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FIG. 5.6: Illustration to show the effect of sideband frequencies driving otherwise stable
modes unstable.
for which, as previously mentioned, the sidebands produced by the 2106 MHz mode
lie exactly at 1786.206 MHz and 1881.481 MHz.
Figure 5.7 shows a plot of the HOM voltage squared of four modes in zone 3.
The simulation was performed with an average beam current that exceeds the thresh-
old current. Therefore the voltage corresponding to the 2106 MHz mode grows
rapidly. The other modes plotted correspond to 1786 MHz, 1881 MHz and 2114 MHz
(which is pseudo-stable because M ∗ sin(ωTr) > 0). While they exhibit growth, the
magnitude of their voltages are 14, 12 and 9 orders of magnitude, respectively, less
than that of the 2106 MHz mode after 15 ms.
Figure 5.8 shows the results of repeating the simulation while changing only the
bunch frequency from 1497 MHz to 37.425 MHz. Modes that correspond to the side-
band frequencies generated by the 2106 MHz mode are now resonantly driven. After
15 ms, the magnitude of the voltages for the 1786 MHz and 1881 MHz modes are
now only 5 and 3 orders of magnitude, respectively, less than that of the 2106 MHz
mode while the magnitude of the voltage for the 2114 MHz mode remains nearly 9
orders of magnitude less than that of 2106 MHz. Thus only HOM frequencies that
correspond to these sideband frequencies are affected.
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