Coupling computation

To achieve a strong coupling control, a precise computation and measurement of the coupling ratio are required. Beam emittances with linear coupling can be analyzed using perturbation theories. The coupling measurement considers the closest tune distance during tune-crossing predicted by the linear coupling resonances as the magnitude of the coupling coefficient κ, as an important coupling measure method, and the coupling ratio is therefore $\frac{{ϵ}_y}{{ϵ}_x}=\frac{{\kappa }^2}{{\text{Δ}}_{\text{r}}^2+{\kappa }^2}$, where ${\text{Δ}}_{\text{r}}=v_{0x}-v_{0y}-l$. However, when the linear coupling is sufficiently strong, these analyses of the beam parameters may not be accurate, and the envelope method based on the equilibrium beam distribution [31] provides accurate results for the emittances, despite the presence of a strong coupling. The equilibrium beam distribution is obtained via particle tracking considering radiation damping and quantum excitation as follows: $\text{Σ}=\left(\begin{array}{cccccc}\langle x^2\rangle & \langle x{p}_x\rangle & \langle xy\rangle & \langle x{p}_y\rangle & \langle xz\rangle & \langle x\delta \rangle \\ \langle p_{x}x\rangle & \langle p_x^2\rangle & \langle p_{x}y\rangle & \langle p_x{p}_y\rangle & \langle p_{x}z\rangle & \langle p_x\delta \rangle \\ \langle yx\rangle & \langle y{p}_x\rangle & \langle y^2\rangle & \langle y{p}_y\rangle & \langle yz\rangle & \langle y\delta \rangle \\ \langle p_{y}x\rangle & \langle p_y{p}_x\rangle & \langle p_{y}y\rangle & \langle p_y^2\rangle & \langle p_{y}z\rangle & \langle p_y\delta \rangle \\ \langle zx\rangle & \langle z{p}_x\rangle & \langle zy\rangle & \langle z{p}_y\rangle & \langle z^2\rangle & \langle z\delta \rangle \\ \langle \delta x\rangle & \langle \delta p_x\rangle & \langle \delta y\rangle & \langle \delta p_y\rangle & \langle \delta z\rangle & \langle {\delta }^2\rangle \end{array}\right)$ (1) The horizontal and vertical emittances, local beam tilt, and beam size were determined once the six-dimensional distribution matrix ∑ was calculated. The anti-symmetric matrix S is defined as follows: $S=\left(\begin{array}{cccccc}0& 1& 0& 0& 0& 0\\ -1& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& -1& 0& 0& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& -1& 0\end{array}\right)$ (2) The eigenvalues of ∑S are ±iεk, where εk indicates the equilibrium emittance of modes I, II, and III. The rotation of the phase space in the horizontal, vertical, and longitudinal planes provide the orientation of the normal modes compared with the axes of the physical system. Thus modes I, II, III are defined as the eigenmode motions in the “eigen-directions". In the simulation, the envelope matrix is computed by using the Accelerator Toolbox [32] function ohmienvelope developed by the formalism proposed by Ohmi, Hirata, and Oide and the Elegant function moment_out [33]. Referring to SSRF-U as the update of SSRF, a feasible and cost-effective plan is to employ skew quadrupole components induced by winding on a sextupole magnet. A total of 120 sextupole magnets of SSRF-U were grouped into families for chromaticity correction, with six sextupoles installed in each 7BA cell. Thus, the beam coupling was adjusted by manipulating 120 sextupole winding skew quadrupoles and then determining the corresponding accurate coupling ratio using the envelope method. Meanwhile, all chromaticity correction sextupoles were located in the dispersive region; therefore, these skew quadrupoles generated a significant vertical dispersion and strong betatron coupling, leading to a growth in emittances in mode II.

NSGA-II algorithm and objective functions

Genetic algorithms have been widely demonstrated to be useful techniques in many optimization problems of accelerators, especially those that are complex, such as non-linear optimization [33, 34]. NSGA-II is an evolutionary algorithm used for multi-objective optimization problems that can be applied to our study. It ranks solutions based on their non-dominated status using the Pareto dominance. It incorporates elitism to preserve the best solutions, uses genetic operators for exploration and exploitation, and maintains diversity along the Pareto front. NSGA-II converges towards the true Pareto front, providing decision-makers with a range of trade-off options. There are six sextupoles in each cell of the SSRF-U lattice, as shown in Fig. 1. A total of 120 independent skew quadrupole components coiling on two families of sextupoles in the dispersive region with a limitation of the magnet strength can be selected as a set of variables devoted to coupling control in a minimization algorithm.

