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.