Undulators, Wigglers and Their Applications

Undulators, Wigglers and their Applications

Undulators, Wigglers and their Applications

Edited by Hideo Onuki and Pascal Elleaume

First published 2003 by Taylor & Francis 11 New Fetter Lane, London EC4P 4EE Simultaneously published in the USA and Canada by Taylor & Francis Inc, 29 West 35th Street, New York, NY 10001 Taylor & Francis is an imprint of the Taylor & Francis Group This edition published in the Taylor & Francis e-Library, 2004.

© 2003 Taylor & Francis All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Every effort has been made to ensure that the advice and information in this book is true and accurate at the time of going to press. However, neither the publisher nor the authors can accept any legal responsibility or liability for any errors or omissions that may be made. In the case of drug administration, any medical procedure or the use of technical equipment mentioned within this book, you are strongly advised to consult the manufacturer’s guidelines. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data A catalog record for this book has been requested

ISBN 0-203-27377-X (Adobe eReader Format) ISBN 0-415-28040-0 (Print Edition)

Contents

List of contributors Preface

vii ix

PART I

Undulators and wigglers

1

1

3

Electron beam dynamics LAURENT FA RVACQUE

2

Generalities on the synchrotron radiation

38

PAS CAL ELL E AUM E

3

Undulator radiation

69

PAS CAL ELL E AUM E

4

Bending magnet and wiggler radiation

108

RI CHARD P. WAL KE R

5

Technology of insertion devices

148

J OEL CHAVA NNE AND PASCAL E L L E AUM E

6

Polarizing undulators and wigglers

214

HIDEO ONUKI

7

Exotic insertion devices

237

S HI GEMI S A SAKI

8

Free electron lasers MARIE- EMMANUE L L E COUPRIE

255

vi

Contents

PART II

Applications 9 Impact of insertion devices on macromolecular crystallography

291 293

S OICHI WAKAT SUKI

10 Medical applications – intravenous coronary angiography as an example

322

W.- R. DIX

11 Polarization modulation spectroscopy by polarizing undulator

336

HI DEO ONUKI, TORU YAM ADA AND KAZ U TO S H I YAG I - WATA NA B E

12 Solid state physics

349

TS UNEAKI M IYAHARA

13 X-ray crystal optics

369

WAH- KEAT L E E , PAT RICIA FE RNANDE Z A N D D E N N I S M. MI L L S

14 Metrological applications

421

TERUBUMI SAITO

Index

435

Contributors

Joel Chavanne is at the European Synchrotron Radiation Facility, Grenoble. Marie-Emmanuelle Couprie is at the LURE (and CED/DSM/DRECAM), Orsay. W.-R. Dix is at HASYLAB at DESY, Hamburg. Pascal Elleaume is at the European Synchrotron Radiation Facility, Grenoble. Laurent Farvacque is at the European Synchrotron Radiation Facility, Grenoble. Patricia Fernandez is at the Advanced Photon Source, Argonne National Laboratory, Argonne. Wah-Keat Lee is at the Advanced Photon Source, Argonne National Laboratory, Argonne. Dennis M. Mills is at the Advanced Photon Source, Argonne National Laboratory, Argonne. Tsuneaki Miyahara is at the Department of Physics, Tokyo Metropolitan University, Tokyo. Hideo Onuki is at the National Institute of Advanced Industrial Science and Technology (the former Electrotechnical Laboratory), Ibaraki. Terubumi Saito is at the National Institute of Advanced Industrial Science and Technology (the former Electrotechnical Laboratory), Ibaraki. Shigemi Sasaki is at the Advanced Photon Source, Argonne National Laboratory, Argonne. Soichi Wakatsuki is at the Institute of Materials Structure Science, High Energy Accelerator Research Organization, Ibaraki. Richard P. Walker is at Diamond Light Source Ltd, Rutherford Appleton Laboratory, Oxfordshire. Kazutoshi Yagi-Watanabe is at the National Institute of Advanced Industrial Science and Technology (the former Electrotechnical Laboratory), Ibaraki. Toru Yamada is at the National Institute of Advanced Industrial Science and Technology (the former Electrotechnical Laboratory), Ibaraki.