Fig. 1Full-screen View

(Color online) Linear optics and structures of one SSRF-U storage ring lattice cell; the blue rows indicate the skew quadrupole (SQ) families in the dispersion-free region

Beause the goal of coupling control is apparently the coupling ratio standing the ratio of the vertical and horizontal emittances, the achievement of the eigenemittance coupling ratio defined by ε_II/ε_I is a simple task under global skew quadrupole settings, whereas the arbitrary change of skew quadrupoles can severely affect the beam parameters, such as beam optics, especially at a large coupling. Therefore, we considered the coupling ratio as a constraint, where the computed coupling ratio was close to the target coupling ratio; otherwise, the set of skew quadrupole components was considered invalid. Both eigenemittances, ε_I and ε_II, as well as the RMS beta-beating computed via the parameterization proposed by Edwards and Teng (ET) [36], were considered as the optimization objectives in NSGA-II. The constrained condition is defined as the difference between the expected and calculated coupling ratios, and the vector space for searching for the best solution is a 2-dimensional vector of skew quadrupole strengths. Each solution in the population is considered a child in the parameter space. In NSGA-II, we initialized a population of solutions, selected parents via a binary tournament, and then generated a child-simulated binary crossover (SBX). After non-dominated sorting, the best solutions were selected, and the features were inherited by the next generation. The crowding distance was used to maintain the diversity of the best solutions in the objective space.

Skew quadrupoles setting with optimization algorithm

A population of 100 children and 100 iterations was initialized for the optimization until the convergence was satisfactory, and a set of optimal configurations was obtained. A 5% to 20% target coupling ratio was used to test the reliability and fit various requirements for future lattice designs. To consider the lattice symmetry, two families of skew quadrupoles applied with their respective strengths were chosen to feed into the optimization algorithm. The strength of the normalized skew quadrupoles was limited to less than 0.1 m^-2.

The results of the calculation are shown in Table 2. A good agreement was achieved between the setting of the beam coupling and optimal results under a significantly low RMS beta-beating in both the horizontal and vertical directions to maintain a stable linear solution, indicating that we can achieve a large beam coupling without additional quadrupole adjustments for beam optics correction. Meanwhile, the total emittance of the beam increasing owing to the larger beam coupling can be interpreted as the vertical dispersion generated in the dispersion area. Therefore, if full coupling is required, additional large emittances nearly double the total emittances are unacceptable. A significantly fast convergence was also observed during the optimization; the best solution was obtained in a few iterations. Because the fractional tune changes owing to the optical distortion and can be corrected to the original tune by tuning the correction quadrupoles, tuning can avoid the difference in resonance, and the closed orbit distortions can be corrected owing to the unnoticeably sensitive coupled lattice.

Equilibrium emittances based on the tracking, global beam size, and effective emittances

The skew quadrupoles setting of the 10% coupling was examined by tracking using the Elegant code. A total of 5000 particles were tracked with 20,000 turns with synchrotron radiation and quantum excitation. The tracking results from Elegant demonstrate that both transverse-beam distributions reach equilibrium, and the horizontal and vertical emittances remain stable around the emittances predicted (ε_x=72.5 pm·rad, ε_y=7.3 pm·rad) with the aforementioned computations after 10,000 turns with no particle loss.

The beam size can be determined both by the computed beam envelope or by the Sands formalism [37] using the ET parameterization. Considering the given equilibrium emittances and energy spread, the beam size along the ring can be computed using the formula given by the linear coupled analysis [38] and proven in [30]. Here, ET parameterization is preferred owing to the negligible mode II β function under low optical distortion. Because all skew quadrupoles are situated in the dispersion area, the vertical dispersion is not negligible and contributes to the vertical beam size, as shown in Formula 3. ${\sigma }_{x\mathrm{,}y}=\sqrt{{\beta }_{x\mathrm{,}y}{\epsilon }_{x\mathrm{,}y}+{\sigma }_{\delta }^2{\eta }_{x\mathrm{,}y}^2}$ (3) In Fig. 2a and b, a comparison of the envelope method and Sands formalism for the beam-size computation of the 10% coupling setting demonstrates a good agreement between the two computation methods, and a significant vertical beam-size growth to approximately 7.5 μm, leading to an improvement in the Touschek lifetime. Figure 2b also shows that the vertical beam size is of the same order of magnitude as the horizontal beam size in a dispersion-free region.

