Design of a triple-bend isochronous achromat with minimum coherent-synchrotron-radiation-induced emittance growth

Using a 1D steady-state free-space coherent synchrotron radiation (CSR) model, we identify a special design setting for a triple-bend isochronous achromat that yields vanishing emittance growth from CSR. When a more refined CSR model with transient effects is included in the analysis, numerical simulations show that the main effect of the transients is to shift the emittance growth minimum slightly, with the minimum changing only modestly.


I. INTRODUCTION
It is desirable for acceleration and transport of highbrightness electron bunches to occur without degradation of the beam quality. Unfortunately, a number of processes can spoil the beam transverse emittance and among these one of the most prevalent is coherent synchrotron radiation (CSR). As the electrons in a bunch travel through a bend, synchrotron radiation at the low end of the frequency spectrum is emitted coherently, perturbing the particle energy, inducing transverse offsets both in the spacial and angular coordinates, and therefore causing projected emittance growth.
One way to contain the adverse effects of CSR is to reduce overall bending; however, to eliminate bending altogether is usually not an option. For instance, in single-pass systems for free electron lasers (FELs) dipole magnets are required for bunch compression and often to distribute the electrons to off-axis beamlines. In multipass systems, such as energy recovery linacs, bending is integral to the machine topology.
Here we consider the problem of minimizing CSR effects on the transverse emittance in a triple-bend isochronous achromat, a lattice unit widely used in accelerator design. We adopt a 1D steady-state free-space model of CSR [1] and a method of analysis first introduced in [2] for the study of CSR in bunch compressors. We show that within the approximation of the model it is possible to specify a lattice design that yields vanishing CSR-induced emittance growth.
Our approach has some similarities with [3,4] and in particular [5]. We refer to the Introduction in [5] for a review of various approaches to the problem of minimizing CSR-effects on the emittance. For additional related work see also [6][7][8][9][10][11][12].

II. FORMALISM
Consider a dispersive beam line from s ¼ s i to s ¼ s f with bending occurring in the horizontal plane and no acceleration. In a 1D approximation, the effect of CSR on a particle of the bunch at location s along the beam line is to induce a relative-energy change δ s ðzÞ depending on the arclength coordinate s and particle longitudinal coordinate z.
In the linear approximation, the particle orbit in the horizontal plane following the energy kick evolves according to where R ij are the entries of the linear transport matrix while x i , x i 0 and x, x 0 are the particle coordinates at the entrance and exit of the beam line respectively. Notice that the entries R 11 , R 12 , R 21 , and R 22 are for the transport matrix from s i to s f , whereas the entries R 16 an R 26 are for transport starting from s i ≤ s ≤ s f , where the CSR energy kick occurs.
Integrating the effect of CSR through the whole dispersive section, the particle coordinates at the exit of the beam line become x ¼ x β þxðzÞ and If the beam is initially centered, hx i i ¼ hx i 0 i ¼ 0, where h·i represents averaging over the bunch population in phase-space, we can think of (xðzÞ,x 0 ðzÞ) as the centroid Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
coordinates of the beam thin slice centered at z. If the initial beam has vanishing transverse emittance (x β ¼ x β 0 ¼ 0) at the exit of the beam line the beam is represented by a onedimensional curve in the horizontal phase space described parametrically by the (xðzÞ,x 0 ðzÞ) pair.
Assuming that the beam at s i is centered, we average over the full 6D phase-space at s f to obtain the 2ndmoment of the beam distribution in the horizonal spacial coordinate: having introduced the notation Δx ≡x − hxi, and exploited the lack of correlation (justified in a 1D model) between the CSR-induced energy kick and the transverse coordinate, hx β Δxi ¼ 0. We have where λðzÞ is the beam longitudinal density normalized to unity. Similar expressions hold for hðΔx 0 Þ 2 i and hΔxΔx 0 i. From now on, to avoid notational clutter we will drop the parentheses and write hΔx 2 i instead of hðΔxÞ 2 i, etc. We now specialize these expressions to the case of interest here and assume that (i) the longitudinal density of the bunch is stationary or close to stationary, (ii) CSR is represented by a steady-state model neglecting transient effects through entrance and exit of the bends, and (iii) all bends (modeled in the hard-edge approximation) have the same nominal magnetic field. Under these assumptions the z dependence is factored out and we have dδ s ðzÞ ds ≡ h B ðsÞδ 0 ðzÞ where h B ðsÞ ¼ 1 for s in a bend and h B ðsÞ ¼ 0 elsewhere, and In the above expression R is the dipole radius of curvature, N b the bunch population, m the electron mass, r c the electron classical radius, and γ the relativistic factor. Notice that we are adopting the convention that a particle in the bunch head has z < 0; δ 0 ðzÞ > 0 denotes energy gain. The above expression (6) simplifies to where hΔδ 02 i ¼ hδ 02 i − hδ 0 i 2 ≡ ðdσ δ =dsÞ 2 is the square of the rms relative-energy spread per unit arc length induced by CSR along a dipole. Similarly The subscript "B" on the integral signs emphasizes that the integrals extend only over the bending magnets. The rms projected (geometric) emittance is defined as the determinant of the covariance matrix Observed at the exit s f of the beam line the emittance reads where ε 2 x0 is the beam unperturbed emittance, α x , β x , γ x are the Twiss functions at s f , and Notice that from (8), (9), and (10), it follows that hΔxΔx 0 i 2 ¼ hΔx 2 ihΔx 02 i and therefore Δε 2 x ¼ 0. We should point out that this is not a general result: it strictly depends on the assumption that the longitudinal beam density remains stationary, in which case the set (xðzÞ, x 0 ðzÞ) describes a zero-area straight segment in the horizontal phase space (see Fig. 5). In contrast, downstream of a magnetic chicane compressor Δε 2 x can be significant. We can reduce emittance growth by either choosing α x , β x , γ x appropriately (see Appendix A) or minimizing the second moments hΔx 2 i, hΔx 02 i and hΔxΔx 0 i. Equation (12) indicates no emittance growth if hΔx 2 i and hΔx 02 i can be set to zero simultaneously, i.e., Z As shown in the next section, in a three-bend isochronous achromat these integrals can indeed be made vanish simultaneously through second order in the bend angle.

