Suppression of emittance variation in extremely low emittance light source storage rings

A passive method based on an optimized quasiachromatic condition effectively suppresses emittance variation during the independent tuning of insertion devices (IDs), which is one of the major obstacles for conducting precise experiments in extremely low emittance light source storage rings. This method, simple and easy to apply to any light source storage ring, reduces the emittance variation to potentially less than 1% by carefully optimizing dispersion leakage at the straight sections where the IDs are installed according to an actual condition of the ID independent tuning.

Figure 1

Definition of the objective function $F$ and division of the operational range of IDs.

Figure 2

The dependence of the optimum dispersion function ${\eta }_{\mathrm{opt}}$ on the number of divisions obtained from Eq. (27) when the natural emittance is 115 pmrad. $N$ on the horizontal axis is the maximum number of division points: $N=2,3,4,\dots$ corresponding to the case where the number of divisions is $1,2,3,\dots$, respectively.

Figure 3

Objective function $F$ (a) and its derivative with respect to the dispersion function ${\eta }_{x\_\mathrm{ST}}$ (b) for the case of $N=3$. The red dots indicate the local minimum point where $\frac{\partial F}{\partial {\eta }_{x\_\mathrm{ST}}}=0$.

Figure 4

Dependence of emittance (a) and effective emittance (b) on the average ID peak field for a given value of the dispersion ${\eta }_{x\_\mathrm{ST}}$ leaked into the straight section. The red solid curves show the minimized variations by the optimum value of ${\eta }_{\mathrm{opt}}=15.0\mathrm{mm}$. The black arrows represent a typical range of $B_{av}$ ($0.54\mathrm{T}\le B_{av}\le 0.74\mathrm{T}$) expected at SPring-8-II (see the next section for details).

Figure 5

Dependences of the suppression effect of emittance variation on natural emittance. The maximum (a) and relative (b) values of emittance variation are plotted. The red filled and blue open circles show calculation results under the achromatic and optimized quasiachromatic conditions, respectively. The green crosses in Fig. 5 show values for the optimum linear energy dispersion ${\eta }_{\mathrm{opt}}$ at the straight section under the quasiachromatic condition.

Figure 6

The same as Fig. 4 but for the natural emittance of 10 pmrad.

Figure 7

Dependence of the average effective emittance (a) and suppression effect of emittance variation (b) on the operating range of $B_{av}$. The vertical axis in the left figure represents the average of the effective emittance taken over a given range of $B_{av}$, and that in the right figure represents the normalized maximum variation of the effective emittance in the same range of $B_{av}$. The red filled circles, triangles, and squares indicate calculations under the achromatic condition for the ranges of $0.4\mathrm{T}\le B_{av}\le 1.0\mathrm{T}$, $0.8\mathrm{T}\le B_{av}\le 1.0\mathrm{T}$, and $0.5\mathrm{T}\le B_{av}\le 0.7\mathrm{T}$, respectively. The blue open circles, triangles, and squares indicate calculations under the optimized quasiachromatic condition for the ranges of $0.4\mathrm{T}\le B_{av}\le 1.0\mathrm{T}$, $0.8\mathrm{T}\le B_{av}\le 1.0\mathrm{T}$, and $0.5\mathrm{T}\le B_{av}\le 0.7\mathrm{T}$.

Figure 8

The variation of the peak magnetic field of thirteen undulators over ten days of user operation (a) and their average (b). The ID gap data for the present SPring-8 ring is scaled to the SPring-8-II ring, and the peak magnetic field is calculated.

Figure 9

The variation of the effective emittance for several values of the dispersion leakage. The peak field data in Fig. 8 are used in the calculation. The standard deviations for ${\eta }_{x\_\mathrm{ST}}=0$, 5, 10, 15, 20, and 25 mm are, respectively, 1.8, 1.7, 1.3, 0.6, 0.8, and 2.1 pmrad. The standard deviation is minimized with the optimized dispersion leakage of 15 mm.