Beam emittance preservation using Gaussian density ramps in a beam-driven plasma wakefield accelerator

The luminosity demanded of a future high-energy lepton collider will require the delivery of beams with nm-rad normalized emittance and energies upwards of 1 TeV [1,2]. The plasma wakefield accelerator (PWFA) is an advanced particle accelerator that has the potential to meet these demands at a reduced size and cost compared with metallic structure-based accelerators [3,4]. To date, multi-GeV energy gain in a single-stage beam-driven PWFA [5] has been repeatedly achieved in experiment for both electron and positron beams [6–9]. Beam emittance preservation, however, has yet to be demonstrated by the plasma accelerator community. Here, a simple empirical formula is presented that describes the precise beam and plasma conditions required to provide near-perfect matching of a beam into a PWFA stage using a Gaussian plasma density ramp. In addition, numerical studies are carried out to show that the experimental tolerances are well within the reach of next-generation research facilities.

This study considers a PWFA operating in the nonlinear blowout regime [10], wherein a plasma wake is driven by a dense, ultrarelativistic electron beam. The plasma wake-driving beam is referred to as the drive beam and the beam that is accelerated in the plasma wake (also ultrarelativistic) is referred to as the witness beam. The witness beam must be matched to the plasma of a PWFA in order to preserve its rms normalized emittance ε_n≡γ_b ε, where γ_b is the relativistic Lorentz factor of the beam, and $\epsilon \equiv \sqrt{⟨x^2⟩⟨x^{\mathrm{\prime }2}⟩-{⟨x{x}^{\mathrm{\prime }}⟩}^2}$ is the root-mean-square (rms) geometric emittance [11]. x represents the horizontal position of an individual particle in the beam and x′≡p_x/p_z is the angle of the trajectory of an individual particle. p_x and p_z are the particle's horizontal and longitudinal components of momentum, respectively. A beam is considered matched to the plasma when its transverse envelope avoids the characteristic beating associated with betatron oscillations [12,13]. The transverse envelope is characterized by the rms spot size along each transverse dimension, defined as ${\sigma }_x\equiv \sqrt{⟨x^2⟩-{⟨x⟩}^2}$ in the x-plane, and similarly for the y-plane. Matching occurs when the linear focusing force provided by the ion column in a blowout plasma wake is balanced against the divergence of the beam.

A mismatched beam with a finite energy spread undergoes chromatic filamentation inside a PWFA, leading to emittance growth and transverse profile distortion. Fortunately, filamentation can be avoided by focusing the beam to the properly matched size at the start of the PWFA plasma source. Typical electron beam sizes in a conventional accelerator are much larger than the matched size required of a PWFA, however. There have been studies that aim to design conventional electromagnetic beam delivery systems that can effectively transport a beam from one plasma stage to the next [14–16], but it remains a significant challenge to provide the necessary focusing within a reasonable distance. To address this challenge, the strong focusing forces arising inside the plasma wake itself can be used in a plasma density ramp in order to match the beam into the plasma accelerator [17–24]. The same technique can be employed to control the divergence at the exit of the plasma source in order to prevent normalized emittance growth during free diffraction [25,26].

1. Beam matching

An electron beam is matched to a uniform-density plasma if the beam envelope experiences no oscillations, requiring that α =  − (1/2)(dβ/dz) = 0, which occurs when β = β_m = k⁻¹_β, α = α_m = 0 and γ = γ_m = k_β, where k_β is the betatron wave number. Here, β, α and γ represent the Courant–Snyder parameters [27] of the beam, while β_m, α_m and γ_m represent the parameters that satisfy the matching condition inside the plasma. The matched rms transverse size of the beam is then given by ${\sigma }_m=\sqrt{\epsilon {\beta }_m}=\sqrt{\epsilon /k_{\beta }}$. The betatron radial frequency is ${\omega }_{\beta }={\omega }_p/\sqrt{2{\gamma }_b}=c\hspace{0.17em}{k}_{\beta }$, where ω_p = (n_pe²/ϵ₀m_e)^1/2 is the plasma electron angular frequency, n_p is the plasma electron number density, ϵ₀ is the vacuum permittivity, m_e is the electron mass, e is the elementary charge and c is the speed of light in a vacuum.

