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Preparation of the concerned sectors for educational and R&D activities related to the Hungarian ELI project Free electron lasers Lecture 2.: Insertion devices Zoltán Tibai János Hebling 1

Outline Introduction and history of insertion devices Dipole magnet Quadropole magnet Chicane Undulators Pure Permanent magnet Hybrid design Helical undulator Electromagnet Planar undulator Electromagnet Helical undulator Examples 2

Introduction Whenever an electron beam changes direction it emits radiation in a continuous frequency band. The most conspicuous example is the intense radiation produced by electron in a synchrotron orbit. It is sometimes concentrated in a certain frequency range by wiggling the beam as it leaves the machine with the help of a few magnets so as to follow a shape like the outline of a camel s back. Such a device is called wiggler. Many wiggler in succession, say 50 or more, serve to concentrate the radiation spatially into a narrow cone, and spectrally into a narrow frequency interval. The beam is made wavy and waves are produced, and for this reason a multi-period wiggler is called an undulator. 3

History of insertion devices 1947 Vitaly Ginzburg showed theoretically that undulators could be built. 1951/1953 The first undulator was built by Hans Motz. 1976 Free electron laser radiation from a superconducting helical undulator. 1979/1980 First operation of insertion devices in storage rings. 1980 First operation of wavelength shifters in storage rings Today few tens of 3 rd generation synchrotron radiation light sources (SASE FEL) 4

Dipole magnet A dipole magnet provides us a constant field, B. The field lines in a magnet run from North to South. The field shown at right is positive in the vertical direction. In an accelerator lattice, dipoles are used to bend the beam trajectory. The set of dipoles in a lattice defines the reference trajectory: 5

Quadrupole Partice focusing magnet system Quadrupole Quadrupole has 4 poles A quadrupole magnet imparts a force proportional to distance from the center. According to the right hand rule (the force on a particle on the right side of the magnet is to the right, and the force on a similar particle on left side is to the left.) This magnet is horizontally defocusing. A distribution of particles in (x) would be defocused! What about the vertical direction? A quadrupole which defocuses in one plane focuses in the other. 6

Focus-Drift-Defocus-Drift with quadrupole Quadrupoles focus in one plane while defocusing in the other. So, how can this be used to provide net focusing in an accelerator? Consider the optical analogy of two lenses, with focal lengths f 1 and f 2, separated by a distance d: 1 f 12 = 1 f 1 + 1 f 2 d if f 1 = f 2 1 = f 1 f 2 f 12 d f 1 f 2 The key is to alternate focusing and defocusing quadrupoles. This is called a FODO lattice (Focus-Drift-Defocus-Drift). : 7

Chicane The most widely used longitudinally dispersive element is a chicane Typically consists of four dipole magnets Particles with lower energies are bent more and have longer path lengths, while particles with higher energies are bent less and have shorter path lengths One primary application of a chicane is to compress the beam to obtain high peak currents The process of bunch compression, to first order, can be described as a linear transformation 8

Chicane z = R 56 δ 1 + T 566 δ 1 2 + U 5666 δ 1 3 + R 56 2θ 2 2 L + D 3 T 566 3 2 R 56 U 5666 2R 56 9

Outline Introduction and history of insertion devices Dipole Quadropole Chicane Undulators Pure Permanent magnet Hybrid design Helical undulator Electromagnet Planar undulator Electromagnet Helical undulator Examples 10

Undulator general structure Undulator structure consist of a sequence of magnet pairs. Magnetic field along the axis is nearly sinusoidal. Spatial period of the magnetic field is λ u. Electron velocity: v. Amplitude of the electron s transverse motion (in the x-z plane): A. Electron coordinates (approximately): x = x 0 + A sin 2πz λ u, y = y 0, z = z 0 + v z t. 11

Type of undulators Synchrotron radiation emitted by relativistic particle travelling through various periodic magnetic field. This magnetic field configuration is generated by different types of insertion device, which are based on two kind of magnets: permanent magnets pure permanent magnet device hybrid device helical design electromagnets planar undulator design helical magnet 12

Pure permanent magnet undulator A magnet which does not contain iron (i. e. iron poles) or current carrying coils is called a pure permanent magnet (PPM). The ideal undulator would have a sinusoidal magnetic field along the direction of the electron beam. To achieve this field an ideal PPM undulator would have two array of permanent magnet (with the axis of the material smoothly rotating through 360 per undulator period) This can be approximated by a series of M rectangular homogeneous block per period. 13

Pure permanent magnet undulator Magnet block periods: 4 Material of magnet blocks: NdFeB, SmCo 14

Pure permanent magnet undulator Magnetic fields (of undulators having infinite width in x direction): B y = 2B r i=0 cos nkz cosh nky sin nεπ M nπ M e nkg 2 1 e nkh, B z = 2B r i=0 sin nkz sinh nky sin nεπ M nπ M e nkg 2 1 e nkh, where k = 2π λ u, n = 1 + im and ε is a filling factor. If only the first harmonic makes a significant contribution, the on-axis field components reduce to: B y = 2B r cos kz sin επ M π M e kg 2 1 e kh, B z = 0. 15

