F. Effects of dipole field error on orbit length
III.3 Application of Dipole Field Error
where � =s2−s1. In thin lens approximation, the transfer matrix of Eq. (2.103) becomes
whereθ= ΔBz�/Bρandf= 1/K�are respectively the dipole kick angle and the focal length of the perturbing element. Dipole field error can also arise from quadrupole misalignment. Let Δyq be the quadrupole misalignment. The resulting extended transfer matrix in the thin-lens approximation is
Mquad=
The 3×3 extended transfer matrix can be used to obtain the closed orbit of beta-tron motion. For example, the closed-orbit equation (2.90) is equivalent to
⎛
whereM’s are matrix elements of 2×2 one-turn transfer matrix for an ideal machine, andθis the dipole kick angle. Similarly, the 3×3 extended transfer matrix can be used to analyze the sensitivity of the closed orbit to quadrupole misalignment by multiplying the extended matrices along the transport line.24
III.3 Application of Dipole Field Error
Sometime, we create imperfections in an otherwise perfect accelerator for beam ma-nipulation. Examples are the local-orbit bump, one-turn kicker for fast extraction, rf knock-out, etc.
24S.Y. Lee, S. Tepikian,Proc. IEEE PAC Conf., p. 1639, (IEEE, Piscataway, N.J., 1991).
III. EFFECT OF LINEAR MAGNET IMPERFECTIONS 87 A. Orbit bumps
To facilitate injection, extraction, or special-purpose beam manipulation,25the orbit of beams can be bumped to a desired transverse position at specified locations. In this example, we discuss the four-bump method facilitated by four thin dipoles with kick anglesθi (i= 1,2,3,4). Using Eq. (2.92), we obtain
yco(s) = β(s) 2 sinπν
4 i=1
βiθi cos(πν− |ψ−ψi|),
whereθi = (ΔBΔs)i/Bρand (ΔBΔs)i are the kick-angle and the integrated dipole field strength of thei-th kicker. The conditions that the closed orbit is zero outside these four dipoles areyco(s4) = 0, y�co(s4) = 0, or
√β1θ1cos[πν−ψ41] +√
β2θ2cos[πν−ψ42] +√
β3θ3cos[πν−ψ43] +√
β4θ4cosπν = 0,
√β1θ1sin[πν−ψ41] +√
β2θ2sin[πν−ψ42] +√
β3θ3sin[πν−ψ43] +√
β4θ4sinπν = 0, whereψji=ψj−ψiis the phase advance fromsitosj. Expressingθ3andθ4in terms ofθ1 andθ2, we obtain
β3θ3=−(
β1θ1sinψ41+
β2θ2sinψ42)/sinψ43, β4θ4= (
β1θ1sinψ31+
β2θ2sinψ32)/sinψ43. (2.105) The orbit displacement inside the region of orbit bumps can be obtained by applying the transfer matrix to the initial coordinates. Using four bumps, we can adjust the orbit displacement and the orbit angle to facilitate ease of injection and extraction, to avoid unwanted collisions, and to avoid limiting-aperture in accelerator.
The three-bump method (see Exercise 2.3.4) has also been used for local orbit bumps. Although the slope of the bumped particle orbit can not be controlled in the three-bump method, this method is usually used for the local orbit correction because of its simplicity. Occasionally, two bumps can be used at favorable phase-advance locations in accelerators. Figure 2.14 shows an example of a local orbit bump using three dipoles. Since the two outer bumps happen to be nearly 180◦ apart in the betatron phase advance, the middle bump dipole has negligible field strength.
B. Fast kick for beam extraction
To extract a beam bunch from accelerator, a fast kicker magnet is usually powered in about 10–100 ns rise and fall times in order to bump beam bunches into the extraction
25Other examples are orbit bump at the aperture restricted area, internal target area, avoiding unwanted collisions in colliders, etc. For example, the counter-circulatinge+ande−beams, or the
¯
pandpbeams in a collider can be made to avoid crossing each other in a common vacuum chamber with electrostatic separators.