Fig. 2Full-screen View

(Color online) (a) and (b) Comparison of the envelope method and Sands formalism with ET parametrization for beam-size computation of the 10% coupling setting. (c) Comparison of the effective vertical emittances, projected vertical emittances, projected betatron emittance on the vertical axis when the energy spread and dispersion effect are ignored

With the same goal of quantifying the inevitable dispersion effect on the emittances, the effective emittance defined [39] by Formula 4 can be computed. Considering the brilliance, the effective emittance which represents the total spread of circulating beams at a source point is a more crucial physical quantity. The vertical emittances from the contributions of the betatron coupling effect and vertical dispersion are also examined. $\begin{array}{c}{ϵ}_{x_{\text{eff}}\mathrm{,}{y}_{\text{eff}}}\left(s)\right)=\sqrt{{ϵ}_{x\mathrm{,}y}^2+{\mathcal{H}}_{x\mathrm{,}y}\left(s\right){ϵ}_{x\mathrm{,}y}{\sigma }_{\delta }^2}\\ {\mathcal{H}}_{x\mathrm{,}y}\left(s\right)={\beta }_{x\mathrm{,}y}(s){\eta }_{x\mathrm{,}y}^{\text{'}2}\left(s\right)\\ +2{\alpha }_{x\mathrm{,}y}\left(s\right){\eta }_{x\mathrm{,}y}\left(s\right){\eta }_{x\mathrm{,}y}^{\text{'}}\left(s\right)+{\gamma }_{x\mathrm{,}y}\left(s\right){\eta }_{x\mathrm{,}y}^2\left(s\right)\end{array}$ (4) Although the envelope method provides the projected emittances on the horizontal, vertical, and longitudinal axes whether or not the energy spread is present, the projected emittances vary along the storage ring and can be determined by the envelope matrix along the ring. Because there are beamlines using the bending magnet SR sources, the beam size and effective emittances in the arcs as well as the minimum brightness requirement must be considered and verified, respectively. Figure 2c shows that the results agree with the two methods and demonstrate the global dispersion effect on the vertical emittances in the matching section, which may cause an unwanted reduction in the brightness and other operational side effects.

Betatron coupling and vertical dispersion are simultaneously generated by the skew quadrupoles in the arc; therefore, it is difficult to suppress betatron coupling while maintaining the vertical dispersion, and vice versa, which may lead to significantly larger vertical emittances than expected in the arc. Therefore, the skew components in the dispersion-free region can be considered to solve this problem.

Skew quadrupoles setting with optimization algorithm

For the ultra-low-emittance of 4th generation synchrotron light sources, certain injection schemes have been proposed to match the significantly small dynamic aperture (DA), such as the swap-out injection, replacing the preceding pulsed-bump injection in 3rd generation light sources. The use of different injection schemes, divided into on-axis and off-axis injection, require various storage ring performances and beam parameters. Off-axis injection, which requires a bumped orbit, is impossible with a strong betatron coupling of the horizontal and vertical planes as in a Mobius accelerator, whereas vertical dispersion can generate unwanted vertical emittances in the case of a large coupling ratio. Thus, a large coupling induced mainly by betatron coupling is necessary and worth further consideration.

To prevent unwanted vertical dispersion, two additional skew quadrupoles (SQ family) were used in each straight section, as shown in Fig. 1, inspired by the generalization of the Mobius accelerator proposed by [13]. The skew quadrupoles with a length of 0.2 m were situated at the connection of the straight and matching sections to save the expensive space for insertion devices (Table 3).

All skew quadrupoles are powered independently; thus, 40 variables can be fed into the optimization algorithm. Instead of skew quadrupoles in the arc, emittances along the storage ring are well-distributed if the beta function distortion is under control. The objective function and constrained condition retained the total eigenemittances, RMS beta-beating, and coupling ratio. We initialized a population of 100 children in the NSGA-II algorithm for optimization.

