Effect of Insertion Devices The IDs are normally made of dipole magnets ith alternating dipole fields so that the orbit outside the device is un-altered. A simple planer undulator ith vertical sinusoidal fields, that satisfies Maxell s equation, is The nonlinear magnetic field can be neglected if the vertical betatron motion is small ith k z 1. The horizontal closed orbit becomes cos here k = π/λ is the iggler ave number, λ is the iggler period, and B is the magnetic field at mid-plane. The corresponding horizontal and longitudinal magnetic fields, and the vector potential are B x = 0, B s = B sinh k zsin k s, and The Hamiltonian of particle motion is sin ks z z 0 z 1 4 sin ks z ds sin k z s z 0 1 4 sin ks z ds z Effect of IDs on beam dynamics A. Effects of insertion devices on the beam emittances are the energy losses in the storage ring dipoles and in an undulator or iggler in one revolution respectively. Here ρ 0 and ρ are the bending radii of storage ring dipoles and iggler magnets, L is the length of the undulator or iggler, and E is the beam energy. The emittance ill decrease slightly hen the undulators field is B (3πf h /8) B 0, and the natural emittance ill increase for strong field igglers
B. Effects of IDs on momentum spread C. Effect of the ID induced dispersion functions and its effect on emittance We consider a simple ideal vertical field iggler (Fig. 4.0), here ρ = p/eb is the bending radius, θ = Θ = L /ρ is the bending angle of each dipole, and L is the length of each iggler dipole. Since the rectangular magnet iggler is an achromat (see Exercise.4.0), the iggler, located in a zero dispersion straight section, ill not affect the dispersion function outside the iggler. If B 3πB 0 /8 for the planer undulator, the beam momentum spread ill decrease. On the other hand, the high field IDs can increase momentum spread, particularly important for the lo main dipole field design. The dispersion functions in the insertion region induced by a sinusoidal vertical iggler field D. Effect of IDs on the betatron tunes The IDs ith rectangular magnets is achromatic, and the edge defocusing in the rectangular vertical field magnets cancels the dipole focusing gradient of 1/ρ. Thus there is no net focusing in the horizontal plane. The focal length of the vertical betatron motion and the tune shift resulting from the rectangular iggler dipole are respectively here L,total = 4N L is the total length of the undulator, and <βz> is the average betatron amplitude function in the iggler region. The vertical sinusoidal field undulator generates average vertical focusing strength and vertical betatron tune shift: For a given U /U 0 =(ρ 0 /ρ )N θ ith a constant N θ, the factor F ε is smaller for a smaller θ. Hoever, igglers ith very large K values may increase the beam emittance. The momentum spread of the beam ill increase or decrease depending on hether the magnetic field B of the planer iggler is larger or smaller than (3π/8)B 0 of the main dipole field.
