Design of double- and multi-bend achromat lattices with large dynamic aperture and approximate invariants

A numerical method to design nonlinear double- and multi-bend achromat (DBA and MBA) lattices with approximate invariants of motion is investigated. The search for such nonlinear lattices is motivated by Fermilab's Integrable Optics Test Accelerator (IOTA), whose design is based on an integrable Hamiltonian system with two invariants of motion. While it may not be possible to design an achromatic lattice for a dedicated synchrotron light source storage ring with one or more exact invariants of motion, it is possible to tune the sextupoles and octupoles in existing DBA and MBA lattices to produce approximate invariants. In our procedure, the lattice is tuned while minimizing the turn-by-turn fluctuations of the Courant-Snyder actions $J_x$ and $J_y$ at several distinct amplitudes, while simultaneously minimizing diffusion of the on-energy betatron tunes. The resulting lattices share some important features with integrable ones, such as a large dynamic aperture, trajectories confined to invariant tori, robustness to resonances and errors, and a large amplitude-dependent tune-spread. Compared to the nominal NSLS-II lattice, the single- and multi-bunch instability thresholds are increased and the bunch-by-bunch feedback gain can be reduced.


I. INTRODUCTION
The Integrable Optics Test Accelerator (IOTA) [1], whose design is based on an integrable Hamiltonian system with two invariants of motion [2,3], paves the way for a new class of highly nonlinear storage rings. Experiments using a lattice design with one invariant of motion have also been performed, both at IOTA and in the University of Maryland Electron Ring (UMER) [4]. In each case, the lattice is tuned to provide one or more analytically known invariants of motion, resulting in a dynamic aperture (DA) that is large and robust to the presence of resonances.
The storage rings used as dedicated synchrotron light sources are designed in a different way: a linear achromat lattice with a desired beam emittance is designed first, and then the nonlinear dynamics is optimized with sextupoles and/or octupoles. The nonlinear magnets are often tuned to control the low order resonance driving terms of the one-turn map [5] to obtain sufficient dynamic aperture. Under these conditions it is generally difficult, if not impossible, to optimize the nonlinear dynamics to produce a one-turn map with one or more exact invariants. However, it is sometimes possible to produce approximate invariants, or quasi-invariants (QI), in these achromat lattices. This paper describes a procedure for designing near-integrable double-bend achromat (DBA) * email: yli@bnl.gov † currently at Michigan State Uni. ‡ email:chadmitchell@lbl.gov and multi-bend achromat (MBA) lattices with two QI. The motivation for constructing such lattices is that, although they are not completely integrable, the DA is large and robust to the presence of resonances. While crossing the resonance lines, their stop-band widths are observed to be narrow. Like nonlinear integrable lattices such as the one at IOTA, these lattices also have a large amplitude-dependent betatron tune-spread which can increase instability and space charge thresholds due to improved Landau damping [6,7]. As a result, the requirements on the feedback system's gain can be reduced significantly. This research was motivated by related studies such as the square matrix method [8] and the constant Courant-Snyder invariant method [9,10].
The remainder of this paper is outlined as follows: Section II explains the concept of Poisson-commuting invariants in integrable Hamiltonian systems, and describes a numerical approach for optimizing the nonlinear lattice to produce approximate invariants using symplectic tracking. Sections III and IV describe the properties of two such lattices, both of which have been constructed: the existing National Synchrotron Light Source-II (NSLS-II) DBA storage ring, and a diffractionlimited MBA storage ring (whose design is preliminary). Some detailed studies of the DBA lattice are described in Sect. V. Section VI describes the simulation of a kicked beam to illustrate the decoherence effect that results from a large nonlinear tune-spread. Some discussion and a brief summary are given in Sect. VII. A technique to modify the action-like invariants to reshape the invariant tori (and the resulting DA) is described in the Appendix.

II. LATTICE DESIGN PROCEDURE
A Hamiltonian system is Liouville integrable if it possesses a maximal set of independent Poisson commuting invariants of motion. For a system described by a symplectic map on a phase space of dimension 2n, this means that there exist n functions f j (j = 1, . . . , n) on the phase space such that: i) each f j is invariant under the map, ii) the Poisson brackets satisfy [f i , f j ] = 0, and iii) the set of gradient vectors {∇f j : j = 1, . . . , n} is linearly independent [11,12]. The behavior of trajectories for a completely integrable system is well-known, i.e., all its trajectories are confined to tori with well-defined and stable tunes.
