Free Electron Lasers Lecture 2.: Insertion Devices

„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

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 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 TÁMOP-4.1.1.C -12/1/KONV-2012-0005 projekt 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.

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 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 3rd generation synchrotron radiation light sources (SASE FEL)

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 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:

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 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.

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 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 f1 and f2, separated by a distance d:

1 1 1 푑 𝑖푓 푓1=−푓2 1 푑 = + − = 푓12 푓1 푓2 푓1푓2 푓12 푓1푓2

The key is to alternate focusing and defocusing quadrupoles. This is called a FODO lattice (Focus-Drift-Defocus-Drift). :

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 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

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 8 Chicane

2 3 ∆푧 = 푅56훿1 + 푇566훿1 + 푈5666훿1 + ⋯

2 푅 ≈ −2휃2 퐿 + 퐷 56 3

3 푇 ≈ − 푅 566 2 56

푈5666 ≈ 2푅56

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 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 TÁMOP-4.1.1.C -12/1/KONV-2012-0005 projekt 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 휆푢. • Electron velocity: 푣. • Amplitude of the electron’s transverse motion (in the x-z plane): 퐴. • Electron coordinates (approximately): 2휋푧 푥 = 푥0 + 퐴 sin , y = 푦0, 푧 = 푧0 + 푣푧푡. 휆푢

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 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 electromagnets pure permanent magnet device planar undulator design hybrid device helical magnet helical design

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 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.

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 13 Pure permanent magnet undulator

• Magnet block periods: 4 • Material of magnet blocks: NdFeB, SmCo

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 14 Pure permanent magnet undulator

Magnetic fields (of undulators having infinite width in x direction): ∞ sin 푛휀휋 푀 퐵 = −2퐵 cos 푛푘푧 cosh 푛푘푦 푒−푛푘푔 2 1 − 푒−푛푘ℎ , 푦 푟 푛휋 푀 𝑖=0

∞ sin 푛휀휋 푀 퐵 = 2퐵 sin 푛푘푧 sinh 푛푘푦 푒−푛푘푔 2 1 − 푒−푛푘ℎ , 푧 푟 푛휋 푀 𝑖=0

2휋 where 푘 = , 푛 = 1 + 𝑖푀 and 휀 is a filling factor. 휆푢 If only the first harmonic makes a significant contribution, the on-axis field components reduce to: sin 휀휋 푀 퐵 = −2퐵 cos 푘푧 푒−푘푔 2 1 − 푒−푘ℎ , 푦 푟 휋 푀

퐵푧 = 0.

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 15 Pure permanent magnet undulator

Maximum on-axis field can be achieved, when 푔 휆푢 → 0, 푀 → 0, ℎ 휆푢 → 0.

−푘푔 2 ⟹ 퐵푦0 = 1.72퐵푟푒 . 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

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Permanent magnet + Fe-pole: P. Elleaume et al., Nucl. Instr. 푔 푔 2 −5.07 +1.52 퐵 ≈ 3.69 ∙ 푒 휆푢 휆푢 . and Meth. in Phys. Res. A 푦0 455 (2000) 503-523

- Magnets must be taller and wider than the poles. - Scheme:

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 17 Comparison of the fields with PPM and hybrid device

Hybrid undulator has advantage at longer periods. [1]

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 18 End poles

An insertion device is required to produce, no net change: - in angle (∆푥’), R. P. Walker. Advanced insertion devices. In - in position of the beam (∆푥). Proceedings of the

The changes are given by the following integrals: European Particle Accelerator Conference, 푒 ∆푥’ = 퐵 푑푧, London, pages 310-314, 훾푚푐 푦 1994

푒 ∆푥 = − 푧퐵 푑푧, 훾푚푐 푦

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.

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 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.

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 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.

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 21 Rectangular geometry

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

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 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: 2휋푧 퐵푥 = 퐵푥0 sin + 휙 , 휆푢 3 independent 2휋푧 variables 퐵 = 퐵 sin , 푦 푦0 휆 푢 ⇒ with this 3 variables, we can define any polarization state.

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 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]

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 24 Planar geometry

Where the magnetic fields of the undulators are created as:

2휋푧 퐵푎푥 = 퐵푥0 sin , 휆푢 휙 2휋푧 휙 ’a’ 퐵 = −2퐵 sin cos + , 푥 푥0 2 휆 2 2휋푧 푢 퐵푎푦 = 퐵푦0 sin , 휆푢

2휋푧 퐵 = −퐵 sin + 휙 , 푏푥 푥0 휆 푢 휙 2휋푧 휙 ’b’ 퐵 = −2퐵 cos sin + . 푦 푦0 2 휆 2 2휋푧 푢 퐵푏푦 = −퐵푦0 sin + 휙 , 휆푢

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 25 Planar geometry

From 퐵푥 and 퐵푦 ⇒

. Fixed phase difference between the two fields 휙 . 2 . If • 휙 = 0, field is vertical, and polarization is linear in horizontal, • 0 < 휙 < 휋, electron travel around ellipse,

(when 퐵푥 = 퐵푦 and 휙 = 휋/4 ellipse will become a circle) • 휙 = 휋, field is horizontal, and polarization is vertical linear. . The circular polarization occur, when

휙 휙 퐵 sin = 퐵 cos . 푥0 2 푦0 2

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 26 Outline

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. TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 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):

퐵푦 = 퐵푚 sin 푚푘푧 cosh 푚푘푦 , 푚

퐵푧 = 퐵푚 cos 푚푘푧 sinh 푚푘푦 . 푚

where 푘 = 2휋 휆푢.

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 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 b) electromagnet polarizing undulator with crossed and retarded magnetic fields. a)

[2] [2] [2]

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 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

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 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!

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