Designing linear lattices for round beam in electron storage rings using the solution by linear matrices analysis

For some synchrotron light source beamline applications, a round beam is preferable to a flat one. A conventional method of obtaining a round beam in an electron storage ring is to shift its tune close to a linear difference resonance. The linearly coupled beam dynamics is analyzed with perturbation theories, which have certain limitations. In this paper, we adopt the solution by linear matrices (SLIM) analysis to calculate exact beam sizes to design round beam lattices. The SLIM analysis can deal with a generally linearly coupled accelerator lattice. The effects of various coupling sources on beam emittances and sizes can be studied within a self-consistent frame. Both the on- and off-resonance schemes to obtain round beams are explained with examples. The SLIM formalism for two widely used magnet models: combined-function bending magnets, and planar wigglers and undulators, is also derived.

Figure 1

NSLS-II DBA lattice magnet layout in a supercell. The beam sizes are represented with the 2nd moments in the SLIM analysis (dots) and the Courant-Snyder parameters (lines) for the uncoupled motion agrees with each other. Note that the vertical axes’ scales and units for the horizontal (left) and vertical (right) beam sizes are different.

Figure 2

Beam sizes obtained with the SLIM analysis (dots) and Ripken parametrization (line) in the presence of linear coupling. The locations of two skew quadrupoles are marked. By assigning them the same $|K_1|$ value, but alternating polarities along the ring, the vertical emittance and overall beam sizes will blow up. The dotted lines are the second moments obtained with the SLIM analysis, and the solid lines are computed with the coupled Twiss functions in Ripken parametrization. Nevertheless, for both cases, the equilibrium emittances ${\epsilon }_{I,II}$ and energy spread ${\sigma }_{\delta }$ need to be computed with the SLIM analysis first.

Figure 3

The correlation between the transverse emittances and the skew quadrupoles’ strength $|K_1|$. The vertical emittance (the dot-dash line) grows faster than the reduction of the horizontal emittance (the dashed line). The total emittance is not constant when the coupling becomes strong, which leads to the breakdown of Eq. (1).

Figure 4

Vertical beam size decomposition: coupled vertical dispersion (1) generates a considerable amount of mode II emittance contribution, which contributes ${\beta }_{II,y}{\epsilon }_{II}$ (blue), (2) introduces local energy oscillation, which contributes ${\eta }_y^2{\sigma }_{\delta }^2$ (yellow); (3) the coupled ${\beta }_{I,y}$ function couples some mode I emittance, which contributes ${\beta }_{I,y}{\epsilon }_I$ (green).

Figure 5

Correlation between the emittances and RMS vertical orbit distortions $y_{\mathit{co},\mathit{rms}}$. With gradually increasing orbit distortion, a very flat beam becomes a near-round one. When the closed orbit distortion is sufficiently large, it becomes flat (and tilted) but with larger emittances. The total emittance is observed to be blown up, albeit rather slowly.

Figure 6

Emittance variation with quadrupole roll errors when the machine’s tune ($0.22/0.26$) is off a difference resonance.

Figure 7

On-resonance scheme: correlation between the beam eigenemittances with r.m.s. quadrupole roll errors when the ideal machine’s tune is on a difference resonance. Each error bar is the standard deviation for 50 random seeds.

Figure 8

On-resonance scheme: correlation between the quadrupole roll errors and beam sizes at short straight centers when the ideal machine’s tune is on a difference resonance. The error bar is the standard deviation of 50 random seeds.

Figure 9

On-resonance scheme: four coupled $\beta$ functions of one supercell for one random seed with $500\mu \mathit{rad}$ quadrupole roll errors. The local ${\beta }_{(I,II),x}$ at short straight sections are tuned close to ${\beta }_{(I,II),y}$, to obtain a geometric round beam there.

Figure 10

On-resonance scheme: horizontal and vertical beam sizes of one supercell for one specific random seed with $500\mu \mathrm{rad}$ quadrupole roll errors. Such data can be used for the beam lifetime estimation.

Figure 11

On-resonance scheme: round beam profile can be maintained with gradually increased closed orbit errors when the machine’s tune stays near a difference resonance.

Figure 12

Off-resonance scheme: horizontal and vertical beam sizes of one supercell.

Figure 13

Off-resonance scheme: ratio of the vertical and horizontal beam sizes varies with the vertical closed orbit distortions. The initial ratio of transverse beam sizes is ${\sigma }_y/{\sigma }_x\approx 0.97$, then a closed orbit that is increased gradually can slightly stretch the beam $x-y$ profile vertically. Each error bar represents the standard deviation of 20 random seeds.

Figure 14

Schematic diagram of a hybrid flat-round beam mode in a storage ring. A pair of solenoids are used to form a local closed section with a round beam (red arc), while maintaining a flat beam in the rest part of the ring (blue arc.)

Figure 15

Optimized on- and off-momentum dynamic apertures for the off-resonance scheme as listed in Table 2.