, and Meanwhile,γ Commonly in a plasma wakefield |ζ 0 | < ∼ 1. Note ω β 1, the second term is negligible, thus where is γ averaged in the ω −1 β time scale. We apply Eq. (14) to Eq. (11) and multiply both sides by γ 2 0 . By collecting the "slow" varying parts one obtains which can be applied to Eq. (12) for the averaged longitudinal velocity β z0 . Note . γ z0 is regarded as the phase-locking Lorentz factor. If γ w = γ z0 and ζ 0 = 0 initially, ζ 0 will remain 0 and there will not be net acceleration or deceleration.

III. BETATRON AMPLITUDE DURING ACCELERATION OR DECELERA-TION
In this section we consider the effect of slowly varying γ 0 . We only keep the first order variation. Within a time scale t ∼ ω −1 As a result, the amplitude and frequency of BO also change Note the factor 1/2 in Eq. (21) comes from the fact that a linear chirp contributes twice to the frequency shift. BO still follows Eq. (7) but with slow-varying x 1 and ω β where φ ≡ ω β + 1 2ω β t t. According to Eqs. (1) and (3), we havė Meanwhile, we write down whereγ 0 γ 0 t,ẋ 1 x 1 t andω β ω β t are in the same order, and higher order terms are neglected. Then we take derivative one more timė We substitute left-hand-side of Eq. (23) by Eq. (25), the 0th order of the "sin" term retrieves Eq. (17). The 1st order of the "sin" and "cos" terms construct two equationṡ which lead toω By integral, we obtain the long-term dependencies of ω β and x 1 on γ 0 The amplitude of p x oscillation is As a result, the area encircled by the trajectory of the particle in the x-p x phase space, is a constant with varying γ 0 , where A is defined as the normalized BO amplitude We check the scaling Eq. (31) using PTracker. Two cases with significant acceleration and deceleration are shown in Fig. 2. Because ζ 0 = 0 and γ z0 < γ w at t = 0, the electron firstly drifts to the acceleration phase, and γ 0 increases so that γ z0 > γ w at some point.
Later it crosses the ζ = 0 point, enter the deceleration phase and γ 0 starts to decrease. The two numerical cases show exact scaling of x 1 = Aγ −1/4 . Note we have replaced γ 0 by γ here, because their difference is not distinguishable in these plots, and γ can be obtained much more easily than γ 0 from numerical result.

IV. RADIATION REACTION EFFECTS ON THE BETATRON OSCILLATION
The Lorentz-Abraham-Dirac equation in the same normalized units as in Sec. II is with the metric (−1, 1, 1, 1), where τ is the proper time (γdτ = dt), r e is the classical electron radius (also normalized to k −1 p ), p µ = (γ, p) is the four-momentum, and F rad µ is the four-force of RR. The space component of Eq. (35) is The three-dimensional RR force f rad is decomposed to With the RR force, the equation of motion Eq. (3) is modified aṡ

A. Radiation reaction force of betatron oscillation
We assume RR is perturbation to the electron BO which has the form discussed in Sec. II, Note Eq. (15), as long as According to Eqs. (37) and (38), the RR force terms are f rad2 Use Eqs. (31) and (34), we have x 2 1 γ 0 1 and conclude f rad2 x . Use Eq. (17) and We also have f rad2

B. Perturbation correction and phase-locking with longitudinal forces
In Sec. II we have obtained the phase-locking Lorentz factor γ z0 in Eq. (16). With RR, extra longitudinal deceleration force exists. Define the phase-locking ζ 0 to be the summation of the longitudinal external and RR forces is written by using Eqs. (2), (8), (18) and (50) If ζ 0 = ζ 0l , the longitudinal momentum gain averaged in the ω −1 β time scale is zero. If γ w = γ z0 is also satisfied, the phase-locking will occur and there will not be net acceleration or deceleration. The 2nd term, coming from BO, is already included in Eqs. (43) and (45).
In cases the 3rd term, coming from RR, is comparable or more significant than the 2nd term, i.e. r e γ where the γ 0 definition Eq. (15) is modified as In the regime r e γ 0 1, Eq. (46) still holds, and in Eqs. (47) -(50), only Eq. (48) has to be modified f rad1 Note f rad2 z f rad1 z still holds, the above discussion is self-consistent. One may compare our Eqs. (56) and (57) with Eqs. (30) and (32) of Ref. [29] by taking K 2 = k 2 p /2 and find that our model keeps more details of the BO, while they have omitted the difference betweeṅ γ andṗ z . These details, as shown in Sec. IV C, are important for the long term RR effect.
We also estimate the maximum γ 0 achievable in PWA, due to the limited size of the blowout wakefield structure, by setting ζ 0l ∼ −1 in Eq. (52) and using Eq. (33): . (60) One should note that this maximum energy is not due to dephasing or pump-depletion effects. It is the power of radiation loss approximately equals to the maximum power of acceleration in the wakefield.

