Transverse emittance-preserving arc compressor for high-brightness electron beam-based light sources and colliders

The transverse emittance of ultra-relativistic high-brightness electron beams, normally generated in photo-injectors, may be degraded by emission of coherent synchrotron radiation (CSR) in dipole magnets if the bunch peak current exceeds few hundreds of ampere [1]. For this reason, if such a peak current is needed, for instance, at the end of recirculating accelerators like energy-recovery linacs (ERLs) driving free-electron laser (FEL) and/or ERL-based particle colliders, the photo-injected beam is recirculated in isochronous beam lines and magnetically time-compressed only after the very last stage of acceleration [2], e.g. with a magnetic chicane [3,4]. Although this approach tends to preserve beam brightness during recirculation [5], it may put an upper limit to either the compression factor or the beam charge or both, thus to the final peak current, because of CSR-induced emittance growth in a single-stage compression [6].

In the past, several ERL designs [2,7–9] attempted to use recirculating arcs for bunch length compression in the energy range 0.2–7 GeV, while keeping the normalized emittance growth below $0.1\ \mu \text{m}\ \text{rad}$. Double- and triple-bend achromatic cells were tested for compression factors C < 30. The highest beam charge compatible with the target emittance control was set to 150 pC at the highest energy, and to 77 pC at 3 GeV. In [8,9] some degree of optics control was exercised in order to minimize the CSR effect following prescriptions in [10], but in all these designs CSR was mainly suppressed by a low beam charge. In the following we show, for the first time to the authors' knowledge, that it is feasible to redistribute a compression factor of up to 45 for a 500 pC beam, by means of a periodic 180 deg arc at 2.4 GeV (similar results were also obtained at 1.2 GeV), while keeping CSR transverse kicks under control. The total growth of the normalized emittance does not exceed the $0.1\ \mu \text{m}$ level for peak currents of up to 2 kA. In comparison with the existing literature, our solution allows larger compression factors at higher charges, simplifies ERL lattice designs (since, in principle, a dedicated chicane is no longer needed for compression as the arc acts both as final stage of recirculation and compressor) and paves the way for repeated compressions at different stages of acceleration, i.e., at different energies. Although it finds an immediate application to ERLs, as initially suggested in [11,12], the proposed CSR-immune arc compressor promises to be applicable to a more general accelerator design, thus offering the possibility of new and more effective layout geometries of single-pass accelerators and of new schemes for beam longitudinal gymnastic.

We first consider a symmetric double-bend achromat (DBA). The periodic 180 deg arc is made of N identical DBAs, whose number and lattice are chosen on the basis of the required total compression factor. This is defined in the linear approximation as $C=1/(1+hR_{56})$. An $R_{56}\sim 100\ \text{mm}$ is required to shorten an ultra-relativistic electron bunch which is initiated with a linear correlation in the longitudinal phase space $(z,\ E)$, that is, a linear energy chirp. The chirp is defined as $h=\text{d}E/(E\text{d}z)\approx \sigma_{\delta,0}/\sigma_{z,0}$, and it may correspond to an rms fractional energy spread of few 0.1% at GeV beam energies. $\sigma_{\delta,0}$ and $\sigma_{z,0}$ are the rms fractional energy spread and bunch length before compression, respectively. Linear compression is obtained as long as $|T_{566}| \sigma_{\delta,0}\ll|R_{56}|$; T₅₆₆ depends on the linear and the nonlinear optics functions. In the proposed arc design, two families of sextupole magnets, 4 sextupoles per DBA, allow full control of linear chromaticity and of T₅₆₆ to the sub-mm level, similarly to what is reported in [3,13,14]. However, owing to a nonlinear energy chirp induced by CSR, the sextupole strengths were numerically optimized for compensating it [15]: they provide a negative T₅₆₆ in the cm range, the actual value depending on the specific charge distribution, that is, on the CSR-induced energy variation along the bunch. As also shown in the aformentioned literature, such a linearization of the compression process has the advantage of avoiding the need for an RF harmonic linearizer [16,17].