Preface

When a charged particle is subjected to acceleration, it shakes off to radiate an electromagnetic field. If the acceleration is produced by a magnetic field, the radiation is called synchrotron radiation (SR). SR is the intense radiation over a broad spectral range produced by electrons or positrons in a bending magnet of a synchrotron or storage ring. In contrast to the SR produced in a uniform magnetic field, the spectral range can be concentrated around a few frequencies by “wiggling” the electron (or positron) beam. The device used to produce this effect was originally called a wiggler. Many short amplitude wiggles in succession serve to concentrate the radiation spatially in a narrow cone, and spectrally in a narrow frequency interval. Such a multi-period wiggler is called an undulator, a term introduced by H. Motz in 1951. The earliest consideration of undulators goes back to a theoretical paper written by V. L. Gintzburg in 1947. In 1953, Motz and co-workers constructed the first undulator, which was aimed at millimeter- and submillimeter-wave generation, and they succeeded in producing radiation up to the visible region. Undulator and wiggler devices are inserted in a free straight section of a storage ring and are, therefore, generically known as Insertion Devices. The magnetic field produced by undulators consists of many short periods in which the angular excursion of the electron beam is of the order of the natural emission angle of the synchrotron radiation (given by γ −1 = m0 c2 /E, the ratio of the electron rest mass energy to its total energy). Therefore, the radiation produced in each period interferes, resulting in a spectral density that grows proportionally to the square of the number of periods, N 2 , at some particular resonant frequencies and in a narrow cone of emission N −1/2 smaller than the natural emission angle γ −1 . The word “wiggler” now designates a device very similar to an undulator. The difference is that a wiggler has a higher field and longer period, resulting in a larger angular excursion and a lack of phase coherence of the radiation produced in two consecutive periods (essentially due to electron beam size and divergence). A consequence of the lack of interference effects is that the spectral density of the radiation produced by a wiggler is, essentially, the sum of the spectral densities produced by each period of the magnetic fields. Recently, there has been an increased demand for higher brilliance SR sources covering the spectral range from VUV to X-ray. The third generation of SR facilities that have already been built, or are being built, is dedicated to produce high-brilliance, high-energy radiation. These facilities are operated with ultra-low emittance electron beams and equipped with a large number of undulators and multipole wigglers installed in long straight sections. The undulators installed on the recently built high-energy rings can now produce highly brilliant X-rays. This has dramatically changed the type of science being performed with SR. More

x

Preface

advanced insertion devices have been developed, including polarizing undulators generating polarizing radiation of any ellipticity and other exotic insertion devices optimized for a particular application. This volume contains a detailed presentation of the radiation produced by insertion devices, the engineering, the associated beamline instrumentation, and some scientific applications. Examples of the most important and outstanding topics have been selected from a large variety of scientific fields including that of solid state physics, biology, biomedical systems, polarization modulation spectroscopy, optical engineering and metrology. The topics are intended to stimulate the reader’s interest in the many applications of insertion devices. Because of the multidisciplinary aspect of synchrotron radiation, this book is aimed at a wide range of students, researchers and engineers working in the field of synchrotron radiation. Some background knowledge of electromagnetism and the theory of relativity will prove helpful. Hideo Onuki Pascal Elleaume

Part I

Undulators and wigglers

1

Electron beam dynamics Laurent Farvacque

1

Introduction

The properties of a photon beam from a synchrotron radiation source are primarily defined by the electron beam parameters at the radiation source points, namely bending magnets or insertion devices. This chapter describes the basics of accelerator physics and points out the main parameters relevant to the use of synchrotron radiation: beam dimensions, positional stability, intensity limitations, beam lifetime etc. We shall first describe, in Section 2, the motion of a single particle, electron or positron, along the circumference of a storage ring and check its stability conditions. We shall then consider in Section 3 a beam composed of a large number of particles. The beam dimensions in space and time will be deduced from the statistical distributions on the particles. In Section 4 we shall look at the various unavoidable imperfections on a real accelerator and see how they affect the predictions of the previous theory. Then, when increasing the beam intensity, and therefore the particle density, we shall be confronted with intensity limitations resulting from the interaction between the particles and their environment. Finally, we shall identify some causes of particle losses, resulting in the finite lifetime of the particle beam.

2

Equations of motion

Generally speaking the motion of an electron (or positron) in an electromagnetic field is governed by the Lorentz equation: dp   × B) = e(E + βc dt

(1)

 with e the charge of the particle, c the velocity of light, R the position of the particle, R˙ = βc  the momentum of the particle. With this the velocity of the particle and p = mγ R˙ = mγ βc single tool we want to guide the particles on a well defined closed trajectory and give them back the energy which is radiated. In a pure magnetic field (E = 0) the energy variation is null. On the other hand, in the energy range of interest for synchrotron radiation, the magnetic  We shall therefore use:  × B is much stronger than the electric force eE. force eβc • •

magnetostatic fields for guiding the particles along the desired trajectory; electric fields for acceleration.

