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Insertion Devices Insertion Devices Johannes Bahrdt BESSY, Berlin, Germany Abstract The interaction of an insertion device with the electron beam in a storage ring is discussed. The radiation property including brightness, flux and polarization of an ideal and real planar and helical / elliptical device is described. The magnet design of planar, helical, quasiperiodic devices and of devices with a reduced on axis power density are resumed. 1 Introduction Undulators and wigglers, also called insertion devices (IDs), are periodic magnetic arrays which are installed at linear or circular accelerators. The electrons are forced to move on sinusoidal or elliptical trajectories inside the devices. The strength of the IDs and hence the amplitude of the oscillations can be modified whereas the period length is usually fixed. Outside of the device the trajectory is unchanged. There are basically two areas of applications for insertion devices: 1: IDs can modify electron beam parameters such as momentum compaction factor, damping times, emittance, energy spread, spin polarization time and degree of spin polarization. 2: IDs produce synchrotron radiation with a high brightness (photons / s / mm2 / mrad2) over a wide energy range and a tunable polarization. This paper concentrates mainly on this aspect. The first undulator has been operated at a linear accelerator of 5 MeV (100 MeV) for the generation of radiation in the mm (visible) region [1]. In 1976 with a superconducting helical undulator installed at a linear accelerator for free electron laser operation a finite gain at 10.6 µm has been demonstrated. One year later the system, now equipped with a resonator, operated above threshhold at 3.4 µm [2]. A few years later the first insertion devices for light generation had been installed in storage rings (SSRL [3], LURE [4], VEPP3 [5] [6]). At the same time the first so called wavelength shifters have been installed in storage rings (VEPP3 3.5 T [6], SRS T [7], VEPP2M 7.5 T [8]). These superconducting devices have significantly higher magnetic fields than the bending magnets and hence, the radiation spectrum is shifted to the higher energies. In first and second generation synchrotron radiation facilities the bending magnets played a dom- inant role as radiation sources. Third generation facilities are optimized for the operation of IDs which are superior to bending magnets with respect to brightness (flux) by up to four (two) orders of magnitude (assuming a 1 mrad fan for a bending magnet). Today about fifteen third generation mashines are in operation and the demand for synchrotron radiation is still increasing. Free electron lasers have been developed at linacs and storage rings [9]. The achievable photon energies are restricted to the near VUV because the reflectivity of mirrors decreases dramatically with increasing energy. To overcome this problem SASE-FELs which start from noise and do not need a resonator, have been proposed and successfully built (BNL VISA [10], APS LEUTL [11], DESY TTF [12]. Saturation has been achieved and measurements agree well with theory. The shortest wavelength at which saturation has been demonstrated is 82 nm [12]. The next generation of light sources will be SASE-FEL devices [13]. Two projects aiming for 1 A˚ radiation have been started (DESY TESLA [14], SLAC LCSL [15]). The undulator structures for these facilities will have a length of 100–200 m. 441 J. BAHRDT In chapter 2 the dependence of basic beam parameters on the synchrotron radiation integrals will briefly be summarized. Chapter 3 concentrates on the motion of charged particles in permanent mag- net fields. In chapter 4 the radiation characteristics including polarization will be discussed. Various technologies and undulator designs will be presented in chapter 5. 