Small longitudinal variations of the plasma density n_p and/or the beam Lorentz factor γ_b lead to the approximation k_β≃k_β,0(1 − δ/2), where δ≡δ_n − δ_γ − (3/4)δ²_n + (9/4)δ²_γ − (1/2)δ_nδ_γ to second order in δ_n and δ_γ. Here δ_n≡Δn_p/n_p,0≪1, δ_γ≡Δγ_b/γ_b,0≪1, n_p,0 and γ_b,0 are the nominal plasma density and beam momentum, and k_β,0 is the nominal value of k_β when δ = 0. The relative emittance growth over a distance ∼k⁻¹_β,0 for an initially matched beam traversing a plasma with a density perturbation is then δ_ε≡(Δε(k⁻¹_β)/ε₀)≃(δ²_n + σ²_δγ), to second order in δ_n and σ_δγ, where σ_δγ = (〈δ²_γ〉)^1/2 is the rms relative energy spread of the beam, and where ε₀ is the initial matched emittance. It is evident that beam emittance can be well preserved under an adiabatic transition of the plasma density, providing a method for beam matching into or out of a PWFA plasma source [17,20,21]. The adiabatic condition is (1/2)|(dβ_m/dz)| = (4n_pk_β)⁻¹|(dn_p/dz)|≪1 [22,24], where γ_b,0 is assumed to be a constant of the motion. Under this assumption, the adiabatic condition depends only on the plasma density profile and not on the beam evolution.

Beam matching into a uniform plasma density may also be achieved with plasma density gradients that exceed the adiabatic limit, even though the beam may not be matched to the local plasma density throughout the ramp. This can be leveraged to reduce the overall scale of the beam-matching section of plasma. Similar schemes that use a highly mismatched beam with an adiabatic ramp can also reduce the length scale of the matching section [28,29]. This study concentrates on matching into a uniform plasma with a Gaussian plasma density ramp shape, which is not fully adiabatic. It is chosen because it provides a continuous density profile at the transition to the bulk plasma region and it rapidly reduces to negligible densities away from the bulk. It has also been used to model the heat-pipe oven plasma sources used in PWFA experiments at SLAC National Accelerator Laboratory's Facility for Advanced Accelerator Experimental Tests (FACET) [18]. The Gaussian plasma density ramp takes the form n_p(z) = n_p,0 e^−(z2/2σ2) , where the endpoint of the uniform density region of the bulk plasma is located at z = 0, and the half-width of the ramp is ${\sigma }_{\mathrm{hw}}=\sqrt{2\ln 2}\hspace{0.17em}\sigma$.

The Gaussian plasma ramp satisfies the adiabatic condition for |z|≲2σ. In this region, the beam will remain matched to the local plasma density if it is initially matched. In the region 2σ≲|z|≲4σ, the ramp is non-adiabatic, and the beam is not matched to the local plasma density. In the region 4σ≲|z|, the local matched betatron wavelength is much larger than the beam's beta function, βk_β≪1, and the beam evolution can be approximated as being vacuum-like, due to the small, slow effect of the plasma on the beam. Figure 1 shows the evolution of a beam that is matched to a uniform density plasma by a Gaussian ramp, and identifies the three qualitatively different regions of beam evolution.

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A mismatched beam will experience chromatic emittance growth in a plasma via filamentation, reaching a saturated limit after many betatron periods. In a uniform density, the parameter B_m≡(1/2)(β_iγ_m − 2α_iα_m + γ_iβ_m) gives the ratio of the saturated normalized emittance of a fully filamented beam to the beam's initial normalized emittance [19,28–32]. Here, β_i, α_i and γ_i represent the initial Courant–Snyder parameters of the beam. In addition to the rms emittance growth, the distribution of the beam particles in transverse phase space will become distorted, causing halo formation around a strongly peaked, non-Gaussian core.

A beam that is bi-Gaussian in transverse phase space (x, x′) has a density function f(J) = ε⁻¹e^−J/ε, where J is the betatron action amplitude. The phase space coordinates (x, x′) are related to the generalized action-angle coordinates (J, ϕ) by the transformations $x=\sqrt{2J\beta }\cos \varphi$ and $x^{\mathrm{\prime }}=-\sqrt{2J/\beta }(\sin \varphi +\alpha \cos \varphi )$. The kurtosis of J for a bi-Gaussian beam is κ(J) = 9. The process of filamentation in a plasma acts to increase κ(J) [30]. For nearly matched beams (B_m≃1), the saturated kurtosis is given by κ(J)≃B_m 9. For significantly mismatched beams (B_m≫1), the kurtosis approaches an asymptotic upper limit of 15. When the beam is at a waist, the kurtosis of J will be most evident in the x′ projection of the beam; when it is far from a waist, it will be most evident in the x projection of the beam. The effect is to create a sharp peak at the centre of the distribution, with wide, halo-like tails extending well beyond the rms spot size.