Given an undulator length L, the optimal β* is about L/. at the center of the undulator ith a large tolerance. The basic lattice has been designed to provide optimal performance of each specific undulators. 1/ 1/ 0.8 0.7 0.6 0.5 0.4 0.3 0. Nonlinear beam dynamics The vertical chamber size in Insertion devices (IDs) is ±5 mm! Detailed studies on nonlinear beam dynamics has been carried out and good dynamic aperture has been found! [H.P. Chang, P.J. Chou, C.C. Kuo, G.H. Luo, H.J. Tsai, and M.H. Wang, EPAC06, 3430.] 0.1 0 0.00 1.00.00 3.00 4.00 5.00 / 16 nd Workshop on Nonlinear Beam Dynamics in Storage Rings, November, 009 C.C. Kuo: TPS DA in the presence of Undulators H.P. Chang: nonlinear field representation in undulator. The TPS design handbook (009) Nonlinear beam dynamics associated ith igglers The nonlinear magnetic field can be neglected if the vertical betatron motion is small ith k z 1 or z λ u /π. This is clearly ok for most 3 rd GLS! The field profile depends on the poleidth of the undulator. Name λ(mm) U80 80 EPU100 100 EPU70 70 SW48 48 EPU46 46 IU SU15 15 CU18 18 Nonlinear dynamics in a SPEAR iggler J. Safranek, et al., PRSTAB 5, 010701 (00) BL11, the most recently installed iggler in the SPEAR storage ring at the Stanford Synchrotron Radiation Laboratory, produces a large nonlinear perturbation of the electron beam dynamics, hich as not directly evident in the integrated magnetic field measurements. Measurements of tune shifts ith betatron oscillation amplitude and closed orbit shifts ere used to characterize the nonlinear fields. Because of the narro pole idth in BL11, the nonlinear fields seen along the iggling electron trajectory are dramatically different from the magnetic measurements made along a straight line ith a stretched ire. This difference explains the tune shift measurements and the observed degradation in dynamic aperture. Because of the relatively large dispersion (1. m) at BL11, the nonlinearities particularly reduced the off-energy dynamic aperture. Because of the nature of these nonlinear fields, it is impossible, even theoretically, to cancel them completely ith short multipolecorrectors. Magic finger corrector magnets ere built, hoever, that partially correct the nonlinear perturbation, greatly improving the storage ring performance. dynamic field integral correction
BL11 has 50 mm pole idth results in the fields rolling off quickly at ±5 mm. It has significant 3 rd and 5 th harmonics. The TOSCA model of BL11 By(x) is shon at right. If a particle is launched at the entrance to the iggler ith (x, x )=(xi,0), it ill follo a iggling trajectory of The integrated field seen along the iggling trajectory is here L is the iggler length. Generally, xp is small (for BL11, the iggler period, π/k is 17.5 cm and the peak field is T, so x p is 155 μm), but ith narro iggler poles db y /dx can be large, generating a strong perturbation. The large change in the linear term of ν x vs x β from BL11 indicates a strong octupole-like x 3 component in the horizontal equation of motion. The amplitude is limited hen BL11 is closed, a reduction of dynamic aperture. These to measurements ere made ith all other igglers closed. With all igglers open, the beam could be kicked to x β ~ 45 mm, so other igglers had already compromised the dynamic aperture before BL11 as installed. With BL11 closed, the bump range as limited to the range shon in the Figure. The lifetime drops to minutes. The horizontal tune as also measured as a function of the rf frequency ith BL11 gap opened and closed. The dispersion at BL11 is relatively large (1. m), so varying the rf frequency is simply another ay to vary the horizontal closed orbit. This measurement gave results similar to the Figure at right. The derivative of the field integral according to the electron beam measurements compared to the stretched ire magnetic measurements and the dynamic field integrals. A polynomial as fit to the stretched ire field integrals, and the derivative of this polynomial. The first item to note is that the dynamic field integral is very large. The negative peaks in the dynamic integral at ±19 mm are 9 kg, far off scale. A 9 kg field integral ould generate a horizontal tune shift of 0.15 in SPEAR. The BL9 poles are 95 mm ide, hile the BL11 poles are only 50 mm ide. Figure 6 shos the derivative of the field integral for BL9 according to tune measurements and stretched ire measurements. The field roll-off data ere not readily available for BL9, so the dynamic field integrals are not shon in Fig. 6. The good agreement beteen the tune and stretched ire measurements, hoever, indicates that the dynamic integrals are much smaller in BL9 than BL11. The field integrals from construction tolerances ( stretched ire in Fig. 5) are negligible in comparison. The field integrals from tune measurements sho good qualitative agreement ith the dynamic integrals (ith some vertical offset that can be attributed ith uncertainty in the β function at BL11, and ith some horizontal offset due to the uncertainty in the electrical centers of the beam position monitors).