By ignoring radiation and longitudinal acceleration, a charged particle's transverse motion in a storage ring is a 4-dimensional Hamiltonian system, described by a symplectic one-turn map M. If we let the canonical coordinates of the system be denoted z = (x, p x ; y, p y ), a quantity f (z) is an invariant of the map M if: If two such invariants f i , (i = 1, 2) exist, if they are independent: and if they Poisson-commute: the lattice is Liouville integrable. When the map M is linear and uncoupled, the Courant-Snyder actions J x and J y form the most commonly-used Poisson commuting pair of invariants, where: in the horizontal plane, with a similar expression for J y .
Here α x , β x , and γ x are the horizontal Twiss parameters [13] at the longitudinal location where the Poincaré section is observed. The canonical action-angle coordinates are (Φ x , J x , Φ y , J y ), where Φ x,y denotes the betatron phase in each plane, and the one-turn map is determined by the phase advance completed in a single revolution: with a similar expression for φ y . Here, k is the integer part of the betatron tune. The phase advance values φ x , φ y are independent of the actions J x and J y . In a realistic storage ring, once the linear lattice and the nonlinear magnet locations are fixed, the one-turn map M depends on the nonlinear magnet strengths K i , with i ≥ 2. It is difficult, if not impossible, to tune the K i so that M possesses even one exact invariant. However, we can imitate the linear case by constructing a nonlinear system in which the Courant-Snyder actions J x , J y form a pair of approximate invariants, as illustrated in Fig. 1. Unlike the linear case, however, the phase advance values φ x and φ y can depend on the actions J x and J y . The procedure is as follows. To optimize the behavior of the Courant-Snyder action J x within the available DA, multiple particles with different values of J x,0 are launched. Element-by-element tracking of this set of particles is used to compute the turn-by-turn evolution of J x . The tracking is implemented with a kick-drift symplectic integrator [14] to preserve the geometry of the Hamiltonian system. The available nonlinear knobs are simultaneously tuned to minimize the turn-by-turn fluctuations of J x for each particle, as illustrated in Fig. 2.
At the same time, we minimize the turn-to-turn variations of the horizontal phase advance. Instead of directly calculating the phase advance for each turn with Eq.(5), the complex turn-to-turn evolution ofx ± ip x [15] was analyzed in the frequency domain. One reason for using such a spectral method is to determine whether the fractional tune is below or above the half integer. The amplitudes of the two leading frequencies were computed utilizing the Numerical Analysis of Fundamental Frequencies (NAFF) technique [16]. By tuning the nonlinear knobs, the ratio between the two leading frequencies r = A2 A1 was minimized. As a consequence, the smaller amplitude frequencies were also suppressed ( Fig. 3). In principle, the summation of all non-fundamental peaks should be compared against the fundamental frequency peak, but the computational cost will become high. Since NAFF outputs the peaks in descending order, we only use the two leading peaks to get a quick but rough approximation. This procedure is performed independently for several initial conditions of varying amplitude. As a result, the tune diffusion of each particle is suppressed, but the tunes may be amplitude-dependent. The same procedure is repeated for the vertical plane. . Schematic illustration of the spectrum obtained from turn-by-turn trajectory datax ± ipx. The ratio between the amplitudes of the two leading frequencies A 2 A 1 is the objective to be minimized to suppress the orbit tune diffusion.
Since the goal is to minimize the fluctuations of four different quantities for different initial conditions simultaneously, the construction of such a nonlinear lattice becomes a typical multi-objective optimization problem: • given a set of nonlinear knobs K i within their allowed ranges; • subject to some constraints, such as maintaining certain desired chromaticities; • simultaneously minimize the objective functions, i.e., ∆Jx,y Jx,y and r x,y = A2;x,y A1;x,y of multi-particles launched from different initial conditions.
Multi-objective optimization techniques are now widely used in the accelerator community. Here, the nondominated sorting genetic algorithm (NSGA-II) [17] was used. Five virtual particles with gradually increasing initial values of J 0;x,y were launched, and for each initial condition, four objectives were used. The total number of objectives was therefore 5 × 4 = 20.
