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suppress BBU. As mentioned in Section 6.4.2, a true 90 ◦ rotation was not achieved in the FEL Upgrade. For the sake of the simulations, a fictitious optics was generated to simulate the effect of a rotator such that a 90 ◦ rotation is achieved from cavity 7 back to itself. As the mode polarizations deviate from 0 ◦ and 90 ◦ , a rotation remains more effective at suppressing BBU than the reflector. Discussion Several important conclusions can be drawn from the aforementioned simulations. The first is that varying the angle at which the two orthogonal polarizations deviate from 0 ◦ and 90 ◦ did not improve the effectiveness of the suppression techniques. In fact, in the special case of the local reflector, as the deviation increased the threshold current became rapidly smaller, from approximately 25 mA to 3 mA when the rotation angle was 45 ◦ . Although such a decline does not occur with the rotator, neither does changing the polarization angle bring any significant advantage. The primary objective of this simulation study, however, was to bring attention to the fact that HOM orientations can play an important role in the choice of suppression techniques, which until recently, has been overlooked [89]. One of the salient conclusions is that a global rotation is clearly a more robust suppression technique for arbitrarily polarized higher-order modes. This can be understood by recalling that the threshold current is inversely proportional to M ∗ , defined in Eq. (4.19). For a local reflection, M14 = M32 which will increase M ∗ . A rotator on the other hand, with M14 = −M32 will, in general, tend to decrease the value of M ∗ and thereby increase the threshold current. It is important to keep in mind that these simulations were performed for the Jefferson Laboratory FEL Upgrade Driver, which is comprised of 3 cryomodules 159 with a small number of dangerous HOMs which are well separated in frequency. As
TABLE 6.1: Summary of the measured effects of suppression techniques on the 2106 MHz mode. Suppression Technique Effect on 2106 MHz mode Point-to-Point Focusing Stabilized Local Reflector 5.1 × Ith Rotator Stabilized the number of HOMs increases and/or better damping is obtained, the likelihood of overlapping modes increases and optical suppression techniques become less effective. In fact, for large accelerators with many cryomodules, using these beam optical schemes in the presence of many HOMs can cause further destructive mode interference thus rendering these suppression techniques ineffective altogether. Recently, it has been proposed to fabricate elliptical RF cavities to deliberately break the cylindrical symmetry, thereby lifting the dipole HOM degeneracy and gen- erating a large frequency spread between the two polarizations [90]. Assuming the dipole modes are sufficiently separated in frequency from cavity to cavity, then using elliptical cavities in conjunction with coupled optical suppression techniques can be effective at increasing the threshold current in machines with many cryomodules [91]. 6.5 Summary Beam optical suppression techniques proved to be very effective at increasing the threshold current in the FEL Upgrade Driver. A summary of the various meth- ods and their effect on the 2106 MHz mode is displayed in Table 6.1. Using point-to-point focusing and the rotator, the 2106 MHz mode could be stabilized. However since point-to-point focusing cannot be arranged for each indi- vidual cavity in an accelerator with an extended linac, the usefulness of this method 160 is restricted to smaller machines or machines where dangerous HOMs are well local-
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STUDIES OF ENERGY RECOVERY LINACS A
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DEDICATION To my wife Danielle and
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2 CEBAF with Energy Recovery . . .
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5.2 HOM Power . . . . . . . . . . .
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ACKNOWLEDGMENTS First and foremost
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6.1 Summary of the measured effects
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2.10 Illustration of quadrupole sca
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5.1 Successive frames in time (prog
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6.8 A plot of 1/Qeff versus average
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ABSTRACT An energy recovering linac
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CHAPTER 1 Introduction An increasin
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FIG. 1.1: Schematic of a generic li
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FIG. 1.2: A CEBAF 5-cell cavity wit
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The solution to Eq. (1.3) is U(t) =
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y reducing the impedance of HOMs, a
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Despite its success, this method of
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design parameters, most notably ach
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1.4.2 Machine Optics The second cat
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analytic model elucidates many impo
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CHAPTER 2 CEBAF with Energy Recover
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FIG. 2.1: Energy versus average cur
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FIG. 2.3: Additional hardware insta
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FIG. 2.4: A picture of the energy r
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dipoles and beam diagnostics such a
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FIG. 2.7: Horizontal (red) and vert
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FIG. 2.8: Illustration of the cryom
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linac and θNL is the RF phase. The
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2.4 Transverse Emittance One of the
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where σ2 is the rms beam size meas
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eams. The effects of varying the qu
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FIG. 2.12: A typical wire scan near
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quadratic fit and a multiple regres
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ting the data is difficult. Without
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primary source of error is measurin
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identified, although the phase dela
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TABLE 2.3: Comparison of Twiss para
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the results of the fits. The vertic
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FIG. 2.18: Schematic illustrating t
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FIG. 2.19: The GASK signal measured
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FIG. 2.20: The measured normalized
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CHAPTER 3 The Jefferson Laboratory
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FIG. 3.1: Schematic of the 10 kW FE
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FIG. 3.2: Layout of the DC photocat
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accelerating gradient at the front
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eason for making the endloops achro
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FIG. 3.7: Illustration of path leng
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3.5 Longitudinal Dynamics This sect
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FIG. 3.9: The effect of a thin focu
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Under the constraint that each orde
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form of beam breakup not only occur
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4.1 The Pillbox Cavity Although the
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FIG. 4.2: Electric field (red) and
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where the full 4×4 transfer matrix
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The threshold is inversely proporti
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4.3 BBU Simulation Codes: Particle
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6. The second pass beam bunch then
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which excites it. The BBU instabili
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Equation (4.41) is a dispersion rel
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FIG. 4.4: Output from MATBBU showin
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FIG. 4.5: Setup for measuring cavit
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Consequently, depending on the exte
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The projection of the beam displace
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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 and 164: 6.2.3 Discussion The method of poin
Page 165 and 166: FIG. 6.3: Beam envelopes (horizonta
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: FIG. 6.14: Threshold current utiliz
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
Page 215 and 216: BIBLIOGRAPHY [1] M. Tigner, Nuovo C
Page 217 and 218: [22] L. Merminga, in Proceedings of
Page 219 and 220: [50] C. Hernandez-Garcia et al., in
Page 221 and 222: [79] L. Merminga et al., in Proceed
Page 223 and 224: VITA Christopher D. Tennant Christo
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