In the horizontal bending plane, the single-particle motion is described analytically through the first-order transport matrix formalism [18]. The transverse CSR kick is modelled as being only due to the longitudinal component of the CSR electric field at the middle of each dipole magnet. CSR emission is assumed for simplicity in the one-dimensional (1-D) "short bunch, long magnet" approximation [19] and its effect on the particle coordinates is expressed through the formalism introduced by Jiao et al. in [20]:

$$\begin{equation} \left(\begin{array}{c} x_{k}\\ x^{\prime}_{k}\\ \end{array}\right) =\left(\begin{array}{c} {\rho^{4/3}k\left[{\theta \cos \left({\theta/2}\right)-2\sin \left( {\theta/2}\right)}\right]}\\ {\sin \left({\theta/2}\right)\left({2\delta_{0} +\rho^{1/3}\theta k}\right)} \end{array}\right), \end{equation} \tag{ 1 }$$

where ρ and θ are the dipole bending radius and angle, respectively, $\delta_{0}$ is the particle initial relative energy deviation, and $k=0.2459r_{\mathrm{e}}Q/(e\gamma \sigma_{z}^{4/3})$ is function of the number of electrons per bunch Q/e, the electron classical radius $r_{\text{e}}$ and the beam energy Lorentz factor $\gamma$. It relates the transverse CSR effect to the rms bunch length $\sigma_{z}$, and applies to a Gaussian longitudinal charge distribution emitting CSR in the steady-state regime. In general, a different dependence of k on $\sigma_{z}$ could be considered for different current profiles [19]. The novelty of the present approach is that the optics is balanced at each DBA so that, even with bunch length varying from one dipole magnet to the next, the emittance is protected from asymmetric CSR kicks. Optics balance has been adopted so far in transfer lines that include up to 4 dipole magnets, bending the beam by a few degrees per dipole, while keeping the bunch length constant along the line in order to equalize CSR kicks (but with opposite sign) at each dipole, as originally proposed by Douglas [21,22]. Here, we propose to control CSR in a longer line of 12 dipoles, each bending by 15 deg. The variation of the bunch length along the line is inlcuded in the model of the CSR kicks. By accepting a negligible emittance growth in our bunch length-dependent optics analysis, we derived a few constraints on the values of both the betatron function in the dipole magnets and the compression factor per cell. Transient CSR fields at the dipole magnets' edges and in drift sections following the dipoles are computed with the Elegant code [23,24]. We also included incoherent synchrotron radiation (ISR) and up to third-order nonlinear transport matrices. 3-D CSR effects [25,26] are neglected both in the analysis and in the simulations: for the beam parameters considered in this report, the Derbenev criterion [25], which sets a rule-of-thumb for the validity range of 1-D CSR field approximation, is satisfied along most of the line.

The initial coordinates of a particle relative to the reference trajectory are $x_{0}=0,\ x_{0}^{\prime}=0,\ \delta_{0}=0$, and the particle initial Courant-Snyder (C-S) invariant [27] is zero. The DBA is sketched in fig. 1 and, henceforth, numerical indexes of particle coordinates and Twiss parameters refer to the numbers in fig. 1.

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According to eq. (1), the particle coordinates after a CSR kick in the first bend are [20]

$$\begin{equation} \left\{\begin{array}{l} x_{1} =\rho^{4/3}k_{1} \left( {\theta C_{\theta } -2S_{\theta } } \right),\\ x_{1}^{\prime} =\rho^{1/3}k_{1} \theta S_{\theta},\\ \delta_{1} =\rho^{1/3}k_{1} \theta, \end{array}\right. \end{equation} \tag{ 2 }$$

where $C_{\theta}=\cos(\theta /2)$ and $S_{\theta}=\sin(\theta /2)$. The beam mean energy, in the range of hundreds of MeV to few GeV, is assumed constant along the cell (arc), as it is much larger than the CSR-induced mean energy loss. The particle coordinates are then computed along the lattice according to the linear transport matrix formalism, as depicted in [22]. Since the DBA is symmetric, we have Twiss functions $\beta_{1} = \beta_{2}$ and $\alpha_{1}=-\alpha_{2}$ at the dipole magnets; these are assumed to be short with respect to the total DBA length, and thus the horizontal betatron phase advance between the centres of the two dipoles approaches π [28]. After the CSR kick in the second dipole —different from the first kick because the bunch length has changed— we find

