Genetic algorithm for chromaticity correction in diffraction limited storage rings

A multiobjective genetic algorithm is developed for optimizing nonlinearities in diffraction limited storage rings. This algorithm determines sextupole and octupole strengths for chromaticity correction that deliver optimized dynamic aperture and beam lifetime. The algorithm makes use of dominance constraints to breed desirable properties into the early generations. The momentum aperture is optimized indirectly by constraining the chromatic tune footprint and optimizing the off-energy dynamic aperture. The result is an effective and computationally efficient technique for correcting chromaticity in a storage ring while maintaining optimal dynamic aperture and beam lifetime.

Figure 1

The crossover operation takes the parent values $x_i^{(p1)}$ and $x_i^{(p2)}$ and a random number $q$, and returns two offspring values $x_i^{(c1)}$ and $x_i^{(c2)}$. Larger values of the parameter $\kappa$ result in “near-parent” offspring.

Figure 2

The mutator operation takes a random real $m$ and adjusts an offspring variable by an amount $M\times {\beta }_m(m)$. $M$ is a settable parameter which can be customized for each variable type. Larger values of $\kappa$ make large mutations less likely.

Figure 3

Dynamic aperture is calculated using an element-by-element tracking code. Particles are tracked for 200 turns. The aperture is found using a binary search for particle loss. In this plot $N_{\text{angle}}=7$.

Figure 4

Dynamic aperture for SLS upgrade at 0%, $-3\%$, and 3% energy offset. Dashed lines are dynamic aperture, solid lines are linear aperture. At the injection point ${\beta }_x=3.3\mathrm{m}$ and ${\beta }_y=6.5\mathrm{m}$. The maxima throughout the machine are ${\beta }_x=8.7\mathrm{m}$ and ${\beta }_y=11.7\mathrm{m}$.

Figure 5

Momentum aperture of SLS upgrade. Linear aperture is calculated using an element-by-element linearization of the optics. The 6D aperture is calculated with full 6D tracking including radiation damping and synchrotron oscillation for 2.5 periods of the linear synchrotron tune. The asymmetry of the 6D momentum aperture is the influence of nonlinear momentum compaction. The rf voltage is set such that the linear calculation gives a $\pm 5\%$ bucket.

Figure 6

Horizontal and vertical phase space portraits for SLS upgrade on-energy, and at $-3\%$ and $+3\%$ energy defect.

Figure 7

SLS upgrade: Chromatic footprint from $-5\%$ to $+5\%$ and ADTS along $x$ from the $-x$ DA to the $+x$ DA. Blue points are 1% increments. Resonance lines are labeled $(p,q,r,n)$ where $p{Q}_x+q{Q}_y+r{Q}_s=n$. Low order resonance lines and higher order lines are plotted. Higher order resonance lines excluded by periodicity 3 are not shown.

Figure 8

Dynamic aperture (DA) and linear aperture (LA) for CANDLE at 0%, $-2\%$, and 2% energy offset. At the injection point ${\beta }_x=3.3\mathrm{m}$ and ${\beta }_y=1.8\mathrm{m}$. The maxima throughout the machine are ${\beta }_x=9.1\mathrm{m}$ and ${\beta }_y=8.4\mathrm{m}$.

Figure 9

CANDLE: Chromatic footprint from $-3\%$ to $+3\%$ and ADTS along $x$ from the $-x$ DA to the $+x$ DA. ADTS is calculated with a small vertical offset to allow for accurate calculation of vertical tune. Higher order resonance lines excluded by periodicity 16 are not shown.

Figure 10

Momentum aperture and linear momentum aperture for one period of the CANDLE lattice. Linear momentum aperture is calculated from Twiss optics linearized about the on-energy particle. The rf voltage is set such that the linear calculation of the rf bucket depth is $\pm 3\%$.

Figure 11

Horizontal and vertical phase space portraits for CANDLE on-energy, and at $-2\%$ and $+2\%$ energy defect.

Figure 12

On-energy dynamic aperture for ideal lattice and 30 misaligned and corrected lattices. At the injection point ${\beta }_x=3.3\mathrm{m}$ and ${\beta }_y=6.5\mathrm{m}$. The maxima throughout the machine are ${\beta }_x=8.7\mathrm{m}$ and ${\beta }_y=11.7\mathrm{m}$.

Figure 13

Momentum aperture from 6D tracking for ideal lattice and 30 misaligned and corrected lattices.

Figure 14

Area of the $+y$ dynamic aperture versus percent beta beating induced by quadrupole gradient errors. The solid enclosure is the convex hull containing 50% of the seeds nearest the mean, the dashed enclosure contains 90% of the seeds.