Pure permanent magnet undulator Maximum on-axis field can be achieved, when g λ u 0, M 0, h λ u 0. B y0 = 1.72B r e kg 2. Now, reaching up to 1.5 T is possible, but - requires very high remanent field material, - small magnet gap, - relatively long period. To reach higher field level is possible using permanent magnets if ferromagnetic poles are included in the design. Hybrid design 16

Hybrid insertion devices Permanent magnet + Fe-pole: B y0 3.69 e 5.07 g λ u +1.52 g λ u 2. P. Elleaume et al., Nucl. Instr. and Meth. in Phys. Res. A 455 (2000) 503-523 - Magnets must be taller and wider than the poles. - Scheme: 17

Comparison of the fields with PPM and hybrid device Hybrid undulator has advantage at longer periods. [1] 18

End poles An insertion device is required to produce, no net change: - in angle ( x ), - in position of the beam ( x). The changes are given by the following integrals: x = e γmc B y dz, R. P. Walker. Advanced insertion devices. In Proceedings of the European Particle Accelerator Conference, London, pages 310-314, 1994 x = e γmc zb y dz, The requirement for all insertion devices is that both the first and second field integral should equal zero under all operating conditions. The most common solution is a suitable selection of the end terminations for the magnet at the entrance and exit of the device. 19

End poles The simplest solutions: a) Symmetric design, Strengths: 1,-3, 4, -4,, 4, -4, 3, -1 b) Antisymmetric design, 1,-3, 4, -4,, -4, 4, -3, 1 c) and Symmetric design with longer termination. 20

Helical design If we need more circularly polarized magnetic field shapes, and variable polarization, the solution is the helical undulators. The helical design can be generated with: a) rectangular, b) circular, c) or planar geometry. 21

Rectangular geometry The rectangular geometry: two conventional undulator mounted perpendicular to each other. 22

Planar geometry This undulator consist of four standard PPM arrays. We want two orthogonal fields of equal period but of different amplitude and phase, the field components are: B x = B x0 sin B y = B y0 sin 2πz λ u + φ, 2πz λ u, 3 independent variables with this 3 variables, we can define any polarization state. 23

Planar geometry Here the a arrays of undulators moving together. D determine the phase shift between the a and b undulator arrays. a b [1] 24

Planar geometry Where the magnetic fields of the undulators are created as: a B ax = B x0 sin B ay = B y0 sin 2πz λ u, 2πz λ u, B x = 2B x0 sin φ 2 cos 2πz λ u + φ 2, B bx = B x0 sin 2πz λ u + φ, b B y = 2B y0 cos φ B by = B y0 sin 2πz 2 + φ, λ u sin 2πz λ u + φ 2. 25

Planar geometry From B x and B y Fixed phase difference between the two fields φ. 2 If φ = 0, field is vertical, and polarization is linear in horizontal, 0 < φ < π, electron travel around ellipse, (when B x = B y and φ = π/4 ellipse will become a circle) φ = π, field is horizontal, and polarization is vertical linear. The circular polarization occur, when B x0 sin φ 2 = B y 0 cos φ 2. 26

Electromagnetic planar undulator Electromagnetic undulator has the key advantage of the ability to generate rapidly time varying magnetic field. Electromagnetic undulator is made up of two steel yokes (upper and lower). Each yoke is made up of a series of poles connected to each other by a base plate, and a set of coils, which drives the field in each pole with alternating polarization. 28

Electromagnetic planar undulator Important thing in the undulator built-up: The termination of such a structure for a vanishing field integral is usually done by applying the sequence 1, 3/4, 1/4 of ampere turns on the poles of the ends. The magnetic fields can be defined in a two-dimensional approximation (by two infinite sums): B y = m B m sin mkz cosh mky, where k = 2π λ u. B z = m B m cos mkz sinh mky. 29

Helical undulator Two types of helical undulator family: a) Bifilar helix: bifilar coil winding carrying current in opposite directions, produce a helical magnet field along axis. b) Elliptical wiggler: Planar type electromagnet polarizing undulator with crossed and retarded magnetic fields. b) a) [2] [2] [2] 30

References [1] J. A. Clarke, The science and technology of undulators and wigglers, Oxford University Press, 2009 [2] H. Onuki and P. Elleaume, Wigglers, Undulators and Their applications, Taylor & Francis, 2004 [3] P. Luchini and H. Motz, Undulators and Free-Electron, Oxford University Press, 1990 31

Controlling questions 1. What does FODO mean? 2. How does the chicane work? 3. Write down the two common alternative type of undulators (examples)! 4. Which equation describes the magnetic field of the pure permanent magnet? 5. How it is possible to reach higher field level than 1.5 Tesla? 6. Compare the magnetic fields created with PPM and Hybrid device! 7. What do you know about the end poles of the PPM? 8. Write down the 3 types of the end poles! 9. What are the advantages of the electromagnetic undulator compared to PPM? 10. Define the magnetic fields in a two-dimensional approximation in the electromagnetic undulator! 32