Measuring the Electron Beam Energy in a Magnetic Bunch ... - DESY

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where d 1 is the drift space between the first and second dipoles and d 2 is the drift space between the second and third dipoles. If these equations are rewritten in terms of the magnetic field and the momentum of the electrons (Eqs. 1.1 and 1.3), we have l 4 p ⎛ l eB ⎞ 2d ( ⎜ ⎟ d eff 1 BC p) = arcsin eB ⎜ p ⎟ + + 2 ⎝ ⎠ 1− ( leff eB / p) and (1.6) 2B 2 leff eBd1 xBC ( p) = ⎜⎛ 1− 1− ( leff eB / p) ⎟ ⎞ + . ep ⎝ ⎠ 2 p 1− ( l eB / p) Because high-momentum particles’ trajectories are bent less than those of lowmomentum particles, the high-momentum particles will travel a shorter path through the chicane and will arrive at the end of the chicane earlier than the low-momentum particles. We would like to know how a momentum change of a group of particles will affect the path-length through the chicane and the horizontal position in the chicane. To find this, we will Taylor expand Eq.s 1.5 in terms of a small change in momentum δ=Δp/p. This is given by l x BC BC 2 2 ∂lBC p ∂ lBC 2 ( p(1 + δ )) = lBC + p ⋅δ + ⋅δ 2 ∂p 2 ∂p eff 2 + ... 2 = l BC + R56 ⋅δ + R566 ⋅δ + ... (1.7) 2 2 ∂x BC p ∂ xBC 2 ( p(1 + δ )) = xBC + p ⋅δ + ⋅δ 2 ∂p 2 ∂p = x BC + R δ 2 16 ⋅δ + R166 ⋅ + where R 56 , R 566 R 16 and R 166 are functions of the magnetic field and the effective length of each magnet. They are named after their locations in a transfer matrix used to calculate beam transport and they are used to predict the arrival-time and x position changes of particles traveling through dipole fields for given momentum changes. For short bunches, like those in FELs, the R 566 and R 166 terms are typically small compared to the R 56 and R 16 terms. So, for the majority of the calculations in this thesis, only the first-order terms will be used. The first order terms are frequently referred to as momentum compaction (R 56 =ά c ) and linear dispersion (R 16 =D). Doing the first order derivatives of Eqs. 1.6, one finds R 56 = − 4l 1− ( eBl eff eff / p) 2 − 2d 1 ( eBl / p) eff 1− ( eBl / p) 3 2 eff 2 ... + ... 4 p ⎛ eBl + arcsin ⎜ eB ⎝ p eff ⎞ ⎟ ⎠ (1.8) 2
R 16 ( eBl / 2) d ⎛ ⎞ 1 eff 2 p ⎜ 1 = + ⎟ ⎜ −1 3 2 2 ⎟ 1− ( eBl / p) eB ⎝ 1− ( eBleff / p) ⎠ which, by substituting cosα wherever the square root of 1-(eBl eff /p) 2 arises, can be written more simply in terms of the bend angle and the effective dipole length. 2 tan α 4l eff R 56 = 2d1 + (tanα −α) cosα sinα R 16 tanα 2leff ⎛ 1 ⎞ = d1 + ⎜ −1 2 ⎟ cos α sinα ⎝ cosα ⎠ (1.9) Both R 56 and R 16 increase when the length of the chicane increases and when the bending angle increases. With a rough, small-angle approximation, R 56 ≈ 2 ⋅ R16 ⋅α . Although the results for the single-chicane are the same as those presented here, a different and more complete treatment of dispersion and momentum compaction in different types of chicanes is given in [1]. Because the particles in the chicanes are relativistic, we can use the approximation Δp/p ≈ ΔE/E and will use the term ‘relative momentum changes’ synonymously with ‘relative energy changes’. This allows us to write the useful formulas for the change in path length, Δl, and change in horizontal position, Δx, that occur as a result of momentum compaction and dispersion, Δl = R56 ΔE E ΔE Δx = R16 (1.10) E Because the momentum compaction and dispersion in a chicane are non-zero, changes in the x position of the beam within the chicane correlate with changes in the energy of the beam. Likewise, the path-length changes in and after the chicane will result in beam arrival-time changes which are correlated with beam energy changes. If the electron beam after a chicane is used to generate a photon beam, as in the case of a Free-electron Laser (FEL), the arrival-time stability of the photon beam will be directly affected by the stability of the beam energy, a quantity which can be measured through the position or arrival-time of the beam in or after a dispersive section. To detect the position of an electron beam, one could look at the optical transition radiation that is produced as the particles travel through a metal film, but the beam is significantly disturbed by the film and it cannot be used parasitically. Instead, one could look at the synchrotron light that is produced by the beam as its trajectory is changed by the magnetic field. The resolution of the synchrotron light based beam position measurement would then depend on how much light there is and how well it can be detected. If, instead, the position of the beam is detected with a metal antenna in which a current is induced as the beam passes close to it, the resolution would depend on how 3
Page 1 and 2: Measuring the Electron Beam Energy
Page 3 and 4: Abstract Within this thesis, work w
Page 5 and 6: 9 Synchrotron Light Monitors 9.1 Pr
Page 7 and 8: List of Figures 1.0.1 Magnetic bunc
Page 9 and 10: 5.5.10 Dependence of the slope of p
Page 11 and 12: 8.3.6 RF phase measurement drift wi
Page 13: where α is the bending angle intro
Page 17 and 18: 2 Free-electron Lasers The facility
Page 19 and 20: arrival-time and energy prior to a
Page 21 and 22: electrons travel faster than low-en
Page 23 and 24: efore bunch compressor energy chirp
Page 25 and 26: corresponds to terms that are linea
Page 27 and 28: spread will be larger and the trans
Page 29 and 30: (a) (b) Figure 2.3.4 Single chicane
Page 31 and 32: 3 Beam Arrival-time Stabilization A
Page 33 and 34: changes in the amplitude of the kly
Page 35 and 36: The problem of cavity field measure
Page 37 and 38: creates a single point-of-failure s
Page 39 and 40: accelerator section and it was firs
Page 41 and 42: ACC1 ~Gun 3rd ACC2 ACC3ACC4 ACC7 Ac
Page 43 and 44: • The lack of a normal-conducting
Page 45 and 46: ( ) E − E ⎛ E − E ⎜ ⎝ 0 0
Page 47 and 48: 3.7 Second Accelerator Section Jitt
Page 49 and 50: 4 Beam Shape in the Bunch Compresso