Thus far, we have only discussed uncoupled linear lattices. When linear coupling is present, a different parameterization, such as the one described in [18], is needed.

III. APPLIED TO DOUBLE-BEND ACHROMAT
In this Section, we introduce a nonlinear DBA lattice for the NSLS-II main storage ring [19], which is presently in operation at Brookhaven National Laboratory. It is a 3 rd generation medium energy (3 GeV) light source. The storage ring's lattice is a typical DBA structure with its main parameters listed in Table I. Its linear optics for one cell is illustrated in Fig. 4. The whole ring is composed of 30 such cells. In this configuration, three families of chromatic sextupoles are used to correct its chromaticity to +7. Then, six families of harmonic sextupoles in dispersion-free sections are used as tuning knobs for the multi-objective optimization described in Section II. Below, we present the nonlinear lattice performance of an optimized solution using the tracking simulation code elegant [20]. All the tracking simulations in this paper were performed with this code unless stated otherwise. Fig. 5 illustrates on-momentum dynamic aperture (DA through 1,024 turns of particle tracking) observed at the center of the long straight section, i.e., s = 0 in Fig. 4. Each stable initial condition is colored with its tune diffusion log 10 (∆ν 2 x + ∆ν 2 y ) [16] obtained from turn-by-turn data using the NAFF algorithm. The nominal operation lattice's DA at chromaticity ξ = 2 is also shown for comparison. Although stronger sextupoles are needed to correct the chromaticity in the QI lattice, the obtained DA is comparable with the nominal operation lattice.  The turn-by-turn evolution of the Courant-Snyder actions J x,y for particles at 5 distinct amplitudes are shown in Fig. 6. The size of the visible fluctuations increases gradually with the amplitude of the initial condition. The spectral analysis of tracking data using the NAFF technique indicates that some non-dominant frequencies gradually become stronger as well. The amplitude ratio between the two leading frequencies increases as shown in Fig. 7. One of the features of an integrable system is that the trajectories are confined to tori in the phase space. This Figure 7. Ratio of the two leading NAFF components increases with the initial particle amplitude in the horizontal (left) and vertical (right) planes for the DBA lattice.
is apparent in the turn-by-turn tracking data shown in Fig. 8. Although trajectories begin to gradually deviate from the Courant-Snyder ellipse when the amplitude increases, they are still confined to deformed tori. It therefore appears that this lattice possesses two QIs whose values near the reference orbit are quantitatively close to the Courant-Snyder actions. This feature can also be observed from the symmetry of DA in the horizontal plane as shown in Fig. 5. Like the IOTA ring, this lattice also provides a large amplitude-dependent tune-spread as illustrated in the left subplot of Fig. 9. This property can increase instability and space charge thresholds through improved Landau damping. Even while crossing the low order resonance lines such as ν x = 1/3, the stop-band widths are observed to be much narrower than those in conventional nonlinear lattices [21,22]. The nominal operation lattice's tune footprint (as shown in the right subplot) can provide a smaller tune spread within the bunch when an instability occurs.
It is interesting to directly compare the fluctuations in actions for the QI lattice and the nominal lattice. Although their DAs are comparable, the fluctuations ∆J in the QI lattice are better suppressed as illustrated in Fig. 10. When a small physical aperture exists in a ring, such as an undulator's gap, a trajectory with large action variation might be blocked. Then its realistic DA will be smaller than the physical aperture. In this case, a less fluctuating trajectory will be preferable. Starting from the same initial conditions (x, px, y, py), the action in the QI lattice has smaller fluctuations, despite the fact that the lattice has a high chromaticity ξ = 7. The same feature can also be observed for the horizontal plane.
The robustness of this lattice has been confirmed with a beam test. After loading its sextupoles settings, a near 100% off-axis injection efficiency was achieved, which indicates its DA is sufficient for routine operation. Then, the horizontal DA was scanned by kicking the beam transversely to observe its loss rate as shown in Fig. 11. Although the measured DA is about 10 ± 1 mm, which is worse than the simulation prediction due to various realistic errors, it still can satisfy the requirement (> 8.5 mm) for off-axis injection. The nominal operation lattice's DA is also shown for comparison.