4 2.1

L. Farvacque Reference frame

The simplest magnetic structure used in guiding a particle is a uniform magnetic field. The trajectory is then an arc of a circle with radius ρ=

p eB

(2)

In the case of storage rings, the motion will then be studied for small deviations from a reference trajectory defined by a succession of: • •

straight sections (no magnetic field); arcs of a circle defined by bending magnets.

The coordinate system used in the following refers to a reference particle with the nominal momentum p0 travelling along this trajectory. Figure 1.1 shows the conventional orientation of the axis. When studying particle dynamics one usually also refers to the phase space defined by the position R of a particle and its momentum p.  In accelerator jargon these are replaced by the following set of coordinates in the local system:


dx dz x, x = , z, z = ds ds

2.2





Equations of motion

The description of the motion can be simplified by assuming the following conditions: 1

The trajectories have small deviations from the reference particle: x and z are small; the transverse velocities vx and vz are small compared to the longitudinal velocity vs ; the momentum p deviates slightly from the nominal value p0 . This condition is easily verified considering the dimensions of synchrotron radiation sources.

z

s

x O




Figure 1.1 Coordinate system.

Reference trajectory

Electron beam dynamics 2

5

No acceleration – we assume that there is no energy loss due to radiation, and no accelerating electric field: p is constant, vx2 + vz2 + vs2 = v 2 is constant.

3

Anti-symmetric magnetic field: Bx (x, z, s) = −Bx (x, −z, s) Bz (x, z, s) = Bz (x, −z, s) Bs (x, z, s) = −Bs (x, −z, s) This condition is verified for planar horizontal machines with a mid-plane symmetry.

Within the conditions 1 and 3, the magnetic field can be expanded in a Taylor series up to the second order in the vicinity of the reference trajectory. The normalisation with respect to the momentum is introduced by defining the following quantities: h =

B0 p0 /e

curvature of the reference trajectory

k =

∂Bz /∂x p0 /e

normalised field gradient

m=

1 ∂ 2 Bz /∂x 2 2 p0 /e

normalised field second derivative

(3)

We obtain the field expansion up to the second order in x and z: Bx = kz + 2mxz + · · · p0 /e Bz 1 = h + kx + mx 2 − (h + hk + 2m)z2 + · · · p0 /e 2 Bs = h z + (k  − hh )xz + · · · p0 /e

(4)

where the  denotes differentiation with respect to s. The equation of motion in the laboratory frame, expressed in the local axis system is: x  − h(1 + hx) − x  (hx  + h x) = z − z (hx  + h x) =

e  2 x + z2 + (1 + hx)2 [z Bs − (1 + hx)Bz ] p

e  2 x + z2 + (1 + hx)2 [(1 + hx)Bx − x  Bs ] p

(5) Considering condition 1, we introduce the momentum deviation δ = (p − p0 )/p0 1. Combining equations (4) and (5) and using p0 /p = 1/(1 + δ) ≈ 1 − δ + δ 2 we obtain the

6

L. Farvacque

development of the equations of motion to the second order: x  + (h2 + k)x = hδ − (2hk + m + h3 )x 2 + h xx  + 21 hx 2 + (2h2 + k)xδ + 21 (h + hk + 2m)z2 + h zz − 21 hz2 − hδ 2 + · · ·

(6)

z − kz = 2(m + hk)xz + h xz − h x  z + hx  z − kzδ + · · · We shall now introduce two additional simplifications: • •

we restrict ourselves to perfect ‘hard edged’ magnetic elements, where the field does not depend on s, so that we have h = h = 0; we keep only the first order in x and z.

The equation of motion then takes the simple form: 

x  + Kx2 x = hδ

with Kx2 = h2 + k

z + Kz2 z = 0

with Kz2 = −k

(7)

Motions in the horizontal and vertical planes are independent. If we first look at the horizontal motion, the solution depends on the sign of Kx2 : 1

 Kx2 >0, kx = Kx2 The equation without the right-hand side (δ = 0, the particle has the nominal momentum) describes a harmonic oscillator. Its solution is of the form x = A cos(kx s) + B sin(kx s). Consequently we have x  = −Akx sin(kx s) + Bkx cos(kx s) and the constants A and B can be obtained from the initial conditions s = 0, x = x0 , x  = x0 . This gives A = 1 and B = 1/kx . After including the term on the right-hand side, the motion can be written as 



 cos(kx s) 1/kx sin(kx s) x0 x h/kx2 (1 − cos(kx s)) = ·  + ·δ −kx sin(kx s) cos(kx s) x0 x h/kx sin(kx s)

