STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA
STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA
⎛ ⎜ ⎝ 0 M M 0 ⎞ 145 ⎟ ⎠ (6.10) The 2 × 2 sub-block transport matrix M is the same for both exchanges (x to y, y to x). Thus, such a reflector cleanly exchanges the horizontal and vertical phase spaces. To see how a reflection can be effective in suppressing BBU, consider Eqs. (6.10) and (4.21). Because M12 = M34 = 0, for a mode oriented at 0 ◦ or 90 ◦ the threshold current becomes infinite. However, if an HOM is rotated at an angle α, not equal to 0 ◦ or 90 ◦ , then the recirculated beam will not come back to the cavity with an angle (α + 90) ◦ and its projection on the HOM will be nonzero. To get an infinite threshold for all HOM polarizations requires that M32 = − M14. The statement concerning an infinite threshold current is made in the context for which Eq. (4.21) was derived, namely that only a single mode is present in the cavity. In reality, dipole HOMs come in pairs of orthogonal modes, and the feedback loop between the beam and the mode will be re-established through the coupled beam motions [72]. Nevertheless, some measure of suppression can be achieved by implementing such a scheme. 6.3.1 Implementing a Local Reflector A practical implementation of a local reflector using 5 skew quadrupoles has been non-invasively embedded in the 3F region of the FEL Upgrade Driver. Each skew quadrupole is simply a normal quadrupole which has been rotated 45 ◦ . Op- erationally, normal quadrupoles upstream and downstream of the module are used as betatron matching telescopes. These allow transverse matching of the phase spaces across the reflector so that the module remains transparent to the rest of the machine [87]. Beam envelopes through the 3F region are shown in Fig. 6.3. From an operational point of view, the local reflector is activated by first loading
FIG. 6.3: Beam envelopes (horizontal in red and vertical in blue) for the 3F region of the FEL with the five skew quadrupoles (blue) activated and illustrating the exchange of horizontal and vertical phase spaces. The central four normal quadrupoles (dotted) are de-excited during reflector operation and the upstream and downstream quadrupoles (black) are used as betatron matching telescopes. in the skew quadrupole strengths as determined from a model of the lattice. Directly translating the model values into the machine is typically sufficient to generate a reflection. To verify that the skew quadrupoles are correctly coupling the beam, four principle rays are launched through the local reflector module and the response of the downstream BPMs are monitored. Results of this process are illustrated in Figs. 6.4, 6.5, 6.6 and 6.7 [88]. Because the BPMs in the linac are not able to resolve two beam passes, the first several BPM readings are nonsensical. Figure 6.4 (Fig. 6.5) shows that a cosine-like (sine-like) trajectory in the horizontal plane is fully out-coupled into the vertical plane through the remaining 60 m of the recirculator to the beam dump. Likewise, Fig. 6.6 (Fig. 6.7) shows that a cosine-like (sine-like) trajectory in the vertical plane is fully out-coupled into the horizontal plane. Thus the skew quadrupole strengths are properly set. 146
- Page 113 and 114: which excites it. The BBU instabili
- Page 115 and 116: Equation (4.41) is a dispersion rel
- Page 117 and 118: FIG. 4.4: Output from MATBBU showin
- Page 119 and 120: FIG. 4.5: Setup for measuring cavit
- Page 121 and 122: Consequently, depending on the exte
- Page 123 and 124: The projection of the beam displace
- Page 125 and 126: TABLE 4.1: Experimental measurement
- Page 127 and 128: FIG. 4.10: A plot showing the effec
- Page 129 and 130: these cryomodules. Modes from these
- Page 131 and 132: CHAPTER 5 Experimental Measurements
- Page 133 and 134: threshold current - preferably with
- Page 135 and 136: occurred at approximately 2 mA of a
- Page 137 and 138: FIG. 5.5: FFT of a pure 2106.007 MH
- Page 139 and 140: FIG. 5.6: Illustration to show the
- Page 141 and 142: 5.4 Measuring the Threshold Current
- Page 143 and 144: for the HOM-beam system and is deri
- Page 145 and 146: FIG. 5.10: Schematic of the experim
- Page 147 and 148: FIG. 5.12: A plot of 1/Qeff versus
- Page 149 and 150: measured HOMs in zone 3, a BTF meas
- Page 151 and 152: FIG. 5.16: HOM voltage measured fro
- Page 153 and 154: FIG. 5.18: A plot of the three valu
- Page 155 and 156: the beam’s response in regions wh
- Page 157 and 158: CHAPTER 6 BBU Suppression: Beam Opt
- Page 159 and 160: FIG. 6.1: Schematic of a FODO cell
- Page 161 and 162: plane [85]. Equations (6.7) and (6.