III. APPLICATION TO 3-BEND ISOCHRONOUS ACHROMATS
Our midpoint symmetric beam line consists of three identical bending magnets equally spaced, see schematic in Fig. 1. We adopt a small bend-angle approximation for the transfer matrix of the three dipoles obtained from the analytical expressions in the hard-edge approximation [13] by Taylor expanding with respect to the dipole bending angle θ (while keeping L B ¼ R=θ constant). To lowest order and in the ultrarelativistic approximation the transfer matrix for motion in the horizontal and longitudinal planes reads To this order R B can be thought of as describing the action of either a rectangular or sector-bend magnet. Incidentally, notice that the entry for the momentum compaction is positive ðR B Þ 16 ¼ L B θ 2 =6 > 0, consistent with the convention that a particle in the bunch head has z < 0.
Of interest is also the transfer matrix through half of the dipole R B=2 , obtained from (16) by replacing L B → L B =2 and θ → θ=2.
In the most general terms we write the transfer matrix M from the exit of the first dipole to the entrance of the second as The transfer map for the whole achromat line can then be written as R s i →s f ¼ A II A I where A I and A II are the transfer matrices through the first and second half of the beam line. We have in particular and similarly A II ¼ R BM R B=2 , whereM has the same entries as M, but with m 11 and m 22 exchanged. The four entries of the M matrix are the variables available for minimizing the emittance growth, whereas the dipole magnet length L B and bending angle θ are set by the lattice designer by considerations unrelated to CSR. One degree of freedom is taken by imposition of the achromatic condition A I 26 ¼ 0 (i.e. vanishing derivative of the dispersion function at the midpoint. Because the dispersion function and its derivative are zero at the entrance of the beam line, midpoint symmetry and A I 26 ¼ 0 imply that they will vanish at the exit of the beam line as well). A second degree of freedom is needed to enforce isochronicity (A I 56 ¼ 0), and a third is taken by the symplectic condition. Remarkably, the one degree of freedom left is sufficient to satisfy both Eqs. (14) and (15).
In more detail, through second order in θ the achromatic, isochronous, and symplectic conditions imposed upon Eq. (18) translate into leaving one free parameter, e.g., m 21 . The next step is to work out the expressions for R We observe that choosing m 21 ¼ 15=2L B makes both (22) and (23) vanish. We should emphasize that in this case the model predicts no emittance growth regardless of the values set for the Twiss functions at the beam line ends.
The result can be generalized to triple bend achromats with midpoint symmetry but with the middle dipole having different length and bend angle from those of the other two dipoles. This relaxes the assumption that the radius of curvature be identical for all dipoles. The expression for m 21 that makes the two above integral vanish is FIG. 1. Schematic of a 3-bend achromat with midpoint symmetry consisting of three identical dipoles with L B length and θ bending angle. Indicated is our notation for the transfer matrices through specified beam line sections. The quadrupoles providing the necessary focusing between the dipoles are not shown.
where lL B and tθ are the middle dipole length and bend angle (L B and θ being length and bend angle of the two outer dipoles). The relevant entries of the transfer matrix for the whole achromat are then and R 12 ¼ ðR 11 R 22 − 1Þ=R 21 can be derived by the symplectic condition.