$$\begin{equation} \left\{\begin{array}{l} x_{3} =-\rho^{4/3}k_{1} \left({\theta C_{\theta } -2S_{\theta}} \right)+\rho^{4/3}k_{2} \left({\theta C_{\theta}-2S_{\theta}}\right),\\ x_{3}^{\prime} =-\rho^{1/3}k_{1} \theta S_{\theta } \!-\!\rho^{1/3}k_{2} \theta S_{\theta } -\displaystyle\frac{2\alpha_{2} }{\beta_{2} }\rho^{4/3}k_{1} \left({\theta C_{\theta}\!-\!2S_{\theta}}\right)\!,\\ \delta_{3} =\rho^{1/3}k_{1} \theta +\rho^{1/3}k_{2}\theta. \end{array}\right. \end{equation} \tag{ 3 }$$

Equation (3) is used to compute the single-particle C-S invariant, J₃, at the exit of the second bend. We do it under the approximation $\theta \ll1$ and expanding $C_{\theta}$ and $S_{\theta}$ to the third order in θ. C is now the bunch length linear compression factor in one DBA so that $k_{2} =C^{4/3}k_{1}$. We also impose a beam waist in the dipoles so that $\alpha_{2}\approx 0$ and we therefore obtain

$$\begin{eqnarray} &2J_{3} = \beta_{2} x_{3}^{\prime2}+2\alpha_{2} x_{3} x_{3}^{\prime} +\left({\frac{1+\alpha_{2}^{2} }{\beta_{2}}} \right)x_{3}^{2} = \nonumber\\ &\cong\left({\frac{k_{1} \rho^{1/3}\theta^{2}}{2}} \right)^{2}\!\left[{\beta_{2}\left( {C^{4/3}\!+\!1}\right)^{2}\!+\!\frac{1}{\beta_{2} }\left({\frac{l_{\text{b}}}{6}}\right)^{2}\!\left({C^{4/3}\!-\!1} \right)^{2}}\right]\!,\nonumber\\ \end{eqnarray} \tag{ 4 }$$

where $l_{\text{b}}=\rho\theta$ is the dipole magnet arclength. The invariant has a minimum for

$$\begin{eqnarray} &&\beta_{2,\min} =\frac{l_{\text{b}}}{6}\left({\frac{C^{4/3}-1}{C^{4/3}+1}}\right), \end{eqnarray} \tag{ 5 }$$

For $\beta_{2}\sim l_{\text{b}}\sim 1\ \text{m}$ in eq. (4), the larger C, the more important the first term in square brackets. At lower values of $\beta_{2}$, the larger C is, the more important the second term becomes. The final rms normalized emittance $\varepsilon_{n,f}$ is estimated as the sum in quadrature of its unperturbed value $\varepsilon_{n,0}$ and the energy-normalized value of the aforementioned invariant:

$$\begin{eqnarray} &&\Delta \varepsilon_{n,f} =\varepsilon_{n,f} -\varepsilon_{n,0}\approx\varepsilon_{n,0} \left({\sqrt{1+\frac{\left({2J_{3}}\right)^{2}\gamma^{2}}{\varepsilon_{n,0}^{2}}}-1}\right). \end{eqnarray} \tag{ 6 }$$

In summary, in order to minimize the CSR-induced emittance growth in a periodic 180 deg arc compressor we prescribe the use of several symmetric DBA cells with a proper tuning of the minimum value of the betatron function in the dipoles, according to eq. (5). It is worth mentioning that as the bunch shortens along the arc, the energy chirp increases, thus C may become much larger in the last cells than in the preceding ones. In general, this may imply a different tuning of $\beta_{2}$ in each cell. However, since the CSR effect is larger for shorter bunches, we might be allowed to relax the condition on $\beta_{2}$ (eq. (5)) in the first few cells, where the bunch is longer, while ensuring optimum $\beta_{2}$-tuning in the last ones. We finally point out that, while in a four-dipoles magnetic chicane strong focusing is required in the last magnet to make the beam angular divergence much larger than the CSR kick [29,30], for the arc compressor an optimum value of $\beta_{2}$ exists that depends both on the dipole length and the compression strength.