A bunch-by-bunch feedback (BBFB) system is frequently used in light source rings to suppress instabilities in the beam centroid motion. Limiting the BBFB gain is critical to avoid a degradation of the beam brightness. For example, a significant vertical emittance increase was observed when an excess gain was applied [23] at the NSLS-II ring. Although the beam centroid instabilities could be well suppressed with the BBFB system, the emittance blowup effect due to a large feedback gain was pronounced, especially at higher chromaticity.
In this study, after accumulating a beam current of 400 mA, the BBFB system was re-optimized to reduce its gain gradually. Each bunch's turn-by-turn data was used for a spectral analysis to identify the appearance of sidebands of betatron motion caused by various instabilities. Compared with the NSLS-II lattice at nominal operation, the required gain for the nonlinear QI lattice was reduced by 50% and 75% in the horizontal and vertical planes, respectively. This appears to be due to the increase in nonlinear tune-spread and chromaticity.
The single bunch instability threshold was also increased in the QI lattice. For the nominal lattice, the charge threshold is around 2-3 mA without using BBFB, and 5-6 mA with BBFB [24]. For the QI lattice, this threshold was observed to be greater than 8 mA without BBFB, and 12 mA with BBFB. The actual threshold might be higher, because the single bunch accumulation was interrupted by high outgassing due to the extensive single bunch radiation pulse. Figure 11. Measured dynamic aperture of the DBA lattice with QI (yellow) and the NSLS-II lattice at nominal operation (blue). The lattice chromaticities are ξ = 7 and ξ = 2, respectively.

IV. APPLIED TO MULTI-BEND ACHROMAT
Low-emittance light source ring design is now entering a new era. Various MBA-type lattices already reach diffraction-limited horizontal emittances to deliver much brighter X-ray beams. Like the DBA case, it is interesting to explore whether it is possible to design a nonlinear MBA lattice with two QIs. The ESRF-EBS type hybrid MBA lattice [25] has been widely adopted by other facilities. It is also being considered as one of the options for future NSLS-II brightness upgrade. A preliminary 7-BA design is shown in Fig. 12, which illustrates the linear optics for one cell. Note that several reverse bends are incorporated [26,27]. The main parameters are listed in Table II.  Figure 12. Linear optics and magnet layout for one cell of an ESRF-EBS type hybrid 7BA lattice. The red blocks represent sextupoles. Six chromatic sextupoles grouped into five families inside two dispersive bumps are used to correct the chromaticity with three extra degrees of freedom. Four harmonic sextupoles and four dispersive octupoles (green blocks) are also available for nonlinear dynamics optimization.
A two-stage optimization has been implemented. First, the settings of the chromatic and harmonic sextupoles were optimized to correct the chromaticities, to minimize the fluctuations of the Courant-Snyder actions, and to maximize the ratio of the two leading frequency components. After this procedure, four octupoles inside the dispersive bumps were optimized to further minimize these objectives. The resulting DA is shown in Fig. 13. The MBA lattice was found to be more challenging than the DBA lattice in constructing QI. The Courant-Snyder actions have larger fluctuations after optimization, especially in the vertical plane. Nevertheless, particle orbits are still confined to tori in the horizontal plane, as seen in Fig. 14. More importantly, a large amplitudedependent tune-spread is observed within the tune footprint of the stable DA, as shown in Fig. 15.  Large amplitude-dependent tune-spread is observed in the MBA lattice constructed with QI.
At large amplitudes, the trajectories in the vertical phase space significantly deviate from the Courant-Snyder ellipse (Fig. 14). However, this does not necessarily indicate that the tori are broken, as the 2D projections of 4D tori are observed. A more detailed analysis, for example, expanding the QI expression to high order polynomials for large-amplitude orbits in the vertical plane might be considered needed.

V. DEPENDENCE ON LATTICE LOCATION
In the previous examples, the optimization procedure of Sect. II was performed using tracking data at a specific lattice location, e.g., at the center of the straight section. Here, the orbits lie on invariant tori that are nearly circular when projected into thex −p x andȳ −p y planes. However, it is not surprising to observe that the invariant tori are distorted at other locations in the lattice. Figure 16 shows an example of distorted tori observed at a location inside an achromat of the NSLS-II DBA lattice. The one-cell map behaves like an "optical anastigmat", which has circular tori only at the periodic locations, i.e., its entrance and exit. Elsewhere in the lattice, distorted (perhaps broken) tori can be observed.

