STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA
STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA
to a condition where M ∗ sin(ωTr) < 0 and the system is pseudo-stable. Fourth, the machine lattice can be changed, which amounts to modifying the quantity M ∗ defined in Eq. (4.19). The use of beam optical suppression techniques is the topic of this chapter and methods for modifying the properties of the mode are covered more fully in Chapter 7. Methods to manipulate the transverse beam optics in order to suppress BBU were first presented in 1980 [84]. The strategy of beam optical control techniques is to modify the machine lattice in such a way that the beam cannot couple as effectively to the dangerous dipole mode. This can be achieved with point-to-point focusing, reflecting the betatron planes about an axis that is at 45 ◦ between the vertical and horizontal axes and a 90 ◦ rotation. While the ability of point-to-point focusing to increase the threshold current was demonstrated at the SCA [11] and at MUSL-2 [27], the latter two methods, which require introducing strong betatron coupling into the system, had never before been tested experimentally. In 2005, the ability to raise the threshold current by each of these methods was successfully demonstrated in the FEL Driver, and these methods are described in the following sections. 6.2 Point-to-Point Focusing Because it does not involve complicated transverse coupling schemes, point-to- point focusing was the first beam optical suppression technique employed to combat the effects of BBU in the SCA and at MUSL-2. With a judicious change in the betatron phase advance, point-to-point focusing can be achieved (M12 or M34 = 0) at the location of the cavity containing a dangerous mode so that an HOM-induced kick on the first pass results in a zero displacement on the second pass. In this 139 way the beam cannot transfer energy to the mode by coupling to the electric field
FIG. 6.1: Schematic of a FODO cell of length ℓ. because it has no on-axis component. In the FEL Driver, point-to-point focusing is achieved by utilizing the properties of the FODO channel in the 3F region of the recirculator. 6.2.1 Implementing Point-to-Point Focusing Consider the transfer matrix for a FODO cell of length ℓ depicted in Fig. 6.1 140 ⎛ ⎜ 1 ⎝ 0 ℓ 2 1 ⎞ ⎛ ⎟ ⎜ ⎠ ⎝ 1 1 f ⎞ ⎛ 0 ⎟ ⎜ 1 ⎠ ⎝ 1 0 ℓ 2 1 ⎞ ⎛ ⎟ ⎜ 1 ⎠ ⎝ − 0 1 f ⎞ ⎟ ⎠ 1 (6.1) Carrying out the matrix multiplication in Eq. (6.1) and equating it with the most general form of a unit matrix using the Twiss parametrization gives [38] ⎛ ⎜ ⎝ 1 − ℓ 2f − ℓ 2f 2 ℓ2 − 4f 2 ℓ 1 + ℓ 4f 1 + ℓ 2f ⎞ ⎟ ⎠ = ⎛ ⎜ ⎝ Equating the trace of each matrix leads to cos ∆ψ + α sin ∆ψ β sin ∆ψ 1 − ℓ2 ∆ψ = cos ∆ψ = 1 − 2 sin2 8f 2 2 −γ sin ∆ψ cos ∆ψ − α sin ∆ψ ⎞ ⎟ ⎠ (6.2) (6.3)
- Page 107 and 108: The threshold is inversely proporti
- Page 109 and 110: 4.3 BBU Simulation Codes: Particle
- Page 111 and 112: 6. The second pass beam bunch then
- 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: CHAPTER 6 BBU Suppression: Beam Opt
- 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 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
FIG. 6.1: Schematic of a FODO cell of length ℓ.<br />
because it has no on-axis component. In the FEL Driver, point-to-point focusing is<br />
achieved by utilizing the properties of the FODO channel in the 3F region of the<br />
recirculator.<br />
6.2.1 Implementing Point-to-Point Focusing<br />
Consider the transfer matrix for a FODO cell of length ℓ depicted in Fig. 6.1<br />
140<br />
⎛<br />
⎜ 1<br />
⎝<br />
0<br />
ℓ<br />
2<br />
1<br />
⎞ ⎛<br />
⎟ ⎜<br />
⎠ ⎝<br />
1<br />
1<br />
f<br />
⎞ ⎛<br />
0<br />
⎟ ⎜ 1<br />
⎠ ⎝<br />
1 0<br />
ℓ<br />
2<br />
1<br />
⎞ ⎛<br />
⎟ ⎜<br />
1<br />
⎠ ⎝<br />
−<br />
0<br />
1<br />
f<br />
⎞<br />
⎟<br />
⎠<br />
1<br />
(6.1)<br />
Carrying out the matrix multiplication in Eq. (6.1) and equating it with the most<br />
general form of a unit matrix using the Twiss parametrization gives [38]<br />
⎛<br />
⎜<br />
⎝<br />
1 − ℓ<br />
2f<br />
− ℓ<br />
2f 2<br />
ℓ2<br />
−<br />
4f 2 <br />
ℓ 1 + ℓ<br />
<br />
4f<br />
1 + ℓ<br />
2f<br />
⎞<br />
⎟<br />
⎠ =<br />
⎛<br />
⎜<br />
⎝<br />
Equating the trace of each matrix leads to<br />
cos ∆ψ + α sin ∆ψ β sin ∆ψ<br />
1 − ℓ2<br />
∆ψ<br />
= cos ∆ψ = 1 − 2 sin2<br />
8f 2 2<br />
−γ sin ∆ψ cos ∆ψ − α sin ∆ψ<br />
⎞<br />
⎟<br />
⎠<br />
(6.2)<br />
(6.3)
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