Table I shos the agreement beteen the octupole-like cubic term predicted by the TOSCA field model and that from three different electron beam measurements! SPEAR PERFORMANCE WITH MAGIC FINGERS The bottom half of the magic fingers attached to the ends of the iggler. The yello arros indicate polarity of permanent magnets. Dimensions in mm. The model of BL11 reasonably predicts the electron beam measurements. The horizontal tune shift hen closing the BL11 magnet gap indicated an integrated field gradient of 0.084 T, hile the TOSCA field model has an integrated gradient of 0.069 T. The dynamic aperture reduction results from nonlinear fields. The quadrupole-like focusing generates a horizontal beta beat of 1% and 1% at 3 and.3 GeV, respectively. This linear perturbation is not enough to explain the reduction in dynamic aperture. A loer ja of the magnetic finger assemblies are designed to cancel the dynamic field integrals. The bolt arrangement allos small changes in the block positions in the shimming process. The jas are attached in pairs to the to ends of the iggler to create normal multipoles and must fit in a narro edge-shaped space beteen the iggler end and the vacuum chamber flange. Question: Can one carry out multipole compensator instead of the ja geometry! The jas are attached in pairs to the to ends of the iggler to create normal multipoles and must fit in a narro edge-shaped space beteen the iggler end and the vacuum chamber flange. To limit the number of magnet blocks needed, it as decided the dynamic field integral on the magnetic mid-plane should be canceled in the interval x < 5 mm. The resulting design has six magnet blocks in each ja. The large blocks take care of the dominating 1- pole component, hile the smaller blocks adjust the 8-pole and 4-pole components. With magic fingers installed, the storage ring performance as restored nearly to that ithout BL11. 1. Before magic fingers at the.3 GeV injection energy, beam injected ith the BL11 gap open ould be lost hile closing the gap.. After magic fingers ere installed, the dynamic aperture as sufficiently improved that beam can be stored and injected at.3 GeV. 3. With the gap closed, hoever, the injection rate is still more sensitive to small variations in machine parameters, so the gap is usually opened several millimeters prior to injection. At the 3 GeV operational energy, closing the gap gives no degradation in the ring performance for synchrotron radiation users. There is no measurable change in the lifetime hen closing the iggler gap at 3 GeV. Both the tune shift ith betatron oscillation amplitude and the tune shift ith a closed orbit bump ere remeasured after the installation of the magic fingers. The data ere made ith all other igglers open. and beam loss for the maximum kick as greater hen measuring the data in Fig. 1. The magic fingers increased by nearly a factor of 3 the range over hich the beam could be bumped before the lifetime decreased to minutes. (The change in the starting horizontal tune from 0.13 to 0.17 beteen Figs. 4 and 13 as a change in the operational value in the 1.5 yr beteen the to measurements and had nothing to do ith BL11.)
Nonlinear beam dynamics orkshop 009: P. Kuske, BESSY, Berlin No Impact of sc IDs on Longitudinal Acceptance Field Optimization ith Magic Fingers U15IDR black: ithout Magic Fingers red: ith Magic Fingers ID-gap open ithout ID-gap = 15.7 mm ith Magic Fingers Dynamic multipoles: second order effect created by the oscillatory motion of the electrons in the 3dim fields of the IDs no straight line integrals, cannot be measured ith moving ire A Ne Approach to the Electron Beam Dynamics in Undulators and Wigglers Pascal ELLEAUME, EPAC9, 661 Important in lo and medium electron energy rings (e.g. BESSY II: 1.7GeV) long period lengths (e.g. BESSY UE11 ith 11mm period length) high fields, large transverse gradients (e.g. high field iggler, APPLE) Shimming of UE56ID3R APPLE Devices -- some success Successful Active Compensation 8 flat ires along the ID- chamber ith 14 PS maximum current: 16A, ire diameter: 3x0.3mm ire separation: 4mm Recommendations: 1. It appears that only the field integrals are important in ID. One needs only to correct the field integral locally. The field integrals can be compensated by magic fingers, or by multipolecoil compensators 3. The representation of H.P. Chang may be extended to the treatment of field integrals 4. Lifetime and DA are tune-dependent! Avoid resonances can provide good DA ith good lifetime!