VI. DECOHERENCE DUE TO NONLINEARITY
At the IOTA ring, its large amplitude-dependent tunespread was found to provide a stabilization mechanism due to improved Landau damping [28]. Since constructed lattices with QI also provide a large nonlinear amplitude detuning, we used the DBA lattice to implement a 4dimensional multi-particle simulation to illustrate this nonlinearity. A large tune spread due to the nonlinearity of betatron oscillations can be indirectly observed by kicking a bunched beam transversely [29]. A Gaussian distributed bunched beam was simulated by 2,000 macroparticles with the horizontal beam size σ x . After being kicked to an amplitude Z = aσ x , the beam centroid will decay to the origin as the betatron phases of particles at different amplitudes decohere, which is described by the decoherence factor, Here, N is the number of turns after being kicked, θ = 4πµN , and µ(Z) = dν da 2 is the local amplitude detuning coefficient at Z. The phase space distribution of the beam filaments from a localized bunch to an annulus which occupies all betatron phases and the observed centroid of the beam will show a decaying oscillation as seen in Fig. 17. From the decoherence factor curve (the red line in Fig. 17), the local tune-shift-with-amplitude coefficient can be determined and confirmed to be consistent with the spectrum analysis of the single particle tracking data.

VII. DISCUSSION AND SUMMARY
So far, only the on-momentum particle motion has been described. In practice, a sufficient off-momentum acceptance is required as well. In the two examples of Sect. III-IV, on-momentum lattices were constructed with two quasi-invariants, and then the off-momentum acceptances were checked with tracking simulations. Both lattices were found to have sufficient local momentum aperture for the Touschek lifetime as shown in Fig. 18. In case the momentum acceptance also needs to be optimized, the method can be expanded to include a set of off-momentum particles. For an off-momentum particle, the dispersive reference orbit needs to be computed first, and then the Courant-Snyder actions J x,y in (4) need to be computed using the momentum-dependent Twiss parameters α(δ, K i ), β(δ, K i ) and γ(δ, K i ). Note that the momentum-dependent Twiss parameters depend on the nonlinear magnets' excitation values K i . For a given nonlinear magnet setting, the reference closed orbit and the corresponding momentum-dependent Twiss parameters need to be updated prior to the normalization.
In this paper, we demonstrated that a conventional DBA or MBA lattice can be re-tuned to possess two approximate invariants by optimizing the settings of only the sextupoles and octupoles. While the resulting DA is large (but finite), most particle trajectories are regular and confined to tori, and the amplitude-dependent betatron tunes are well-defined and stable. Like the lattice of the IOTA ring, a large nonlinear tune-spread exists that can provide enhanced Landau damping. The stop-band widths are observed narrow while the tune crosses low order resonance lines.  After being kicked, the centroid of the beam will show a decaying oscillation. The red line is known as the decoherence factor in Eq. (6), and can be used to determine the local amplitude detuning coefficient µ.
invariant J x to obtain triangle-shaped tori, where δ = ∆Jx Jx and n = 3. The phase dependence (top) of J x and the expected torus in the horizontal phase space (bottom) are illustrated in Fig. 19.
After modifying the optimization procedure to minimize the fluctuations of (7) in the DBA lattice, a new set of sextupole settings were obtained. Tracking simulation confirmed that the tori have been reshaped as expected (Fig. 20). Such a phase space manipulation technique might be useful when a non-symmetric DA is desired. For example, a slightly larger inboard DA might be preferred to capture off-axis injected beams from that region. Particle trajectories are confined to triangleshaped tori, as confirmed by the tracking simulation. A nonsymmetric DA can be obtained with this technique.
However, the phase dependence of (7) can introduce additional frequency components into the betatron spectrum. For example, a weak component sitting near 1/3 is visible through the NAFF analysis. If the betatron phase dependence is weak, it does not appear to spoil the overall performance of the nonlinear lattice. Note that the triangle-shaped tori are not caused by the betatron tune's approaching a third-order resonance, because the main betatron tune is around 0.22, and still far away from 1/3.