IV. NUMERICAL EXAMPLE
For illustration we discuss a numerical example loosely inspired by the lattice in the spreader design for the Next Generation Light Source [14], which motivated this study, see Table I.
We consider a beam with Gaussian density profile λðzÞ, rms length σ z , and specialize the calculation to the case of symmetric beta function through the beam line: To this end, we first write the transfer matrix through the whole beam line R s i →s f : Using the general parametrization for symplectic transport matrices in terms of Twiss functions [13] at the two beam line ends [see also Eq. (A3) in Appendix A], we write Having the freedom to set the phase we write We can now write an analytical expression for the emittance (12) as a function of the matrix entry m 21 ≡ ð1 þ μÞ15=2L B , with μ ¼ 0 corresponding to vanishing emittance growth: In the above expression is the rms energy spread per unit arc length induced by CSR in a bend on a Gaussian bunch [5,15]. We employed the macroparticle code ELEGANT [16] to simulate transport through a three-bend isochronous achromat invoking the same steady-state free-space 1D model of CSR adopted here. See Fig. 2 for the horizontal dispersion and beta function and Fig. 3 for a plot of R s→s f 16 and R s→s f 26 . For simplicity, we built the beam lines for each of the two sections between the dipoles using only 2 (focusing) quadrupoles. (In doing so we lose control of the matching in the vertical plane. Vertical motion, however, is irrelevant here and in the simulations we set the vertical emittance to zero).
The result from the ELEGANT simulations (red dots) and the expected emittance growth based on Eq. (32), red line, are shown in Fig. 4. As predicted by the model, essentially no emitttance growth is observed for μ ¼ 0. We should note that the various data points in the figure correspond to slightly different beam line lengths and different quadrupole settings, as needed to enforce the desired value of m 21 , while the dipole lengths and bend angles are kept fixed.
For comparison, we report the simulation results obtained using a more accurate 1D model of CSR accounting for transient effects (black dots) [17]; CSR was included for 4 m in the drift downstream of the dipoles. We observe that the main effect of the transients is to shift the minimum of the emittance growth slightly, with the minimum changing only modestly.

V. CONCLUSIONS
The main result of this paper is a demonstration of the existence of a triple-bend isochronous achromat design that displays virtually no projected emittance growth under the assumption of a 1D stead-state free-space model of CSR. The existence of this design setting is somewhat surprising as the problem is overconstrained; nonetheless, tuning of one independent variable turns out to suffice to satisfy both Eq. (14) and (15).
In a sense, this is the CSR equivalent of the theoretically minimum emittance (TME) lattice [13] minimizing emittance growth from incoherent synchrotron radiation.
We caution that the special design we discussed comes with somewhat extreme conditions on the lattice functions [α x ðs i Þ ¼ −α x ðs f Þ ≃ 35], which may be difficult to accommodate in practice, but we should add that use of unequal dipoles in the triple-bend achromat, see Eq. (24) would alleviate this difficulty.
We hope that our results may be helpful as a starting point in the search for lattice optimum designs and provide guidance on gauging the trade-off between tolerable emittance growth and desired settings of the lattice functions.

Work supported by Department of Energy Contract
No. DE-AC02-05CH11231. We thank P. Emma for useful discussions and review of the manuscript.

APPENDIX A: ALTERNATE METHOD TO MINIMIZE CSR-INDUCED EMITTANCE GROWTH
Suppose that at least one between hΔx 02 i and hΔx 2 i is nonvanishing, say hΔx 2 i ≠ 0. If we choose (see sketch in Fig. 5) we can write the second term in the right-hand side of (12) representing the CSR-induced emittance growth as Again, we remind the reader that α x , β x and γ x are the lattice functions at the exit of the achromat. The above equation predicts vanishing emittance growth in the limit β x → ∞. Depending on the specifics of the lattice this may be a practical way to minimize the emittance growth, but large values of the beta functions for a number reasons are often not desirable.
Observe, though, that unlike the method of Sec. III the quantities hΔx 02 i and hΔx 2 i representing the rms spread of the beam slices spacial and angular centroids remain finite, while for certain applications (e.g., FELs) it is important that the centroids of individual beam slices stay close to the axis in order to preserve good overall beam matching and therefore favoring hΔx 02 i ¼ hΔx 2 i ¼ 0. Setting aside the drawback posed by an outsized beta function [and possibly the difficulty to set α x appropriately, Eq. (A1)], could we then argue that this emittance minimization method is less preferable than the one discussed in Sec. III? Not necessarily, because as shown below, a large β x at the exit of the achromat will have the effect of reducing the spreads hΔx 02 i and hΔx 2 i as observed at the end of a transport line further downstream.
Consider the most general transfer matrix for transport in the horizontal plane from the exit of the achromat to, e.g., the entrance of an FEL undulator, parametrized in terms of Twiss functions at the exit of the achromat ðα x ; β x Þ and entrance of the FEL ðα F x ; β F x Þ: We use this matrix to propagate the rms spread of the slice centroids to the FEL: Indeed, these two expressions show that for finite α F x and β F x , taking the limit β x → ∞ has the effect of reducing to zero the spreads hΔx 2 F i and hΔx 02 F i of the slices centroids as observed at the FEL. Another way to state this is to say that one can always find a symplectic transfer matrix that maps all points in the grey segment in Fig. 5 into a small region arbitrarily close to the origin of the phase space. No area-preservation theorem is violated since the segment has zero area.

APPENDIX B: SELECT ENTRIES OF R-MATRIX
For completeness we report the explicit expressions valid through 1st order for the entries R