2

 Kx2 <0, kx = −Kx2 A similar solution gives



 cosh(kx s) 1/kx sinh(kx s) x x0 h/kx2 (cosh(kx s) − 1) = ·  + ·δ kx sinh(kx s) x cosh(kx s) x0 h/kx sinh(kx s)

3

(8)

(9)

Kx2 =0 The solution is simply

x 1 = x 0



2  s x hs /2 · 0 + ·δ 1 x0 hs

(10)

A similar resolution may be applied to the vertical motion. The motion in each plane can be described by the matrix expression: X = T · X0 + D · δ

(11)

Electron beam dynamics

7

where X is the particle coordinate vector,
 x X= x and T, the ‘transfer matrix’, has the form


t T = 11 t21

t12 t22



and the property det(T) = 1 2.3

(12)

Magnetic lattice

The desired properties of a storage ring are achieved by aligning on the reference trajectory a succession of such magnetic elements, usually separating the different functions of bending and focusing. These elements are: Drift space: h = 0, k = 0, m = 0 No magnetic field. Dipole: h = constant, k = 0, m = 0 The dipole bends the trajectory with a constant radius. Expression (7) also shows that it gives focusing in the horizontal plane and is equivalent to a drift space in the vertical plane. Additional focusing may be obtained by inclining the dipole entrance and exit faces with respect to the plane perpendicular to the reference trajectory (Figure 1.2). Quadrupole: h = 0, k = constant, m = 0 Such an element gives focusing in one plane and defocusing in the other plane. Alternating focusing and defocusing quadrupoles produces focusing in both planes (Figure 1.3). Sextupole: h = 0, k = 0, m = constant This element is equivalent to a drift space at the first order but is used for higher order corrections. It is also possible to design combined function magnets, for instance, bending + focusing (dipole magnet with field index), or focusing + sextupolar strength.

N

S

Figure 1.2 Dipolar field.

8

L. Farvacque

S

N

N

S

Figure 1.3 Quadrupolar field.

3

Single particle motion

3.1

Transverse motion

We shall first look at the case δ = 0. The differential equations (7) are then similar for both the horizontal and vertical planes, and the solution developed for the horizontal plane also applies in the vertical plane. A storage ring is made up of several identical sequences of magnetic elements called superperiods. As we want the particle to circulate indefinitely in the ring we must now state that Kx2 and Kz2 in Eqn (7) are periodic functions of s with period S, length of the machine’s superperiod. Such a system is known as Hill’s equation. Starting from an arbitrary origin on the circumference, one can build the transfer matrix corresponding to one superperiod, obtained by multiplying all the single element matrices: T = Tn · Tn−1 · · · T2 · T1 3.1.1

β-functions

Now, one has to find conditions for which the motion over n turns is bound. This is usually expressed by writing for each plane the 2 ×2 transfer matrix of one superperiod in the general Twiss form, which results from det(T) = 1:

 t11 t12 cos µ + α sin µ β sin µ T= = (13) −γ sin µ cos µ − α sin µ t21 t22 The stability condition is then given by µ real, or cos µ = 21 (t11 + t22 ) < 1; µ does not depend on the choice of the origin. On the other hand α, β, γ are periodic functions of s with period S (length of the superperiod) and with the properties βγ − α 2 = 1 and α = −β  /2.

Electron beam dynamics 3.1.2

9

Betatron oscillations

If the stability condition is fulfilled, two independent solutions of the equations of motion are  C(s) = β(s) cos(ϕ(s)) (14)  S(s) = β(s) sin(ϕ(s))  where the function ϕ is defined by ϕ(s) = s ds/β(s) and is called the ‘betatron phase advance’. The trajectory of any particle can be written as a linear combination of these two trajectories: x(s) = c · C(s) + s · S(s) · x  is obtained by differentiation and the trajectory can be written in the form:


 √ x/ β √ cos ϕ sin ϕ c = · (15)  (αx + βx )/ β − sin ϕ cos ϕ s The constants c and s are given by the initial conditions s = 0, x = x0 , x  = x0 . The motion along s is a pseudo-harmonic oscillation around the reference trajectory, with local peak √ amplitude of the oscillation proportional to β(s). This motion can be √ turned into a simple harmonic oscillation by √ changing the variables s into ϕ(s), x into X = x/ β and x  into X  = dX/dϕ = (αx + βx  )/ β. As a consequence, we have, all along the trajectory, X 2 + X2 = constant, or in the initial coordinate system γ x 2 + 2αxx  + βx 2 = ε = constant

(16)