Different arc lattice designs may satisfy the optics prescriptions discussed above. The Elettra synchrotron light source [31] DBA cell was used as a starting point and a reliable working sample, since it offers a relatively large R₅₆, a well-established solution to manage nonlinear optics, and demonstrates the feasibility of the arc compressor under well-proven technical and operational aspects. Its magnetic lattice is sketched in fig. 2; the nominal Elettra DBA optics was re-adjusted for our purposes, as shown fig. 3. The 180 deg arc is made of 6 expanded Chasman-Green achromats [32] separated by drift sections that allow optics matching from one DBA to the next. The arc is 125 m long (40 m long radius) and functional up to 2.4 GeV. The arc length can be shrunk to around 100 m by shortening the 3 m long drift sections at the DBA edges, at the expense of significant revisions of the linear and the nonlinear optics (not shown). Scaling the magnets' size to lower energy, the arc length may be shortened to around 30 m at 0.7 GeV. The bending angle per sector dipole magnet is $\theta = 0.2618\,\,\text{rad}$ and the dipole arc length $l_{\text{b}} = 1.4489\ \text{m}$ for the 2.4 GeV case. R₅₆ of one dipole is 17.2 mm, while that of the entire arc is 207.1 mm. If, for example, C = 45 were required at the end of the arc, with local value not larger than ${\sim}10$ in the last cells, $h\approx -4.7\ \text{m}^{-1}$ would be needed at its entrance, which corresponds roughly to a fractional rms energy spread of 0.3% for a 3 ps rms long bunch. Such an energy chirp can be managed directly at 2.4 GeV by means, e.g., of a 1.3 GHz, 40 m long linac with 10 MV/m peak gradient [2], and RF phase close to zero crossing (no acceleration). The horizontal betatron function has a minimum of ${\sim}0.16\ \text{m}$ at the middle of the dipoles. This value is a compromise between desirable optics symmetry and low emittance (see eq. (5)). In fact, a cell-by-cell numerical evaluation of eq. (4) shows that such a value keeps the invariant close to its minimum in the last 3 cells, while it is a factor of up to 3 larger in the first 3 cells. These, however, do not contribute much to the total CSR-induced emittance growth because they are traversed by a longer bunch. For a Gaussian beam time-compressed by a factor 45 and with final peak current of 2 kA, eq. (6) predicts a projected normalized emittance growth at $0.1\ \mu \text{m}$ level at the arc's end, where the CSR-induced rms fractional energy spread reaches the order of $10^{-5}$ (the CSR-induced mean energy is at the same level) [19].

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Table 1 summarizes the arc input and output beam parameters used in Elegant runs. Two sets of initial beam parameters are considered, one for high charge-long bunch, the other for low charge-short bunch. In order to damp the CSR-induced microbunching instability [33], the electron beam uncorrelated energy spread is set up to 40 keV rms, as it would be increased with a laser heater [34–36] at low energy. Five million particles in a bunch were tracked. A quiet start of the electron beam input distribution and filtering were adopted to ensure the suppression of numerical sampling noise at uncompressed wavelengths shorter than $35\ \mu \text{m}$ [37]. The rms normalized projected emittance of the 500 pC beam grows from $0.80\ \mu \text{m}$ to $1.05\ \mu \text{m}$ at the arc's end, with contributions from ISR, chromatic aberrations and CSR shown in fig. 4. Chromatic aberrations are responsible for the emittance modulation along the line, as well as for the (small) horizontal slice emittance growth, shown in fig. 5, top plot. Non-uniformity of the horizontal C-S slice invariant (i.e., the invariant of the slice centroid) along the bunch, shown in fig. 5, bottom plot, reflects the slices misalignment in the transverse phase space due to local CSR kicks. Residual CSR-induced microbunching shows up in the longitudinal phase space at final wavelengths longer than $10\ \mu \text{m}$. The final slice energy spread is around 2 MeV and substantially dominated by the initial uncorrelated energy spread times the total compression factor. Similar compression and emittance performance were obtained with the 100 pC beam. The CSR-induced emittance growth is at the same $0.1\ \mu \text{m}$ level as in the 500 pC case (see table 1), in agreement with eq. (6) and with the scaling of the CSR effect with charge and bunch length [19].