Figure 2.14: A simple orbit bump produced by three dipole kickers marked with symbol X in the AGS booster lattice. Since the first and third kickers are nearly 180◦apart in the betatron phase advance, the local orbit bump is essentially accomplished with these outer two kickers. In this example, there are 3 focusing and 2 defocusing quadrupoles between two outer bump dipoles.
channel, where a septum is located.26 With the transfer matrix of Eq. (2.42), the transverse displacement of the beam is
Δxco(s) =
βx(sk)βx(s) sin(Δφx(s))
θk, (2.106)
whereθk =
Bkds/Bρis the kicker strength (angle), Bk is the kicker dipole field, βx(sk) is the betatron amplitude function evaluated at the kicker location, βx(s) is the amplitude function at locations, and Δφx(s) is the phase advance fromskof the kicker to locations. The quantity in curly brackets in Eq. (2.106) is called thekicker lever arm.
To achieve a minimum kicker angle, the septum is located about 90◦phase advance from the kicker, and the values of the betatron amplitude function at the septum and kicker locations are also optimized to obtain the largest kicker lever arm. Similar constraints apply to the kicker in the transverse feedback system, the kicker array for stochastic cooling, etc.
Figure 2.15 shows a schematic drawing of the cross-section of a Lambertson septum magnet. A beam is bumped from the center orbitxcto a bumped orbitxb. At the time of fast extraction, a kicker kicks the beam from the bumped orbit to the the extraction channel at xk, where the uniform dipole field bends the beam into the extraction channel. The iron in the Lambertson magnet is shaped to minimize the field leakage into the field-free region and the septum thickness, that is of the order of 4-10 mm depending on the required magnetic field strength.
26Thekicker is an electric or magnetic device that provides an angular deflection to charged particle beam at a fast rise and fall times so that it can selectively deflect some beam bunches without affecting others. The electric kicker applies the traveling wave to a stripline type waveguide. The magnetic kicker employs ferrite material to minimize eddy-current effects. The rise and fall times of the kickers range from 10 ns to 100’s ns. Theseptumis a device with an aperture divided into a field-free region and a uniform-field region, where the former will not affect the circulating beams, and the latter can direct the beam into an extraction or injection channels. Depending on the application, one can choose among different types of septum, such as wire septum, current sheet septum, Lambertson septum, etc.
III. EFFECT OF LINEAR MAGNET IMPERFECTIONS 89 Figure 2.15: A schematic drawing of the central orbitxc, bumped orbitxb, and kicked orbitxkin a Lambertson septum magnet. The blocks marked with X are conductor-coils, The ellipses marked beam ellipses with closed orbits xc, xb, and xk. The arrows indicated a possible magnetic field di-rection for directing the kicked beams downward or upward in the extraction channel.
C. Effects of rf dipole field, rf knock-out
In the presence of a localized rf dipole, Hill’s equation is d2y
ds2 +K(s)y=θasinωmt ∞ n=−∞
δ(s−nC), (2.107)
whereθa= ΔB�/Bρandωmare respectively the kick angle and the angular frequency of the rf dipole, C is the circumference, and t= s/βcis the time coordinate. The periodic delta function reflects the fact that beam particles encounter the kicker field only once per revolution.
With coordinate transformation: η=y/√
β, φ=ν1s function at the rf dipole location, ω0 is the orbital angular frequency, and we use δ(s−nC) =|ds/dφ|1 δ(φ−2πn). The solution of the inhomogeneous Hill’s equation is η=Acosνφ+Bsinνφ+ηco,whereAandBare the amplitude of betatron motion determined by the initial conditions, and the particular solutionηco is the coherent time dependent closed orbit, The discrete nature of the localized kicker generates error harmonics n+νm for alln∈(−∞,∞). For example, if the betatron tune is 8.8, large betatron oscillations can be generated by an rf dipole at any of the following modulation tunes: νm = 0.2,0.8,1.2,1.8, . . . .The coherent betatron motion of the beam in the presence of an
where the last approximate identity is obtained by expanding the term in the sum withn+νm≈ν, and retaining only the dominant term. Equation (2.109) indicates that the beam is driven coherently by the rf dipole, and the amplitude of betatron motion grows linearly with time.The coherent growth time of the betatron oscillation is inversely proportional to|νm−ν|(mod 1). Beyond the coherent time, the beam motion is out of phase with the external force and leads to damping. This process is related to the Landau damping to be discussed in Sec. VIII.4.