This constant ε is called the Courant–Snyder invariant of the particle. 3.1.3

Tunes

Since β(s) is periodic, the phase advance over one superperiod ϕ(s +S)−ϕ(s) is independent of the choice of the origin s. From the definitions of µ and ϕ(s) it appears that µ = ϕ(S) : µ is the phase advance per superperiod. This leads to the definition of the tune ν so that N µ = 2π ν, N being the number of superperiods in the machine, ν the number of betatron oscillations per turn. Following Eqn (16), the position xn of a particle in phase space, at turn n, at a given location around the circumference satisfies the relation: γ xn2 + 2αxn xn + βxn2 = ε = constant

(17)

In the phase space (x, x  ) corresponding to the chosen origin, the particle describes the ellipse shown in Figure 1.4. The Twiss parameters α, β, γ describe the orientation this ellipse while π ε is its surface. The particle horizontal x position seen at this location samples at the revolution frequency ω0 an oscillation with a frequency ωβ = ν · ω0 called the ‘betatron oscillation’. 3.1.4

Dispersion

We consider now the case of a particle with δ (momentum deviation) non-zero and constant. There is one particular solution of Eqn (7) so that the position and angle after one superperiod are equal to the initial values. This is obtained by solving the system: (T − I) · X = −D · δ

(18)

10

L. Farvacque x⬘

Turn 2

Turn 1

Initial

Turn 3

x Turn 4 Turn 7 Turn 5

Turn 6

Figure 1.4 Betatron motion.

where I is the identity matrix. The solution is proportional to δ. The periodic trajectory corresponding to δ = 1 is called the ‘dispersion’, the components of which are η(s), η (s). The trajectory of any particle can then be written as the sum of the general solution of the homogeneous equation plus the dispersion:   x(s) = c · β(s) cos(ϕ(s)) + s · β(s) sin(ϕ(s)) + δ · η(s) (19) The off-momentum particle √ describes the betatron pseudo-periodic oscillation, with the amplitude modulated like β(s), around the periodic off-momentum orbit δ · η(s). The constants c, s, δ are again defined by the initial conditions. 3.2

Longitudinal motion

We now look at the acceleration process: in the case of a synchrotron radiation source, it must compensate for energy losses due to radiation. Acceleration is provided by a longitudinal time varying electric field. The definition of the reference particle can be extended by stating that for each revolution (along the reference trajectory) it has to cross the accelerating gap at a time when the voltage compensates exactly the average energy loss per turn. This implies that the voltage frequency must be a multiple h of the particle revolution frequency, and can be written as V = Vˆ sin(hω0 t + φs )

(20)

with ω0 = 2πf0 = βc/R the revolution angular frequency, R = C/2π the machine radius and h the harmonic number. The phase shift φs of the reference particle with respect to the voltage is imposed by the peak voltage and the average energy loss per turn -E. -E = eVˆ sin φs

(21)

The test particle can again be described by its deviations from the reference particle: momentum deviation δ and phase deviation -φ.

Electron beam dynamics 3.2.1

11

Momentum compaction factor

By definition, the momentum compaction αC relates the orbit length of a particle to its momentum deviation: αC =

dC/C δ

(22)

This is a geometric factor related to the dispersion function introduced in Eqn (18): 1 αC = C

 C

η ds ρ

(23)

where ρ is the local radius of curvature of the reference trajectory. When studying the longitudinal motion one needs to relate the period of revolution of the test particle to its momentum deviation. This is expressed by the factor ηC derived from αC : ηC =

df/f 1 = 2 − αC δ γ

(24)

ηC depends on the energy of the particle but for electron machines where γ  1, one often makes the approximation ηC ≈ −αC .

3.2.2

Synchrotron oscillation

Over one turn, the test particle gains energy: d(-E) = eVˆ (sin(φs + -φ) − sin φs ) and drifts in phase with respect to the reference particle: d(-φ) = −2πhηC δ. By dividing by the revolution period one obtains a set of two differential equations in δ and -φ, or, by combining both, a second order differential equation describing the motion: ¨ + -φ

.2s (sin(φs + -φ) − sin φs ) = 0 cos φs

(25)

where

.2s =

eVˆ hω0 ηC cos φs 2πRp0

Integrating Eqn (25) gives an invariant for the longitudinal motion: ˙ 2 -φ cos(φs + -φ) − cos φs + -φ sin φs H = − .2s = constant 2 cos φs

(26)

12

L. Farvacque Linear approximation for small amplitudes

V ∆E ∆

0

 s

Figure 1.5 RF accelerating voltage.

