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

A numerical method to design nonlinear double- and multibend 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, 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 double- and multibend achromat 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 National Synchrotron Light Source-II lattice, the single- and multibunch instability thresholds are increased and the bunch-by-bunch feedback gain can be reduced.

Figure 1

Schematic illustration of a rotating trajectory observed at a Pioncaré section with normalized coordinates ($\overline{x},{\overline{p}}_x$). The fluctuations of the action $J_x$ and phase advance ${\varphi }_x$ observed in multiturn tracking simulations are the objectives to be minimized. A similar picture applied in the vertical plane.

Figure 2

Schematic illustration of the fluctuation of actions $\mathrm{\Delta }{J}_x$ starting from different initial amplitudes. Usually the fluctuations increase gradually with the initial amplitude.

Figure 3

Schematic illustration of the spectrum obtained from turn-by-turn trajectory data $\overline{x}\pm i{\overline{p}}_x$. The ratio between the amplitudes of the two leading frequencies $\frac{A_2}{A_1}$ is the objective to be minimized to suppress the orbit tune diffusion.

Figure 4

Linear optics and magnet layout for one cell of the NSLS-II storage ring. The red blocks represent sextupoles. The strengths of six harmonic sextupoles were tuned during optimization of the nonlinear optics (Sec. 2).

Figure 5

DA of the DBA lattice with QI (left) and the NSLS-II lattice at nominal operation (right). The lattices have chromaticities of $\xi =7$ and $\xi =2$, respectively. Colors indicate the tune diffusion obtained with the NAFF technique.

Figure 6

Evolution of $J_{x,y}$ in the DBA lattice starting from five different initial conditions in the horizontal (left) and vertical (right) planes.

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.

Figure 8

Simulated trajectories of the DBA lattice starting from different initial conditions in the horizontal (left) and vertical (right) phase space. Within the DA, although the trajectories deviate from the Courant-Snyder ellipse, they are still confined to thin tori.

Figure 9

Tune footprint of the NSLS-II DBA lattice with QI (left) and the nominal lattice for routine operation (right). The design working point is $33.22/16.26$. Compared to the nominal lattice, a very large amplitude-dependent tune-spread is observed in the QI lattice, and various resonance lines can be crossed with narrow stop-band widths. Here colors indicate the tune diffusion, as in Fig. 5.

Figure 10

Comparison of the vertical action fluctuations in the QI lattice (left) and the nominal NSLS-II lattice (right). Starting from the same initial conditions $(x,p_x,y,p_y)$, the action in the QI lattice has smaller fluctuations, despite the fact that the lattice has a high chromaticity $\xi =7$. The same feature can also be observed for the horizontal plane.

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 $\xi =7$ and $\xi =2$, respectively.

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.

Figure 13

DA of the MBA lattice in the transverse $x-y$ plane colored with the tune diffusion.

Figure 14

Simulated trajectories of the MBA lattice in the horizontal (left) and vertical (right) phase space. The vertical trajectories begin to smear out from thin tori gradually, but some patterns are visible.

Figure 15

Large amplitude-dependent tune spread is observed in the MBA lattice constructed with QI.

Figure 16

Tori are near-circular only when observed at the long straight centers but could be distorted at other locations. The bottom four subplots are the tori observed at the locations “A-B-C-A,” respectively, as marked in the top subplot.

Figure 17

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 $\mu$.

Figure 18

Local momentum aperture of one supercell (including two mirror symmetric cells) for the constructed DBA (top) and MBA (bottom) lattices.

Figure 19

Adding a periodic phase dependence (top) to the action $J_x$ can reshape a circular torus into a triangle-shaped one (bottom).

Figure 20

Particle trajectories are confined to triangle-shaped tori, as confirmed by the tracking simulation. A nonsymmetric DA can be obtained with this technique.