C. Damping of betatron oscillation due to radiation fraction
The potential and the kinetic energy in the transverse direction transform to each other due to BO, with the maximum potential energy being Meanwhile, this potential energy is slowly lost due to the work of dissipation force averaged in the ω −1 β time scale. The averaged power of the transverse force is and the averaged power of the longitudinal forces by using Eqs. (53), (54) and (55) is where the angle brackets stands for averaging in the ω −1 p time scale. Note the 1st term in the right-hand-side of Eq. (63) contributes to the longitudinal acceleration or deceleration, while the 2nd term, coming from the coupling of the oscillation terms of β z and f rad2 z , contributes to the transverse damping of BO. Thuṡ which can be applied to Eqs. (61) for the damping rate of the BO amplitude due to RṘ Note Eq. (29), the total damping rate of x 1 iṡ Use Eqs. (33) and (66), we find the damping rate of the area encircled by the trajectory of the particle in the x-p x phase spacė or in the integrated form 1 S In the time scale that γ 0 does not change significantly, it can be simplified as Thus the length that S reduces by a half is or in the unnormalized form (note S and A remain in normalized units) For an electron beam which has an distribution of S, the damping rates are different for different S. The reduction rate of the normalized emittance n ≈ S/k p is estimated by Eq. (67), but with S replaced by the largest S among the electrons. Thus, we obtain the engineering formula of the normalized emittance reduction length The numerical solution for the S damping in a long time scale (∆t = 10 9 > L S = 1.8×10 8 ) with the same parameters as Fig. 3 (a) is shown in Fig. 3

V. FROM CLASSICAL TO QUANTUM RADIATION DOMAIN
The correction by nonlinear quantum electrodynamics (QED) has been studied for highintensity laser-electron interactions [30][31][32]. The radiation power with nonlinear QED correction is where P classical is the classical radiation power, and q(χ) is the QED correction factor [33][34][35].
q(χ) ≈ 1 with χ 1 is the classical regime of the radiation process, while q(χ) < 1 with χ > ∼ 1 is the QED regime. We calculate the significance of this correction in our model.
RR is effective when a background force is strong enough and perpendicular to the electron momentum in the relativistic regime. This perpendicular direction is primarily the x-direction in our case. We write down the effective vector potential amplitude of the wake- which is an analogy of the normalized laser vector potential amplitude a 0 in laser-electron interactions. When we consider a radiation process to be nonlinear Compton scattering by a laser pulse, the formation phase (or the formation length) a −1 0 is important for the locally constant field approximation (LCFA) in the semi-classical model [30][31][32][33][35][36][37].
Namely, a single photon is emitted instantaneously in a short phase interval a −1 0 at high intensities. By analogy, the formation phase in the wakefield is O(W −1 0 ) which is short enough if γ 1. Thus, the LCFA of a radiation process can be employed for the high-energy electron case. Also, the quantum correction q(χ) can be applied, with the dimensionless quantum parameter where α is the fine structure constant, F ext Note r e is normalized to k −1 p in the expression. The maximum χ in one BO cycle is For χ max 1, RR is classical, while for χ max > ∼ 1, quantum correction may appear in Eq. (73). Commonly, PWA with internal injections has A ∼ 1. Thus for k p ∼ 10 5 m −1 (r e ∼ 10 −10 as a consequence), RR is classical if γ 10 10 .

VI. SUMMARY AND DISCUSSION
We have developed a classical model of the electron betatron oscillation with radiation reaction (RR) in a plasma wakefield accelerator. We found a phase-locking condition γ w = γ z0 and ζ 0 = ζ 0l using Eqs. (16) and (52), under which the electron does not gain or loss energy. The maximum γ 0 achievable in a plasma wakefield accelerator due to RR is estimated by Eq. (60). We also found the damping rate of S, defined as the area encircled by the electron trajectory in the x-p x phase space, in Eq. (67) and the length that S reduces by a half in Eq. (70). The quantum parameter characterizing the classical and quantum radiation domain is given by Eq. (76). Some examples with different plasma densities n p , initial S and γ 0 are listed in Tab. I.
We can see that the reduction of S (and thus the transverse cooling) is positively related to 1) the plasma density, 2) the Lorentz factor of the electron, 3) the initial betatron amplitude and 4) the total length of the plasma wakefield accelerator. We also note that the quantum parameter χ max is sufficiently smaller than unity for all these cases.
In order to decrease L S without an extremely large γ 0 , we can either increase n p or S.
Although an internally injected electron beam has S ∼ 2, a large S is achievable by an external injection. In the two-stage scheme where the first stage with a smaller n p is used for acceleration and the second stage with a larger n p is used for radiation [38], both n p and S are large for the second stage. Thus RR takes effect in a much shorter distance in this case. However, χ may be not negligible for large S values and quantum RR should be taken into consideration.