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The capability of controlling CSR effects in an arc compressor (not necessarily constrained to a 180 deg total bending angle) —and thus to increase the beam peak current while preserving its 6-D normalized brightness using an approach that goes beyond those offered by the existing literature [38]— quite generally opens the door to new geometries in accelerator design and new schemes of beam longitudinal gymnastic. For example, after single- or multi-pass acceleration in an FEL linac-driver, the beam can be arc-compressed at high energy and counterpropagated into an undulator, which could then lie parallel to the accelerator. At least two advantages are seen: one is that cost savings are achieved in civil construction, the other is that the operation of the system is simplified, as much as the beam does not undergo any manipulation other than acceleration until it reaches the arc compressor. A similar layout may also apply to linear particle colliders. As an example, the CLIC design [39] includes optics matching insertions and a magnetic chicane both devoted to bunch length compression before an isochronous turnaround arc, and a similar configuration (two chicanes aside an arc) is before beam deceleration. On the basis of our findings, the arc could be investigated as either compressor (together with a proper setting of the upstream RF phases to match the arc's positive R₅₆) or a CSR-immune transfer line, if the beam has no energy chirp at its entrance. A similar option might apply to ILC [40], which is currently planning an isochronous turnaround followed by two compressor chicanes.

Our arc compressor design is also recommended for an ERL-, or recirculated linac-driven FEL such as that described in [41]. In this case, the electron beam may be accelerated and recirculated in isochronous beam lines until it reaches the target energy and energy chirp, and eventually compressed. From the entrance to the exit of the arc compressor, the energy spread, normally dominated by the energy chirp, remains substantially unchanged. In order for the FEL amplification process to be efficient, $\sigma_{\delta,0}$ must be matched to the normalized FEL energy bandwidth, ρ [42]. For lasing in x-rays, $\rho \ge 10^{-4}$ and this may require a removal of the energy chirp downstream of the arc, i.e. with a dedicated RF section. With the 500 pc beam parameters of table 1, we estimate [43] lasing at 1.3 nm with 2.1 m long 3-D gain length, $\rho =1.1\times 10^{-3}$ and FEL power saturating at 2.6 GW in a 36 m long undulator. If the incoming energy chirp is imposed to the beam at full energy with a linac running close to the zero-crossing RF phase, some concerns could be raised about the shot-to-shot jitter of the final beam energy, energy chirp and peak current, i.e. compression factor. It is shown below that such concerns are not justified when, e.g., stabilities typical of superconducting linacs are met. In the approximation of linear compression and for $C\gg1$, the relative variation of C is linearly proportional to the relative variations of R₅₆ and h, times C. In a periodic arc made of N DBAs we have

$$\begin{eqnarray} R_{56} &= N\int\limits_0^{l_{\text{b}}} {\frac{\eta_{x} (s)}{\rho }\text{d}s} =N\int\limits_0^\theta {\rho \left( {1-\cos \theta^{\prime}} \right)\text{d}\theta^{\prime}}= \nonumber\\ &\quad2Nl_{\text{b}}\left( {1-\frac{\sin \theta}{\theta}}\right)\propto \theta^{2}, \end{eqnarray} \tag{ 7 }$$