For small amplitudes the differential equation of motion (25) becomes ¨ + .2s -φ = 0 -φ

(27)

and an invariant of the motion is H =

˙ 2 -φ -φ 2 + .2s = constant 2 2

(28)

This can also be written as a function of δ: I = δ2 +

.2s -φ 2 = constant 2 2 2 h ηC ω0

(29)

Again, we find a harmonic oscillation provided .2s is positive. .s , the angular frequency of small amplitude oscillations is called ‘synchrotron frequency’. It is usually much smaller than the betatron frequency. As for transverse motion, the position of the particle in phase space, now using the coordinates (-φ, δ) describes an ellipse. This linear approximation is valid in the domain where one can approximate the sinusoidal RF voltage (Figure 1.5) by its slope in the vicinity of the synchronous phase. On the other hand if the amplitude of oscillation increases, the non-linearities become significant and the trajectory in phase space deviates from an ellipse. Finally, one reaches a limit above which the motion becomes unstable. This separatrix defines a maximum momentum deviation δm from the reference (‘momentum acceptance’) and a maximum phase deviation (‘bucket length’) within which particles can be kept:  1 eVˆ cos φs δm = [2 − (π − 2φs ) tan φs ] (30) β πhηC E

3.3

Non-linearities

Obviously, the linear approximation of the transverse motion is valid in a limited domain: the basic equations of motion (Eqn (6)) have been arbitrarily truncated to the first order, even for

Electron beam dynamics

13

Bz

Defocusing

 < 0

Focusing

 > 0

x

Figure 1.6 Chromaticity compensation.

the simplest magnetic elements. For larger transverse oscillation amplitudes, it is necessary to include higher orders. 3.3.1

Chromaticity

Chromaticity is a measurement of the change in focusing with momentum deviation. It is defined as the relative tune change (horizontal or vertical) per unit momentum deviation: ξx =

-νx /νx δ

(31)

and similarly for the vertical plane. Since the tune control is crucial for the performance of a storage ring, one usually wants to minimise or at least control the chromaticity. This can be done by inserting sextupole magnets in the lattice where the dispersion is large. Figure 1.6 describes the principle of this compensation: particles with different energies oscillate about different orbits and therefore experience a different focusing strength. 3.3.2

Dynamic acceptance

In reality, even the simplest magnets considered until now are not perfectly linear, because of their finite aperture or field imperfections. In addition non-linear magnets, such as sextupoles, are introduced on purpose so that for increasing amplitudes the motion also becomes non-linear. The betatron oscillation frequency then varies with amplitude and, as for the longitudinal direction, a maximum amplitude for stable motion may be reached. This is called the ‘dynamic acceptance’ of the machine. The dynamic acceptance can be optimised primarily by tuning additional sextupole magnets so that the detrimental effects of chromaticity sextupoles can be minimised.

4

Emittances

After studying the motion of a single particle, we shall now look at the behaviour of a bunch of particles. Initially we consider that there is no collective effect, meaning that each particle

14

L. Farvacque

behaves as if it were alone. We know that at a given location along the circumference of the storage ring the position, turn after turn, of any particle in phase space (x, x  or z, z ) describes an ellipse. According to Liouville’s theorem, the particle density in phase space in the vicinity of any particle is a constant, and consequently the surface enclosed in any isodensity curve is a constant. From these two statements we can deduce that any distribution whose isodensity curves are ellipses satisfying γ x 2 + 2αxx  + βx 2 = ε is invariant over one or any number of turns. 4.1

Emittance, beam envelope, beam sizes

The surface enclosed in an isodensity curve being constant, it can be used as a measurement of the beam occupancy. The density level to be used as reference is arbitrary. It is customary for electron (or positron) machines to take one standard deviation of the projected distribution. In the horizontal plane, as the different energies in the bunch follow different trajectories, one has also to take into account the energy spread of the distribution σδ and the dispersion function η. The beam size and divergence can be deduced from Figure 1.7. The beam envelope is entirely defined by knowing ε, σδ (constants) and the functions β(s) and η(s). 4.2

Acceptance

The emittance has been defined as the area in phase space occupied by the beam. Similarly, we define the acceptance as the area in phase space where a particle can have a stable motion. The acceptance may be limited either by the maximum stable amplitude resulting from the non-linearities – this is the dynamic acceptance defined above – or by the dimensions of the vacuum chamber – physical aperture. The acceptance plays a role in the design of the injection scheme and in the lifetime of the beam. 4.2.1

Transverse acceptance

The transverse acceptance is limited by the dimension A of the vacuum chamber (Figure 1.8). It can be quantified in each plane (horizontal or vertical) by the maximum invariant value εm x⬘ √  √/

x Area 

√/

√

Size Divergence

Figure 1.7 Beam emittance.