The results of the optimization demonstrate that a large coupling can still be achieved under a low beta function distortion, whereas full coupling is achievable under a relatively large beta function distortion. In the full-coupling case, the average value of the vertical dispersion is less than 10^-4 m, which is negligible compared to that of the first scheme. The optimization process is illustrated in Fig. 3, which indicates that RMS beta-beating and eigenemittances can be simultaneously reduced with the diversity of the solutions, and finally converge. Because the transverse motion is coupled by skew quadrupoles, the transverse eigenemitances are equalized so as the radiation integrals and damping times respectively, leading to an increase in the total emittances.

Fig. 3Full-screen View

(Color online) Optimization process for the 100% coupling set, objective functions: eigenemittances (horizontal axis) and RMS beta-beating (vertical axis); the points are colored according to their rank from blue to red

The bare SSRF-U lattice without the insertion devices has a natural emittance of ${\epsilon }_x=72.4\text{ pm}\cdot \text{rad}$, and the transverse damping partitions are ${\mathcal{J}}_x=2.2675$ and ${\mathcal{J}}_y=1$. With the exchange process of transverse motion, the damping partition is equalized as follows [40]: ${\mathcal{J}}_{\text{I}}={\mathcal{J}}_{\text{II}}=\left({\mathcal{J}}_x+{\mathcal{J}}_y\right)/2=1.6338$. Thus emittances ${\epsilon }_{\text{I}}={\epsilon }_{\text{II}}={\epsilon }_x/\left(1+\frac{{\mathcal{J}}_y}{{\mathcal{J}}_x}\right)=50.24\text{ pm}\cdot \text{rad}$, which demonstrates a good agreement of our optimization for minimizing the emittances. The total emittances can remain constant only when ${\mathcal{J}}_x={\mathcal{J}}_y$; otherwise, brightness loss is inevitable in the round-beam scheme.

Equilibrium emittances based on tracking, global beam size, and effective emittances

The skew quadrupoles setting of 100% coupling was examined via tracking using the Elegant code. The tracking results show that both transverse beam distributions reach equilibrium, and that the horizontal and vertical emittances are equalized after 10,000 turns when reaching the equilibrium state with the predicted emittances and coupling ratio.

Because there was a relatively large optical distortion compared to that using the first skew quadrupole setting method, indicating that the mode II beta function was induced and needs to be computed, the Sands formalism using the ET parameterization may not be accurate as an approximation for computing the beam size. The Mais and Ripken (MR) parameterization [41] was applied in our computation as follows: ${\sigma }_{x\mathrm{,}y}=\sqrt{{\beta }_{\text{I}\mathrm{,}(x\mathrm{,}y)}{\epsilon }_{\text{I}}+{\beta }_{\text{II}\mathrm{,}(x\mathrm{,}y)}{\epsilon }_{\text{II}}+{\sigma }_{\delta }^2{\eta }_{x\mathrm{,}y}^2}\mathrm{.}$ (5) As shown in Fig. 4a and b, the apparent vertical-size increase is almost entirely contributed to the betatron coupling, whereas the contribution of the local vertical energy oscillation ${\eta }_y{\sigma }_{\delta }$ is negligible. Here, the mode II twiss parameter is given via MR parameterization.

Fig. 4Full-screen View

(Color online) (a) and (b) Comparison of the envelope method and Sands formalism via MR parametrization for beam-size computation of the 100% coupling setting. (c) Comparison of the effective vertical emittances, projected vertical emittances, and projected betatron emittance on the vertical axis when the energy deviation and dispersion effect are ignored

Figure 4c demonstrates that the periodicity of the lattice is not broken under the 40 independent skew quadrupoles setting, and the three lines perfectly coincide with one another, indicating that there was no contribution of the vertical dispersion on this fully coupled lattice scheme. Therefore, compared to the dispersive coupled scheme, more uniform vertical emittances along the ring are obtained even when the energy spread is present.