if $\theta\ll1$. Hence, we may estimate, $\Delta R_{56}/R_{56}=2\Delta \theta/\theta \le 2\times 10^{-4}$ where the bending angle relative stability is the same as the dipole magnetic field relative stability and all dipoles are powered by the same current source. At the same time $h\approx k_{\mathrm{RF}}eV\cos(\phi)/E_{\mathrm{i}}$ for $\phi \to 0$, with $k_{\mathrm{RF}}$ the RF wave number, V the RF peak voltage of the linac section imposing the energy chirp, ϕ the RF phase and $E_{\mathrm{i}}$ the beam mean energy at the linac entrance (maximum acceleration is for $\phi =90\ \text{deg}$). From this we get: $(\Delta h/h)_{V}\approx \Delta\phi tg(\phi)\le3\times 10^{-4}$ for RF set point $|\phi|<20\ \text{deg}$ and RF phase jitter $\Delta \phi \approx 0.05\ \text{deg}$, and $(\Delta h/h)_{\phi}\approx \Delta V/V\le 1\times 10^{-4}$. The uncorrelated sum of the jitters due to magnetic field, RF phase and RF peak voltage results in a peak current jitter smaller than 2% for $C= 45$. The beam mean energy jitter in the linac-chirper is also small, 0.02% rms at 2.4 GeV. This is equivalent to an arrival time jitter of 150 fs rms at the L-band linac entrance.

We have presented a 180 deg arc compressor that notably surpasses the performance, in terms of final beam quality, reported in previous designs. Such an improvement was made possible by a reformulation of the CSR-driven optics balance technique that takes into account the bunch length variations along the arc. The simulated performance of the periodic arc compressor confirms the analytical predictions by limiting the projected emittance growth (final minus initial value) of hundreds pC beams to $0.1\ \mu \text{m}$ level (normalized, rms value), in the presence of optical aberrations and CSR, for compression factors of up to 45 and final peak currents of up to 2 kA, at the beam energy of 2.4 GeV (see table 1). Similarly small emittance growth is obtained at 1.2 GeV with the same compression factor, although CSR-induced microbunching shows up with a deeper modulation of the longitudinal phase space and a final current modulation around $\pm 20{\%}$ (not shown). These tracking results must be treated as pessimistic estimates of the real beam quality: first, the smearing effect of transverse emittance on the microbunching [30,33,44], which may be particularly important as the bunch shortens in the last DBA cells, is ignored; second, the effect of numerical noise gradually diminishes as the number of particles is increased from 10⁵ to $5\cdot 10^{6}$; in addition, we have suppressed numerical noise at final wavelengths shorter than $1\ \mu \text{m}$, a value much shorter than those at which the instability develops $({>}5\ \mu \text{m})$. Finally, further optimization of the arc lattice with shorter drift sections and shorter dipole magnets is expected to reduce the CSR-induced microbunching.

The agreement of theory and simulations on the CSR-perturbed transverse emittance promises further reduction of the transverse CSR effect for smaller compression factors and/or improved tuning of the beam size in the dipole magnets (e.g., $\beta_{2}$ growing from the first to the last cell according to eq. (5)). The optics that minimizes the transverse CSR effect in the arc is typically in conflict with the one cancelling chromatic effects, in analogy with the conflict of low-emittance optics and chromaticity correction in storage rings. This fact leaves room for numerical optimization of both the arc lattice and its optics functions. Asymmetric arc designs with different optics arrangement can alternatively be considered [12], but we find that the periodic solution has several advantages: for example, energy dispersion leakage is easier to suppress if the dipole magnets are all identical and the optics is periodic. Optics symmetry allows tuning of the momentum compaction, thus of the compression factor, with equally spaced sub-families of quadrupoles (two families are available in the present design) and tune splitting to suppress coupling error effects ($\nu_{x}=7.82, \nu_{y} =3.41$ for the proposed arc). It also allows fast check of beam optics matching with screen systems along the line, as the beam sizes are the same at equivalent locations in the DBAs. Finally, periodicity of the lattice geometry allows cost-saving production of identical lattice elements (magnets, power supplies, diagnostics, etc.).

The authors are grateful to D. R. Douglas for encouragements and fruitful discussions on arcs design, and to Y. Jiao for discussions on emittance control in DBAs. The authors acknowledge that during the preparation of this paper, similar studies of isochronous arcs immune to CSR and microbunching instability, and of non-periodic arc compressors have been independently carried out by D. R. Douglas et al. This work was funded by the FERMI project and by the ODAC project of Elettra Sincrotrone Trieste.