Horizontal  εx βx + η2 σδ2  εx γx + η2 σδ2

Vertical √ √

εz βz εz γz

Electron beam dynamics

15

x⬘

x Area m

A

Figure 1.8 Acceptance of the vacuum chamber.

that can be kept within the chamber over an infinite number of turns. εm =

4.2.2

min

Circumference

A2chamber β

(32)

Longitudinal acceptance

The longitudinal acceptance is limited for two reasons: as shown in Section 3.2.2, the momentum deviation is limited by the RF system (Eqn (30)), but as off-momentum particles follow off-centred trajectories it may also be limited by the horizontal aperture of the machine:
δm =

min

Circumference

Achamber |η|

 (33)

This defines the longitudinal acceptance for particles without betatron oscillations. The momentum acceptance in case of sudden momentum jumps is further reduced by the fact that a momentum jump also induces a correlated betatron oscillation if the dispersion is non-zero. The momentum acceptance now depends on the location of the momentum jump. Equation (33) has to be modified to include the betatron oscillation, and the set of equations defining the momentum acceptance becomes:
  Achamber   δm = min √   Circumference |η| + βH ∗     1 eVˆ cos φs   [2 − (π − 2φs ) tan φs ] δm = β πhηC E

(34)

16

L. Farvacque

where the function H is defined by: H = γx η2 + 2αx ηη + βx η2 and the ∗ denotes the value of the function H at the location of the momentum jump. 4.3

Radiation excitation/damping

In addition to the interactions studied until now, the particles emit photons. The theory of synchrotron radiation will be detailed in Chapter 2 but as far as the electron motion is concerned, we shall assume now that the electron may randomly be subjected to a sudden momentum change corresponding to the energy given to the emitted photon. This implies a change of the invariants of the particle. A momentum kick -δ induces a change -I in the longitudinal invariant I defined in Eqn (29): -I = 2δ · -δ + -δ 2

(35)

It also induces a change of the horizontal invariant because the reference trajectory is different for different energies where the dispersion is non-zero:     -εx = −2 γx xη + αx (xη + x  η) + βx x  η -δ + γx η2 + 2αx ηη + βx η2 -δ 2 (36) As the photon emission is not exactly collinear with the electron trajectory, the particle may in addition experience a horizontal kick -x  resulting also in a change of horizontal invariant: -εx = 2(αx x + βx x  )-x  + βx -x 2

(37)

This last effect happens similarly in the vertical plane. On the other hand, the acceleration in RF cavities necessary to compensate for losses will restore momentum in the longitudinal direction only. Therefore, it has a damping effect on transverse oscillations. Because of all these invariant changes, the particle distribution in phase space may vary with time. The evolution of the particle distribution w(ε, t) is governed by the Fokker–Planck equation: ∂w 1 ∂2 ∂ (wA2 ) = − (wA1 ) + ∂t ∂ε 2 ∂ε 2

(38)

with δε δt→0 δt

A1 = lim

and

δε 2  δt→0 δt

A2 = lim

We look for a stationary distribution of particles. Knowing the properties of synchrotron radiation emission we can compute A1 and A2 and look for the condition ∂w/∂t = 0.

Electron beam dynamics 4.4

17

Equilibrium emittances

The average linear and quadratic invariant changes per unit time (A1 and A2 ) can be expressed as functions of a few integrals of the machine functions:    1 1 (1 − 2n)η I2 = ds I3 = ds I4 = ds 2 3 ρ3 C ρ C |ρ| C   γx η2 + 2αx ηη + βx η2 βz I5 = ds Iz = ds 3 3 |ρ| C |ρ| C The average values for the energy loss and radiated power are -E = 23 re mc2 β 3 γ 4 I2

(average energy loss per turn)

(39)

where re is the classical electron radius, re = 2.82 · 10−15 m, or in more practical units -E =

Cγ E 4 I2 2π

(40)

with Cγ = (4π/3)(re /E03 ) = 8.8575 · 10−5 m/GeV3 . We then obtain for each phase space distribution (horizontal, vertical, longitudinal) a damping time and an equilibrium distribution. Starting with the definition of damping partition numbers Jx = 1 − I4 /I2

Jz = 1

Jδ = 2 + I4 /I2

(41)

The damping times are τi =

4π T0 Ji Cγ E 3 I 2

i = x, z, δ

(42)

Horizontally, the contribution of the photon emission angle (Eqn (37)) can be neglected compared to the contribution of energy/dispersion (Eqn (36)) to the invariant growth. The horizontal emittance is εx = Cq

γ 2 I5 J x I2

(43)

√ (
c/mc2 ) = 3.84 · 10−13 m. with Cq = 55 32 3 Vertically, the only excitation comes from the photon emission angle. Usual values of vertical equilibrium emittance are so small that it can be neglected.

