STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA
STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA
The bunch length, energy spread and centroid energy at the end of the arc (denoted by the subscript a) can be written in terms of parameters at the end of the linac as ℓa = ℓl + M56 ∆E + T566 E l 2 ∆E E l 77 (3.9) ∆Ea = ∆El (3.10) Ea(z = 0) = El(z = 0) (3.11) By combining Eqs. (3.6), (3.7), (3.8) with Eqs. (3.9), (3.10), (3.11) the bunch length, energy spread and centroid energy at the undulator (denoted by the subscript u) can be written in terms of bunch parameters from the injector ℓu = ℓinj + M56 ∆ERF Emax + T566 2 ∆ERF Emax (3.12) ∆Eu = ∆ERF (3.13) Eu(z = 0) = Emax (3.14) Recall that the goal is to take the long bunch with low momentum spread and with an appropriate choice of optics, rotate the phase space to produce a short bunch at the undulator. That is, M56 and T566 are chosen to minimize the bunch length ℓu. For small values of ℓ the expression for ∆ERF can be expanded to second order in ℓinj to yield ∆ERF ElinackRF ℓinj sin φo − kRF ℓinj cos φo 2 Plugging Eq. (3.15) into Eq. (3.12) and collecting like powers of ℓinj results in 1 + M56kRF + ℓ2 k inj −M56 2 RF 2 ℓu = ℓinj Elinac Emax Elinac Emax sin φo cos φo + T566k 2 RF Elinac Emax 2 sin 2 φo (3.15) (3.16)
Under the constraint that each order vanishes, the desired M56 and T566 are found to be M56 = − λRF 2π T566 = − λRF 4π π = − λRF Emax Elinac Emax Elinac cos φo sin φo 1 sin φo 2 cos φo M 2 56 sin 3 φo 78 (3.17) (3.18) For a bunch operating at 10 ◦ off-crest, Emax = 145 MeV and Elinac = 135 MeV, the required M56 is −0.2 m and T566 is −3.5 m. In practice the first endloop utilizes trim quadrupoles to generate an M56 of +0.3 m (as opposed to the inherent +0.2 m of the endloop) [61]. This compaction, in addition to that of the optical magnetic chicane (−0.5 m) create the necessary conditions outlined above. In the FEL Upgrade typical rms bunch lengths obtained at the undulator are 200 fs in accord with design requirements, although bunch lengths as short as 120 fs have been achieved [62]. A similar analysis is used to find the conditions to transform a short bunch with a large momentum spread after the undulator to a long bunch with small momentum spread using the momentum compactions in the second recirculation arc.
- Page 45 and 46: FIG. 2.4: A picture of the energy r
- Page 47 and 48: dipoles and beam diagnostics such a
- Page 49 and 50: FIG. 2.7: Horizontal (red) and vert
- Page 51 and 52: FIG. 2.8: Illustration of the cryom
- Page 53 and 54: linac and θNL is the RF phase. The
- Page 55 and 56: 2.4 Transverse Emittance One of the
- Page 57 and 58: where σ2 is the rms beam size meas
- Page 59 and 60: eams. The effects of varying the qu
- Page 61 and 62: FIG. 2.12: A typical wire scan near
- Page 63 and 64: quadratic fit and a multiple regres
- Page 65 and 66: ting the data is difficult. Without
- Page 67 and 68: primary source of error is measurin
- Page 69 and 70: identified, although the phase dela
- Page 71 and 72: TABLE 2.3: Comparison of Twiss para
- Page 73 and 74: the results of the fits. The vertic
- Page 75 and 76: FIG. 2.18: Schematic illustrating t
- Page 77 and 78: FIG. 2.19: The GASK signal measured
- Page 79 and 80: FIG. 2.20: The measured normalized
- Page 81 and 82: CHAPTER 3 The Jefferson Laboratory
- Page 83 and 84: FIG. 3.1: Schematic of the 10 kW FE
- Page 85 and 86: FIG. 3.2: Layout of the DC photocat
- Page 87 and 88: accelerating gradient at the front
- Page 89 and 90: eason for making the endloops achro
- Page 91 and 92: FIG. 3.7: Illustration of path leng
- Page 93 and 94: 3.5 Longitudinal Dynamics This sect
- Page 95: FIG. 3.9: The effect of a thin focu
- Page 99 and 100: form of beam breakup not only occur
- Page 101 and 102: 4.1 The Pillbox Cavity Although the
- Page 103 and 104: FIG. 4.2: Electric field (red) and
- Page 105 and 106: where the full 4×4 transfer matrix
- 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
Under the constraint that each order vanishes, the desired M56 and T566 are found<br />
to be<br />
M56 = − λRF<br />
2π<br />
T566 = − λRF<br />
4π<br />
<br />
π<br />
= −<br />
λRF<br />
Emax<br />
Elinac<br />
Emax<br />
Elinac<br />
cos φo<br />
sin φo<br />
1<br />
sin φo<br />
2 cos φo<br />
<br />
M 2 56<br />
sin 3 φo<br />
78<br />
(3.17)<br />
(3.18)<br />
For a bunch operating at 10 ◦ off-crest, Emax = 145 MeV and Elinac = 135 MeV,<br />
the required M56 is −0.2 m and T566 is −3.5 m. In practice the first endloop<br />
utilizes trim quadrupoles to generate an M56 of +0.3 m (as opposed to the inherent<br />
+0.2 m of the endloop) [61]. This compaction, in addition to that of the optical<br />
magnetic chicane (−0.5 m) create the necessary conditions outlined above. In the<br />
FEL Upgrade typical rms bunch lengths obtained at the undulator are 200 fs in<br />
accord with design requirements, although bunch lengths as short as 120 fs have<br />
been achieved [62].<br />
A similar analysis is used to find the conditions to transform a short bunch with<br />
a large momentum spread after the undulator to a long bunch with small momentum<br />
spread using the momentum compactions in the second recirculation arc.
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