Figure 2.16 shows the measured betatron coordinate (lower curve) at a beam position monitor (BPM) after applying rf knock-out kicks to the beam in the IUCF Cooler Ring, and the fractional part of the betatron tune (upper curve), that, in this experiment, is equal to the knockout tune. The rf dipole was on from 1024 to 1536 revolutions starting from the triggering time. At revolution number 2048, the beam was imparted a transverse kick. Note the linear growth of the betatron amplitude during the rf dipole-on time. Had the rf dipole stayed on longer, the beam would have been driven out of the vacuum chamber, called the “rf knock out.”
Figure 2.16: The lower curve shows the measured vertical betatron oscillations at one BPM in the IUCF Cooler resulting from an rf dipole kicker at the betatron frequency. The rf dipole was turned on for 512 revolutions, and the beam was im-parted by a one-turn kicker after another 512 revolutions. The betatron amplitude grew linearly during the rf knockout-on time. The upper curve shows the frac-tional part of the betatron tune obtained by counting the phase advance in the phase-space map using data of two BPMs.
The fractional betatron tune, shown in the upper trace, is measured by averaging the phase advance from the Poincar´e map (see Sec. III.5), where data from two BPMs are used. This two-kick method can be used to provide a more accurate measurement of the dependence of the betatron tune on the betatron amplitude. The power supply ripple at the IUCF cooler ring gives rise to a betatron tune modulation of the order of 2×10−3 at 60 Hz and its harmonics. On the other hand, the dependence of the betatron tune on the betatron action is typically 10−4per 1πmm-mrad. To measure this small effect in the environment of the existing power supply ripple, the two-kick method was used to measure the instantaneous betatron tune change at the moment of the second kick.27
27See M. Ellisonet al.,Phys. Rev. E50, 4051 (1994).
III. EFFECT OF LINEAR MAGNET IMPERFECTIONS 91 The rf dipole can be adiabatically turned on to induce coherent betatron oscil-lations for betatron tune measurement without causing serious emittance dilution.28 Figure 2.17 shows the vertical beam profile measured at the AGS during the adia-batic turn-on/off of an rf dipole. When the rf dipole was on, the beam profile became larger because the beam was executing coherent betatron oscillations, and the profile was obtained from the integration of many coherent betatron oscillations. As the rf dipole is adiabatically turned off, the beam profile restored to its original shape.
time
2.54cm
Figure 2.17:The beam profile measured from an ionization profile monitor (IPM) at the AGS during the adiabatic turn-on/off of an rf dipole. The beam profile appeared to be much larger during the time that the rf dipole was on because the profile was an integration of many coherent synchrotron oscillations. After the rf dipole was adiabatically turned off, the beam profile restored back to its original shape (Graph courtesy of M. Bai at BNL).
The induced coherent betatron motion can be used to overcome the intrinsic spin resonances during polarized beam acceleration. Furthermore, the measurement of the coherent betatron tune shift as a function of the beam current can be used to measure the real and imaginary parts of the transverse impedance (see Sec. VIII).
This method is usually referred to as the beam transfer function (BTF).
D. Orbit response matrix and accelerator modeling
Equation (2.92) shows that the beam closed orbit in a synchrotron is equal to the propagation of the dipole field error through Green’s function of Hill’s equation. If the closed-orbit response to a small dipole field perturbation can be accurately mea-sured, Green’s function of Hill’s equation can be modeled. The orbit response matrix (ORM) method measures the closed-orbit response induced by a known dipole field perturbation. The resulting response functions can be used to calibrate quadrupole strengths, BPM gains, quadrupole misalignment, quadrupole roll, dipole field inte-gral, sextupole field strength, etc. The ORM method has been successfully used to model many electron storage rings.29
We consider a set of small dipole perturbation given byθj, j = 1, ..., Nb, where Nb is the number of dipole kickers. The measured closed orbit yi at theith beam
28M. Baiet al., Phys. Rev. E56, 6002 (1997); Phys. Rev. Lett. 80, 4673 (1998); Ph.D. Thesis, Indiana University (1999); see also S.Y. Lee, PRSTAB9, 074001 (2006).
29See J. Safranek and M.J. Lee, Proc. Orbit Correction and Analysis in Circular Accelerators, AIP Conf. Proc. No.315, 128 (1994); J. Safranek and M.J. Lee,Proc. 1994 European Part. Accel.