2 Beam parameters and synchrotron radiation integrals The synchrotron radiation integrals [16] describe the basic parameters of a circular accelerator. η(s) I = ds (1) 1 ρ Z 1 I = ds 2 ρ2 Z 1 I = ds 3 ρ 3 Z | | 1 I = ds 3a ρ3 Z (1 2n(s))η(s) I = − ds 4 ρ3 Z H(s) I = ds 5 ρ(s) 3 Z | | 1 2 H(s) = [η + < βη0 0.5 β0 η >] β − ∂ n(s) = ρ2 (1/ρ) ∂x η and β are the dispersion and the betatron function. The integrals are taken along the circumference of the ring. The following parameters depend on the synchrotron radiation integrals: energy loss per revolution 2 E4 ∆E = re 2 3 I2 3 3(mec ) momentum compaction factor α = I1/2πR damping partition numbers J = 1 I /I x − 4 2 Jz = 1 J = 2 + I4/I2 damping times 3 τi = 3T0/r0γ JiI2 i = x, z, 2 442 INSERTION DEVICES energy spread σE 2 2 I3 ( ) = Cqγ E 2I2 + I4 55 ~ Cq = 32√3 mec emittance I = C γ2 5 q I I 2 − 4 horizontal beam size σ2(s) I σ x = C β 5 + η2( )2 β(s) q I I E 2 − 4 spin polarization time 5√3 ~re E0 5 I3 1/τp = ( 2 ) 8 me mec 2πR achievable degree of polarization 8 I3a Pmax = 5√3 I3 The magnetic field of an insertion device modifies the synchrotron radiation integrals. The disper- sion at the location of the ID is the sum of the dispersion of the ring without ID and the dispersion of the ID [17]: 2 η(s) = η0 + η0(s) + cos(ks)/(ρ0k ) (2) with the undulator parameters ρ0 = eB0/γmec and k = 2π/λ0, where λ0 is the undulator period. Using this dispersion and the ID local bending radius, the synchrotron radiation integrals can be integrated and the modification due to the ID can be estimated [17]. Gradient wigglers have been proposed and built in order to enhance the radial damping in com- bined function mashines [18], [19]. These devices are installed in sections of finite dispersion and reduce the integral I4. Jx is enhanced and the emittance is reduced. On the other hand J is reduced and the energy spread is increased. Planar wigglers have been installed in damping rings to reduce the emittance (damping wigglers). It has also been proposed to install damping wigglers in future light sources in order to reduce the emittance below the limits of existing third generation storage rings [20]. Another effect of high field wigglers is the reduction of the spin polarization time in storage rings [21]. Conventional wigglers diminish the achievable degree of polarization. Asymmetric wigglers having positive and negative poles of different field strengths (keeping the net kick of the device zero) increase again the achievable polarization Pmax. At the storage ring LEP 12 asymmetric polarization wigglers had been installed [22] to provide a high polarization rate and degree. We refer to [17] for a detailed overview on the impact of IDs on the beam parameters. 3 443 J. BAHRDT 3 Charged particles in periodic magnetic structures 3.1 The equations of motion The movement of charged particles in magnetic fields is determined by the Lorentz force. The exact equations of motion for arbitray magnetic fields are given by the following differential equations [23]: 2 x00 = C [y0Bs (1 + x0 )By + x0y0Bx] (3) · − 2 y00 = C [x0B (1 + y0 )B + x0y0B ] − · s − x y 2 2 C = ( 1 + x0 + y0 )/(Bρ) The derivatives are taken with respect pto the longitudinal coordinate s. In small emittance accelerators the angles x0 and y0 are small and the equations can be simplified: 1 1 2 3 2 x00 = [x y B (1 + y + x )B + y B ] (4) Bρ 0 0 x − 2 0 2 0 y 0 s 1 1 2 3 2 y00 = [x y B (1 + x + y )B + x B ] − Bρ 0 0 y − 2 0 2 0 x 0 s In linear optics only (vertical) dipole and quadrupole terms are taken into account and the equations of motion reduce to [24]: 2 x00(s) + (1/ρ (s) κ(s)) x(s) = 0 (5) − · y00(s) + κ(s) y(s) = 0 · ρ is the dipole bending radius and κ = (e/γmc) (∂B )(∂x) is the quadrupole term. The coordinates x · y and y now relate to the closed orbit whereas in Eq. (3) a fixed cartesian coordinate system has been used. The field of a planar undulator with k = 2π/λ0 can be described as follows: kx Bx = B0 sinh(kxx) sinh(kyy) cos(ks) (6) ky · · · B = B cosh(k x) cosh(k y) cos(ks) y 0 · x · y · k Bs = B0 cosh(kxx) sinh(kyy) sin(ks) −ky · · · Higher Fourier components of the magnetic field can be included with appropriate wave numbers kn = k n where n is the expansion order. For elliptical or helical devices further field terms have to be added · which are shifted in phase by λ0/4. k˜x B˜x = B˜0 sinh(k˜xx) sinh(k˜yy) sin(ks) (7) k˜y · · · B˜ = B˜ cosh(k˜ x) cosh(k˜ y) sin(ks) y 0 · x · y · k B˜s = B˜0cosh(k˜xx) cosh(k˜yy) cos(ks) k˜y · · This ansatz satisfies Maxwell's equations if the wave numbers ki fulfill the following relation: 2 2 2 kx + ky = k (8) ˜2 ˜2 2 kx + ky = k The parameters kx and ky describe the field variation in transverse direction. The values are real (imagi- nary) if the fields increase (decrease) off axis. The quantities x0, y0, 1/Bρ are small and therefore, the solutions of the equation of motion can be expanded analytically up to a certain order in these quantities. This will be discussed later. 4 444 INSERTION DEVICES 3.2 Focussing of undulators The fringe field of a dipole gives rise to focussing effects in the horizontal and vertical plane.
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