2. Numerical beam-matching studies

The evolution of the beam's beta function as it travels through a plasma in a nonlinear blowout wake is described by the equation

An analytical solution for β(z) cannot generally be found for an arbitrary plasma density profile. It is, therefore, necessary to use numerical methods to study the evolution of the beam through a given plasma density profile. This exercise bears some resemblance to the matching of a high intensity laser pulse into a laser wakefield accelerator plasma source [33].

Figure 2 shows the numerically calculated evolution of a 10 GeV electron beam with an initial normalized emittance of ε_n = 5 mm mrad, a flat relative energy spread with a half-width of σ_δp,hw = 0.01, and a beta function at the virtual vacuum waist of β*_v = 10 cm passing through two different plasma sources. The virtual vacuum waist here refers to the beam waist size and waist position that would be realized in the total absence of plasma. The virtual vacuum waist is numerically calculated from the initial and final beam parameters far away from the plasma source. The initial beam parameters are chosen to approximate those expected at the next-generation experimental facility FACET-II at the SLAC National Accelerator Laboratory [34,35].

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A bi-Gaussian distribution of 10⁶ electrons is numerically propagated through each plasma source in figure 2 using a step size $\mathrm{\Delta }{z}_{\mathrm{step}}=k_{p,0}^{-1}\sqrt{2{\gamma }_{b,i}}/20$, where γ_b,i is the initial relativistic Lorentz factor of the beam. The transverse motion of each electron is calculated using Hill's Equation: x′^′ + K(z)x = 0, where K(z) = (e²/2γ_bm_ec²ϵ₀) n_p(z) arises from the linear focusing force produced by the pure ion column of density n_p(z) inside the nonlinear plasma wake [36]. The centroid energy gain (or loss) rate is taken into account at each point in the calculation using a formula derived from basic physics arguments justified in the appendix:

where G₀ is the constant rate of energy gain inside the bulk region of the PWFA, and k_p,0 represents the nominal plasma wave number in the uniform density region of the plasma. The energy gain rate in the bulk region of the plasma is set to $G_0=(16.7\hspace{0.17em}\mathrm{GeV}\hspace{0.17em}{\mathrm{m}}^{-1}/m_e{c}^2)\sqrt{n_{p,0}/5\times {10}^{16}\hspace{0.17em}{\mathrm{cm}}^{-3}}$, based on results obtained from the three-dimensional particle-in-cell (PIC) simulations described in the appendix. The energy spread is kept fixed in these calculations, as growth in energy spread is beyond the scope of this study. Regardless of the energy spread, the emittance growth is minimized when the centroid energy of the beam is matched to the plasma. For parameters relevant to this study, the total change in centroid energy over the full distance of the Gaussian plasma density ramp is typically less than 10% of the initial beam energy, which is inconsequential for matching purposes.

Figure 2a shows near-perfect matching of the beam into the bulk plasma source, where the normalized beam emittance ε_n and kurtosis of the betatron action amplitude κ(J) grow by significantly less than one per cent, agreeing with the prediction B_m = 1.00. In figure 2b, the beam is under-focused upon entering the bulk plasma, resulting in filamentation that reaches saturation shortly after entering the bulk. In this case, the emittance grows by 43.2% compared with the initial value, agreeing with the B_m = 1.43 prediction. The kurtosis also grows significantly, nearly reaching the asymptotic maximum limit, even while B_m is less than 2.

It is useful to identify the experimental conditions that will yield beam matching for a given input beam and bulk plasma density. It is also important to understand the sensitivity of the emittance growth to the experimental parameters involved. To perform these tasks, B_m is calculated over a range of values for the entrance ramp half-width σ_hw and the virtual vacuum waist location z*_v. The beam beta function is numerically propagated through the plasma ramp to the start of the bulk, where B_m is then calculated. Figure 3 shows a contour map of B_m obtained for the same 10 GeV electron beam used in figure 2 entering a plasma source with a bulk density of n_p,0 = 5 × 10¹⁶ cm⁻³. The matching point is found to be (σ_hw, m, z*_v,m) = (14.0 cm,  − 38.7 cm) and is indicated on the figure with a black dot.