Conf.1027 (1997). J. Safranek,Proc. 1995 IEEE Part. Accel. Conf., 2817 (1995).
position monitors from a dipole perturbation is
yi=Rijθj, j= 1, ..., Nb i= 1, ..., Nm. (2.110) (Nbcan differs fromNm). The response matrixRis equal to the Green’s function Gy of Eq. (2.91) and another term resulting from the orbit length change due to the dipole kick to be discussed in Sec. IV.3.C.
Experimentally, we measure Rij (i = 1,· · ·, Nm) vs the dipole kick at θj (j = 1,· · ·, Nb). The full set of the measured response matrix R can be employed to model the dipole and quadrupole field errors, the calibration of the BPM gain factor, sextupole misalignment, etc. The outcome of response matrix modeling depends on the BPM resolution, the number of BPMs and kickers, and the machine stability during the experimental measurement.
The ORM method minimizes the difference between the measured and model matricesRexp andRmodel. Let
Wk=|Rmodel,ij−Rexp,ij|
σi (2.111)
be the difference between the closed-orbit data measured and those derived from a model, where σi is the rms error of ith measurements. Here the number of index k is Nb×Nm, and the model response matrix can be calculated from MAD[23], SYNCH[24], or COMFORT[26] programs. The measured response matrix needs cal-ibration in the kicker angle and BPM gain, i.e.
Rexp,ij=Rdata,ij fjgi ,
wherefjis the calibration factor of thejth kicker, andgiis the gain factor of theith BPM. The ORM accelerator modeling is to minimize the error of the vectorWby minimizing theχ-square (χ2) defined as
χ2= 1 Nb·Nm
k
W2k.
We consider sets of parameterswm’s that are relevant to accelerator model and or-bit measurement. Some of these parameters are kicker angle calibration factor, the BPM gain factor, the dipole angle and dipole roll, the quadrupole strength and roll, sextupole strength, etc. The ORM modeling is to find a new set ofwm-parameters such that
||W(wm)||= 0. (2.112)
First, we begin with parameterswmand evaluateW(wm). The idea is to find a new set of parameterswm+ Δwmthat satisfies Eq. (2.112), i.e.
Wk(wm+ Δwm)≈Wk(wm) +dWk
dwmΔwm= 0. (2.113)
III. EFFECT OF LINEAR MAGNET IMPERFECTIONS 93 To evaluate Δwm, we invert matrix W ≡ dWdwmk, which has the dimension of is (Nb·Nm)×Np. Here,Np is the number of parameters. In our application to accel-erator physics, (Nb·Nm)�Np. The singular value decomposition (SVD) algorithm decomposes the matrixW into
W=dWk
dwm =UΛVT, (2.114)
where VT is a real orthonormal Np×Np matrix with VVT = VTV = 1, Λ is a diagonal Np×Np matrix with elements Λ11 = √
λ1 ≥ Λ22 = √
λ2· · · ≥ 0, and U = AVΛ−1 is a (Nm·Nb)×Np matrix with UTU = 1.30 Here λ1, λ2,· · · are eigenvalues of the matrixWTW, andVis composed of orthonormal eigenvectors of WTW, i.e. WTW =VΛ2VT. The SVD-method sets all eigenvaluesλi≤λc, (i > r) to λi = 0, (i > r), where λc is called the tolerance level andr is called therankof the matrix W. Setting all λi = 0 (i > r) is equivalent setting Δwi = 0 fori > r.
This means that these dynamical parameters have no relevance to the measured data.
Once the SVD of matrixW is obtained, one finds Δwm as Δwm=−
VΛ−1UT
W(wm), where Λ−1 is a diagonal matrix with Λ−111 = 1/√
λ1,· · ·,Λ−1rr = 1/√
λr and 0 for all remaining diagonal elements withi > r. The iterative procedure continues until
|Δwm|or the change ofχ2 are small.
The response matrix modeling has been successfully implemented in many electron storage rings, where the BPM resolution is about 1∼10μm. The method has been used to calibrate kicker angle, BPM gain, quadrupole strength and roll, sextupole mis-alignment, dipole and quadrupole power supplies, etc. The method is also applicable to proton synchrotrons, where the BPM resolution is usually of the order of 100μm.
In accelerator modeling, the dimension of the matrixW, (Nm·Nb)×Np, can be large. The inversion of a very large matrix may become time consuming. It is advan-tageous to model accelerator parameters in sequences, e.g., (1) kicker angle calibration fj, (2) BPM gaingi, (3) quadrupole strength ΔKi, (4) dipole angle calibration, (5) dipole roll, etc. These steps are sometimes essential in attaining a reliable set of model parameters.