Page 51 and 52: There has been some debate over the
Page 53 and 54: V T = V cos( k ⋅ z + = V T 0 rf
Page 55 and 56: Upstream of the first accelerator s
Page 57 and 58: Positive bump 5 8 deg off-crest in
Page 59 and 60: E (MeV) 4.75 4.74 4.73 4.72 4.71 4.
Page 61 and 62: leads one to suspect that as in the
Page 63 and 64: vacuum feedthroughs, the buttons we
Page 65 and 66:
J image − ⎡ ⎛ ⎞ ⎤ = ⎢ +
Page 67 and 68:
sensitivity of 8 and 20 mm diameter
Page 69 and 70:
1 v phase = . (5.1.21) LC Finally,
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some locations. Nevertheless, it is
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BPM BAM More power at lower frequen
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arrival-time monitor with and witho
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5.2 Cavity BPM Cavity BPMs (Fig. 5.
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t=0 Z 0 t=L/c Z 0 2L Figure 5.3.1 L
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0.2 nC beam charge and can handle b
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T1-T2=T3 X=T3*c T1 T2 Figure 5.5.2
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None of the designs suffered from a
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pickup functions appropriately over
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35 Measured BPM Vertical amplitude
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2.5 Beam position 2 position (cm) 1
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L( x0 , t) U 0 ( t) = λ( x0 , t) (
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1 x beam = ) xdtdx Q ∫∫λ ( x,
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0 BPM beam width dependence arrival
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100 Vertical y position sensitiviy
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Such an x-y tilt was measured with
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10 5 head x [mm] 0 tail -5 -5 -4 -3
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3.5 x 10-3 Horizontal Charge Distri
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The problem that could arise due to
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around their zero-crossings in orde
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v i ( i i i t) = A sin(2π f t + θ
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strengths of the peaks seen in Fig.
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algorithm required to select the ph
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Downmixers for EBPM BPM-L BPM-R BPM
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While this option was not built and
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A Peltier element acts as a heat pu
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For the measurement shown in Fig. 7
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2 1.5 1 BC2 x position (mm) 0.5 0 -
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0 BC2 arrival time (ps) 1.5 1 0.5 0
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eality check BC2 PMT EBPM and PMT p
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EBPM and PM positions (mm) 6 5 4 3
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If a limiter or nothing at all is u
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Depending on the arrival-time of th
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20mW to BAM FARADAY 8m 80/20 60mW W
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can be placed anywhere prior to the
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Insulation Peltier element on metal
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using the RF limiter. This might be
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compared. The length of the cable c
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eplaced with fiber. The advantage o
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Photodetector Photodetector Two las
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RF-Lock Box for FLASH MLO Synchroni
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In Fig. 8.3.4, the spectral noise d
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Mixer Output (fs) 40 20 0 -20 MLO R
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For a frame of reference concerning
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(Fig. 9.1.2). This method proved to
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N λ ∞ ∫ ω / ω Δω 9 3 ( ω)
Page 163 and 164:
Finally, we can use this, together
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The highest energy resolution (ΔE/
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Good agreement between the RF chica
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10.3 Out-of-loop Vector Sum The out
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out. This was done in two ways: by
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1.8 1.6 stripline button 1.4 Upstre
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10 Conclusion and Outlook Six disti
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Appendix A The derivation of Eqs. 3
Page 179 and 180:
Solving Eq. 8 for E 1 ’ provides
Page 181 and 182:
Higher order terms and the 3 rd -ha
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∂t ∂V 2 2 ⋅ ΔV 2 1 = T1 '( E
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References [1] P. Castro. “Beam t
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[28] H. Schlarb et al., “Beam bas
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[55] C. Gerth, personal communicati
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Measuring the Electron Beam Energy in a Magnetic Bunch ... - DESY

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