- Page 163: 6.2.3 Discussion The method of poin
- Page 167 and 168: FIG. 6.6: Beam position monitor rea
- Page 169 and 170: FIG. 6.8: A plot of 1/Qeff versus a
- Page 171 and 172: ⎛ ⎞ ⎜ ⎝ 0 0 0 0 0 −1/K 0
- Page 173 and 174: FIG. 6.11: A plot of 1/Qeff versus
- Page 175 and 176: FIG. 6.12: Threshold current for no
- Page 177 and 178: FIG. 6.14: Threshold current utiliz
- Page 179 and 180: TABLE 6.1: Summary of the measured
- Page 181 and 182: CHAPTER 7 BBU Suppression: Feedback
- Page 183 and 184: FIG. 7.1: A schematic of the feedba
- Page 185 and 186: FIG. 7.4: A coaxial 3-stub tuner us
- Page 187 and 188: All of these considerations are con
- Page 189 and 190: FIG. 7.6: Generic layout for a feed
- Page 191 and 192: in Section 4.2.1, however, in the p
- Page 193 and 194: FIG. 7.7: The threshold current as
- Page 195 and 196: FIG. 7.8: Threshold current versus
- Page 197 and 198: FIG. 7.10: The threshold current as
- Page 199 and 200: CHAPTER 8 Conclusions The work pres
- Page 201 and 202: le were experimentally measured. Du
- Page 203 and 204: APPENDIX A The Pillbox Cavity Start
- Page 205 and 206: FIG. A.1: A pillbox cavity exhibiti
- Page 207 and 208: Ez(ρ, φ) = ψ(ρ, φ) = E0Jm(γρ
- Page 209 and 210: FIG. B.1: Relationship of the S-par
- Page 211 and 212: FIG. C.1: Impedance and frequency o
- Page 213 and 214: FIG. C.5: Impedance and frequency o
FIG. 6.3: Beam envelopes (horizontal in red and vertical in blue) for the 3F region of<br />
the FEL with the five skew quadrupoles (blue) activated and illustrating the exchange<br />
of horizontal and vertical phase spaces. The central four normal quadrupoles (dotted)<br />
are de-excited during reflector operation and the upstream and downstream quadrupoles<br />
(black) are used as betatron matching telescopes.<br />
in the skew quadrupole strengths as determined from a model of the lattice. Directly<br />
translating the model values into the machine is typically sufficient to generate a<br />
reflection. To verify that the skew quadrupoles are correctly coupling the beam,<br />
four principle rays are launched through the local reflector module and the response<br />
of the downstream BPMs are monitored. Results of this process are illustrated in<br />
Figs. 6.4, 6.5, 6.6 and 6.7 [88].<br />
Because the BPMs in the linac are not able to resolve two beam passes, the first<br />
several BPM readings are nonsensical. Figure 6.4 (Fig. 6.5) shows that a cosine-like<br />
(sine-like) trajectory in the horizontal plane is fully out-coupled into the vertical<br />
plane through the remaining 60 m of the recirculator to the beam dump. Likewise,<br />
Fig. 6.6 (Fig. 6.7) shows that a cosine-like (sine-like) trajectory in the vertical plane<br />
is fully out-coupled into the horizontal plane. Thus the skew quadrupole strengths<br />
are properly set.<br />
146
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