Reduction and compensation of the transient beam loading effect in a double rf system of synchrotron light sources

Double rf systems are used to lengthen the beam bunches in synchrotron light sources. In such a system, the performance of the bunch lengthening is limited by the transient beam-loading effect, which is induced by gaps in the fill pattern. To improve its performance, we investigate an application of a normal-conducting harmonic cavity, which is based on the TM020 resonant mode. By using the TM020 mode with low $R/Q$ and high Q, fluctuation of the rf voltage due to the transient beam loading can be reduced significantly. The remaining small fluctuation of the rf voltage can be compensated by using an active feedforward technique. Using these measures, we expect to realize a bunch-lengthening performance that is comparable to that obtained with superconducting cavities under realistic operational parameters of a proposed 3-GeV next-generation light source. We estimate the bunch-lengthening performances using macroparticle tracking simulations together with semianalytical calculations.

Figure 1

Inner shape (in $z\text{−}r$ plane) of the 1.5-GHz TM020-mode cavity. Only a half of the cavity is shown. Lines and circles indicate the electric and magnetic fields, respectively.

Figure 2

Typical phasor diagrams showing rf voltages in the harmonic cavities. (a) At a maximum beam current of 500 mA, (b) at a beam current of 100 mA. Assumed parameters are given in Sec. 2c. Note that the direction of the beam induced voltage at resonance, ${\tilde{V}}_{\mathrm{br}}$, is opposite to that of the $-{\tilde{i}}_{\mathrm{b}}$.

Figure 3

Results of the tracking simulations showing rms bunch lengths along the bunch trains. Abscissa shows the index of the rf buckets. The upper inset indicates the standard fill pattern of the ring. Symbols in different colors indicate the bunch lengths under the conditions: (magenta) without bunch gap or harmonic cavities, (green) without bunch gap and with the third harmonic cavities, (red) with bunch gaps of 60 ns and using the TM020-mode harmonic cavities, and (blue) with bunch gaps of 60 ns and using BESSY-II-type TM010-mode harmonic cavities.

Figure 4

Variations in the third-harmonic rf voltage due to bunch gaps, which were calculated using the semianalytical method (red) and tracking simulation (blue), respectively. Pictures (a) and (b) show the amplitude and phase, respectively. Abscissa indicates the index of the rf buckets, which ranges over one of the bunch trains. We assumed a beam current of 500 mA with the standard fill pattern of the KEK-LS (two bunch trains, $n_{\mathrm{t}}=446$, $n_{\mathrm{g}}=30$).

Figure 5

Comparison of the bunch lengths obtained from the semianalytical calculation (red) and the tracking simulation (blue). Rms bunch lengths along one of the bunch trains are shown. We assumed the same parameters as those in Fig. 4.

Figure 6

Typical longitudinal distributions of particles that were predicted by the semianalytical calculation (red) and the tracking simulation (blue). Abscissa indicates the time delay ($=\varphi /{\omega }_{\mathrm{rf}}$). Assumed parameters are the same as in Fig. 5.

Figure 7

Estimated fluctuations in the harmonic rf voltage as a function of total $R_n/Q$. We assumed bunch-gap durations of 30, 60, and 120 ns, and a beam current of 500 mA. The symbols indicate the results for the 1.5-GHz harmonic cavities given in Table 4. Each dashed line represents the curve which fit the results for many sets of parameters with a fixed bunch gap.

Figure 8

Definition of the maximum deviation, $|\mathrm{\Delta }{\tilde{V}}_{\mathrm{c}}{|}_{\max }$. Red symbols indicate the harmonic voltages on the complex plane during a period of fill pattern. We assumed a stored current of 500 mA and two 60-ns bunch gaps.

Figure 9

Rms bunch lengths along the bunch trains with (circles) and without (triangles) the compensation. The bunch lengths are shown at every five buckets. The half bandwidth of the compensation was determined to be 1.1 MHz for both the main and harmonic cavities. We assumed a stored current of 500 mA, two 60-ns bunch gaps, and the parameters of the KEK-LS.

Figure 10

Rf amplitudes and phases as a function of the bucket index. Solid and dashed lines indicate those with and without the compensation, respectively. The upper two and lower two graphs indicate the main and harmonic rf voltages, respectively.

Figure 11

Required power for the generator ($P_{\mathrm{g}}$) and reflected power from the cavity ($P_{\mathrm{r}}$) during a beam revolution period. Both values are for a single cavity. Solid and dashed lines indicate those with and without the compensation, respectively. Upper two graphs are for the main cavity while the lower two are for the harmonic cavity.

Figure 12

Rms bunch lengths along the bunch trains with compensation (circles) using a separate cavity when compared to those without compensation (triangles). We assumed a stored current of 500 mA, two 60-ns bunch gaps, and a compensation half-bandwidth of 3 MHz.