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The B_m = 1.01 contour in figure 3 signifies 1% emittance and kurtosis growth at saturation and spans a 4.45 cm range of ramp half-widths, each with a corresponding optimal z*_v. In practice, the virtual vacuum waist location will be determined by an electromagnetic beam delivery system with centimetre-scale precision [34]. Therefore, the beam emittance can be preserved by tuning the vacuum waist location to accommodate for error in the plasma ramp width. If β*_v = 10 cm, then z*_v needs to change by 3.24 cm for every 1 cm that σ_hw deviates from the optimal matching value.

3. Empirical matching formula

By expanding the numerical study to explore a wide range of beam energy γ_b,0 and bulk plasma density n_p,0, an empirical matching formula emerges that depends only on the vacuum waist beta function β*_v. The scalings for the optimal Gaussian ramp half-width and virtual vacuum waist location are given by quadratic equations:

All variables are normalized by a factor $k_{\beta ,0}=(1.33\times {10}^{-4}\hspace{0.17em}{\mathrm{m}}^{-1})\sqrt{n_{p,0}\hspace{0.17em}[\mathrm{c}{\mathrm{m}}^{-3}]/{\gamma }_{b,0}}$ (indicated by tildes). Here, n_p,0 is the bulk plasma density, and γ_b,0 is the Lorentz factor of the beam at the start of the bulk plasma. Because the energy gain that occurs in the ramp is small, it is possible to approximate γ_b,0 as constant without significantly impacting the matching quality.

The scope of validity for this formula was found by calculating the degree of matching, B_m, over a wide range of the normalized virtual vacuum waist beta function, ${\tilde{\beta }}_v^{\ast }$. The scan covered a span of 201 × 201 evenly spaced working points in the parameters space defined by 5 cm ≤ β*_v ≤ 85 cm and 2.8 × 10¹⁵ cm⁻³ ≤ n_p,0 ≤ 3.6 × 10¹⁸ cm⁻³, and a 10 GeV beam was assumed. The ramp half-width and vacuum waist location used in each iteration were determined according to equation (3.1) for every computational working point. The value of ${\tilde{\beta }}_v^{\ast }$ was calculated for each point, as was the value of B_m at the peak density location (i.e. end of the ramp). Figure 4 shows the resultant value of B_m that is achieved by the formula presented in equation (3.1) over the considered range of ${\tilde{\beta }}_v^{\ast }$. The formula preserves emittance to a degree of 1% (B_m ≤ 1.01) over a range of ${\tilde{\beta }}_v^{\ast }$ from 9.50 to 185. When the emittance is preserved, the beam's vacuum waist spot size is demagnified by a factor $\sqrt{{\tilde{\beta }}_v^{\ast }}$ at the end of the plasma entrance ramp. Therefore, the empirical matching formula presented here is valid for a range of spot size demagnification from 3.08 to 13.6. The same optimization scalings can be used to tailor the dynamics of the beam as it exits the plasma source, given the desired value of the virtual vacuum waist ${\tilde{\beta }}_v^{\ast }$ at the plasma exit.

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4. Conclusion

Beam emittance and profile quality can be preserved in a PWFA through the use of multi-centimetre-long Gaussian density ramps. Numerical studies reveal an empirical quadratic formula to predict the optimal ramp length and vacuum waist position for matching that depends only on the vacuum waist beta function. The empirical matching formula is shown to work over a wide range of conditions with experimentally favourable tolerances. This, along with the compact and passive nature of the Gaussian plasma density ramps, makes them an attractive solution for beam matching and emittance preservation in PWFA experiments and applications.

Data accessibility

This article has no additional data.

Authors' contributions

All authors contributed equally to this work.

Competing interests

We have no competing interests.

Funding

This work was supported by the US Department of Energy grant number DE-SC0017906.

Footnotes

One contribution of 10 to a Theo Murphy meeting issue ‘Directions in particle beam-driven plasma wakefield acceleration’.