STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA

STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA

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Using this process to verify that the reflector is operating correctly, virtually no remnant coupling is observed through the back end of the recirculator. In fact, the coupling introduced from the RF cavities (see Sections 2.2.3 and 3.3) prove to be more of an operational concern [61]. 6.3.2 Measured Effect on the Threshold Current The threshold current was first established to be 1.8 mA and caused by the 2106 MHz mode without the reflector. The reflector was then activated and opera- tion with 5.0 mA of average beam current was achieved with no indications of BBU. Since direct observation was not possible, a BTF measurement of the 2106 MHz mode was performed. The results of the measurement yielded a new threshold cur- rent of (9.2 ± 0.4) mA; an increase by a factor of 5.1 (Fig. 6.8). Additional BTF measurements were performed for the 2116 MHz mode (also located in cavity 7) and the 2114 MHz mode (cavity 4). The concern was that by producing a reflection in the betatron planes, these modes which were stabilized for decoupled optics may become unstable. However, the results of the measurements indicate that the modes remained stabilized and did not pose a threat for causing the instability. 6.4 Global Rotation The reflector described in Section 6.3 interchanges horizontal and vertical phase spaces, imaging each identically into the other. This will map a horizontal kick into the vertical plane, and a vertical kick to the horizontal plane, and thus is effective for suppressing BBU driven by modes with these specific polarizations. However, it will equally map a diagonal kick back into a similar diagonal displacement, thus failing to suppress BBU driven by an HOM of arbitrary polarization. Therefore a more robust scheme for BBU suppression can be achieved by generating a 90 ◦ 149

FIG. 6.8: A plot of 1/Qeff versus average beam current for the 2106 MHz mode with the local reflector activated. The best fit line is used to determine that the threshold current is 9.2 mA. rotation of the betatron planes from cavity 7 back to itself. The 4×4 recirculation transfer matrix for a 90 ◦ rotation is completely coupled and the off-diagonal 2×2 matrices are of opposite sign ⎛ ⎜ ⎝ 0 M −M 0 ⎞ 150 ⎟ ⎠ (6.11) From Eq. (6.11), M12 = M34 = 0, M32 = −M14 and the threshold current given by Eq. (4.21) is infinite independent of the mode orientation, α. For a single mode the idea is conceptually simple and is illustrated in Fig. 6.9. If on the first pass an offending mode imparts an angular deflection α, to a bunch, then on the second pass (and after a 90 ◦ rotation), the resultant displacement will be orthogonal to the deflection. Thus the bunch will be unable to couple energy to the mode that caused the deflection. A rotation implemented in a two-pass system effectively

Using this process to verify that the reflector is operating correctly, virtually<br />

no remnant coupling is observed through the back end of the recirculator. In fact,<br />

the coupling introduced from the RF cavities (see Sections 2.2.3 and 3.3) prove to<br />

be more of an operational concern [61].<br />

6.3.2 Measured Effect on the Threshold Current<br />

The threshold current was first established to be 1.8 mA and caused by the<br />

2106 MHz mode without the reflector. The reflector was then activated and opera-<br />

tion with 5.0 mA of average beam current was achieved with no indications of BBU.<br />

Since direct observation was not possible, a BTF measurement of the 2106 MHz<br />

mode was performed. The results of the measurement yielded a new threshold cur-<br />

rent of (9.2 ± 0.4) mA; an increase by a factor of 5.1 (Fig. 6.8). Additional BTF<br />

measurements were performed for the 2116 MHz mode (also located in cavity 7)<br />

and the 2114 MHz mode (cavity 4). The concern was that by producing a reflection<br />

in the betatron planes, these modes which were stabilized for decoupled optics may<br />

become unstable. However, the results of the measurements indicate that the modes<br />

remained stabilized and did not pose a threat for causing the instability.<br />

6.4 Global Rotation<br />

The reflector described in Section 6.3 interchanges horizontal and vertical phase<br />

spaces, imaging each identically into the other. This will map a horizontal kick into<br />

the vertical plane, and a vertical kick to the horizontal plane, and thus is effective<br />

for suppressing BBU driven by modes with these specific polarizations. However,<br />

it will equally map a diagonal kick back into a similar diagonal displacement, thus<br />

failing to suppress BBU driven by an HOM of arbitrary polarization. Therefore<br />

a more robust scheme for BBU suppression can be achieved by generating a 90 ◦<br />

149

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