For high-power synchrotrons, beam particles are injected, accelerated and ex-tracted in a short time duration. For example, the proton storage ring (PSR) at Los Alamos National Laboratory accumulates protons for 3000 turns and the beam bunch is extracted after accumulation for high-intensity short-pulse neutron produc-tion. The closed orbit data can be obtained by averaging betatron oscillations in a
30The SVD decomposition of am×nmatrixWin Eq. (2.114) can also be carried out in such a way thatUandVare respectively orthonormal realm×mandn×nmatrices withUTU=UUT=1 andVTV=VVT=1, andΛis am×ndiagonal matrix.
Figure 2.18:Left, digitized betatron oscillation data of one BPM are used to derive betatron amplitude, phase and tune, and closed orbit offset. Right, top and bottom plots show the closed orbit data compared with Green’s function of Eq. (2.91) at a calibrated vertical steerer angle before and after ORM modeling.
single turn injection. The betatron oscillations of each BPM can be used to obtain the betatron amplitude, phase and tune, and the closed orbit (see the left plot of Fig. 2.18). These information can be used in the ORM analysis for accelerator mod-eling.31 The right plots of Fig. 2.18 shows an example of typical fit in ORM modeling.
The success of accelerator modeling depends critically on the orbit and tune stability, the number of BPMs and orbit steerers, proper set of experimental data for attaining relevant parameters.
E. Model Independent Analysis
Using turn-by-turn BPM data excited by resonant pinger discussed in Sec. III.3C, one can also carry out response matrix analysis for accelerator modeling, called Model In-dependent Analysis (MIA). This method has been successfully applied to SLC linac, PEP-II and Advanced Photon Source.32 For the application of MIA in a storage ring, one uses an rf dipole pinger to excite coherent betatron oscillation and measures the response function with turn-by-turn BPM digitizing system (See Sec. III, in Ap-pendix A, where we introduce the independent component analysis (ICA) for beam measurements).
31X. Huanget al.,Analysis of the Orbit Response Matrix Measurement for PSR, Technote: PSR-03-001 (2003).
32J. Irwin, C.X. Wang, and Y.T. Yan, Phys. Rev. Lett. 82, 1684 (1999); C.X. Wang, Ph.D.
Thesis, Stanford University (1999); J. Irwin and Y.T. Yan, Proceedings of EPAC 2000, p. 151 (2000); C.X. Wang, Vadim Sajaev, and C.Y. Yao, Phys. Rev. ST Accel. Beams6, 104001 (2003).
III. EFFECT OF LINEAR MAGNET IMPERFECTIONS 95
III.4 Quadrupole Field (Gradient) Errors
The betatron amplitude function discussed in Sec. II depends on the distribution of quadrupole strengths. What happens to the betatron motion if some quadrupole strengths deviate from their ideal design values? We found in Sec. III.1 that the effect of dipole field error on the closed orbit would be minimized if the betatron tune was a half-integer. Why don’t we choose a half-integer betatron tune?
This section addresses effects of quadrupole field error that can arise from vari-ation in the lengths of quadrupoles, errors in quadrupole power supply, horizontal closed-orbit deviation in sextupoles,33 etc. These errors correspond to theb1term in Eq. (2.19).
A. Betatron tune shift
Including the gradient error, Hill’s equation for the perturbed betatron motion about a closed orbit is
d2y
ds2 + [K0(s) +k(s)]y= 0, (2.115) whereK0(s) is the focusing function of the ideal machine discussed in Sec. II, andk(s) is a small perturbation. The perturbed focusing functionK(s) =K0(s)+k(s) satisfies a weaker superperiod conditionK(s+C) =K(s), whereCis the circumference. Let M0be the one-turn transfer matrix of the ideal machine, i.e.
M0(s) =Icos Φ0+Jsin Φ0, I= 1 0
0 1
, J(s) =
α(s) β(s)
−γ(s) −α(s)
, where Φ0 = 2πν0 is the unperturbed betatron phase advance in one revolution, ν0
is the unperturbed betatron tune, andα(s), β(s), and γ(s) are betatron amplitude
is the unperturbed betatron tune, andα(s), β(s), and γ(s) are betatron amplitude