Dynamic Aperture

A sufficient dynamic aperture along with the injection and beam lifetime must be considered in the strong coupling lattice. The tracking results for checking the DA degradation are shown in Fig. 5. For the 10% coupling set, a slight reduction in DA after the linear chromaticities correction was observed compared to the bare lattice. For the 100% coupling set, the reduction in DA was apparent; however, it still reached a satisfactory value for the on-axis injection. The objective of the NSGA algorithm is to achieve a low RMS beta-beating of the coupled lattice because a large DA is expected to be maintained when the optical distortion is small. However, clear DA degradation was observed in the 100% coupling set, which indicates that a strong coupling effect may cause a non-linear effect owing to the exchange of energy in the horizontal and vertical planes.

Fig. 5Full-screen View

(Color online) Dynamic aperture after correction of the chromaticities with two strong coupling setting schemes compared to original lattice

Energy acceptance

To study the effects that directly lead to beam-loss and therefore a reduced Touschek lifetime, the energy acceptance (EA) for each case of the coupling set was computed. The EA values for the two schemes and the design lattice are shown in Fig. 6a. Degradation of the EAs was observed for each case, especially for the 100% betatron coupling set, leading to a significant reduction in the Touschek lifetime, thus influencing our efforts in improving the beam lifetime by introducing a strongly coupled lattice in vain. The Betatron coupling motion exchanges particle motion in the transverse plane and may excite unwanted resonance and non-linear effects. Octupoles can control the high-order chromaticity terms. With three families of octupoles in the dispersion region, the EA can be optimized by adjusting the position and strength of octupoles according to the amplitude-dependent tune shift (ADTS). The preliminary optimization using the octupoles shown in Fig. 6b did not demonstrate a significant improvement in the low-energy acceptance region; however, an improvement in the global EA was observed.

Fig. 6Full-screen View

(Color online) (a) Energy acceptance of the two strong coupling setting schemes compared to the original lattice; (b) energy acceptance of the 100% coupling set compared to the energy acceptance after octupole optimization

Intra-beam scattering and beam lifetime

IBS and multiple small-angle Coulomb scattering events among the electrons in a bunch lead to electron diffusion in six-dimensional phase spaces. The Elegant code IbsEmittance, developed by implementing the Bjorken-Mtingwa model [39], was used to evaluate the influence of IBS. Based on the analysis of the Bjorken-Mtingwa model, we can conclude that reducing the density of electrons is an appropriate means of weakening the IBS effects. The total transverse emittances increased when the current increased, and the IBS effect of the high beam current was reduced in both coupling sets. Although both coupling schemes provided extra emittances, the total emittances were lower than those of the bare SSRF-U lattices when the beam current was relatively large, according to the calculation of the IBS effect.

Touschek scattering is large-angle Coulomb scattering between the electrons in a beam bunch that causes a momentum transport from the transverse motion to the longitudinal direction. In our case, the Touschek lifetime was calculated using the energy acceptance. Longitudinal stretching by a factor of five can be achieved with a harmonic cavity and applied in the computation of our beam lifetime. The beam lifetime was increased by a factor of 2 to 2.5 under a beam current of 500 ma with 500 bunches, ranging from 0.49 h for the bare lattice, 0.98 h for the 10% coupling set, and 1.21 h for the 100% coupling set, whereas lower total emittances owing to the suppression of the IBS effect in the high-current case and apparent degradation of the energy acceptance are demonstrated in the coupled schemes.

Effect of lattice imperfections

The error lattice was examined to estimate the performance of the global skew quadrupole setting under real-machine conditions. Focusing on the coupled elements, the roll errors on the quadrupoles and displacements of the sextupoles are applied in the error lattice. Fifty random-error seeds gradually increased from 500 μrad to 1000 μrad quadrupole roll errors, and the corresponding 500 μm to 1000 μm vertical offsets of the sextupoles were applied in the two schemes with different error strengths. Figure 7 demonstrates that the global skew quadrupoles setting can remain stable despite relatively large error seeds, proving the robustness of these schemes. On-resonance round-beam schemes always suffer from the challenge of a precise tune control, as well as an unstable beam size and emittances; conversely, this off-resonance scheme presents reduced complexity of the control and a more describable model for a detailed study.

Fig. 7Full-screen View

(Color online) Eigenemittance of the coupled lattice with imperfections. The error bar indicates a standard deviation of 50 random seeds. (a) 100% coupling set by betatron coupling; (b) 10% coupling set with vertical dispersion