εz = Cq  σδ = σs =

1 Iz Jz I2

Cq

γ 2 I3 J ε I2

βc|ηC | σδ .s

(vertical emittance)

(44)

(momentum spread)

(45)

(bunch length)

(46)

18

L. Farvacque

4.5

Time structure

Following Eqn (46) the beam intensity has a Gaussian shape, with a standard deviation in time στ given by: στ =

|ηC | σs = σδ βc .s

(47)

The maximum repetition rate is defined by the harmonic number h chosen for the RF system and is obtained when all the available buckets are filled. This so-called ‘multibunch operation’ gives the maximum average intensity. At the other extreme, a minimum repetition rate may be achieved by filling only one of the buckets: this is the ‘single bunch operation’, providing the maximum peak intensity and giving the possibility of time-resolved experiments. In between, any filling pattern may be envisaged to reach a compromise between average intensity and time resolution. The repetition frequency is chosen between the two extreme cases: ωmin = ω0 =

βc R

(48)

ωmax = hω0 4.6

Matching of β functions

The previous equations set up the main constraints for the design of synchrotron radiation sources: • • •

The horizontal β-function and the dispersion must be optimised to reduce the integral I5 and therefore the horizontal emittance. A basic feature is a small βx value in the dipoles, where radiation occurs. The energy spread cannot be varied significantly for a given bending radius, but the bunch length can be modified through optics tuning (ηC ) or RF parameters (.s ). Dipole field index n allows modifying the sharing of emittances and damping times between horizontal and longitudinal directions through the integral I4 .

In addition, the β-function in both planes can be matched at the radiation source points to best fit the photon beam users. The emittance of the photon beam is the convolution of the single electron photon emission (fixed) and the electron beam emittance (tuneable). For a given emittance value the ratio size/divergence (equal to β) can be chosen so that: •

•

If the emittance is larger than the diffraction limit, the single electron emission can be neglected and the minimum size on the sample (without focusing) calls for large β values (of the order of the distance from the source to the sample). Focusing the electron beam downstream the beamline could even give smaller spot sizes. Minimising the width of harmonics also calls for a large horizontal β (or small angular divergence). If the electron beam emittance approaches the diffraction limit, the spot size becomes independent of the electron optics. Maximum brightness is then achieved when the electron and diffraction emittances are matched. This corresponds to small β values (half the undulator length). This applies to the vertical plane where the emittance is naturally small and also horizontally when the photon beam is focused on the beamline.

These conditions have led to a few basic lattice design.

Electron beam dynamics 40

0.8 x 

0.7

30

0.6

25

0.5

20

0.4

15

0.3

10

0.2

5

0.1

0

5

10

15

20

25

 (m)

x (m)

35

0

19

0

s (m)

Figure 1.9 Expanded Chasman–Green lattice.

4.6.1

Double bend achromat

The Chasman–Green lattice is a compact lattice set to have zero dispersion in the straight sections, for minimising the beam size. Figure 1.9 shows the horizontal β-function and dispersion. The theoretical minimum emittance of such a lattice can be computed. Equation (49) gives the value in the simple case where all the bending magnets are identical: Cq γ 2 εx = √ 4 15Jx




2π Nmag

3 (49)

The theoretical minimum emittance scales with the third power of the deflection angle of one bending magnet: increasing the number of superperiods and consequently the machine length reduces the emittance. If the condition of zero dispersion is relaxed, the theoretical minimum emittance is even smaller: Cq γ 2 εx = √ 12 15Jx




2π Nmag

3 (50)

However, the beam size in the straight sections now depends on the energy spread of the beam and on the dispersion value. A compromise between dispersion and emittance has to be made to get the minimal beam size.

4.6.2

Triple bend achromat

This type of lattice, with the same constraint of zero dispersion in the straight sections has a slightly smaller emittance than the Chasman–Green lattice: it is given by Eqn (51) for

L. Farvacque 16

0.8 x 

14

0.7 0.6

10

0.5

x (m)

12

8

0.4

6

0.3

4

0.2

2

0.1

0

0

2

4

6

8 s (m)

10

12

14

16

 (m)

20

0

Figure 1.10 Triple bend achromat lattice.

identical bending magnets. Its β-function and dispersion are plotted in Figure 1.10. 7Cq γ 2 εx = √ 36 15Jx




2π Nmag

3 (51)

However, for technical reasons the emittance for realistic lattice designs is always much larger than the theoretical optimum, and the choice of the lattice is governed by many other conditions. The triple bend achromat lattice has been used mainly for small rings while on larger rings, its small dispersion value makes the chromaticity correction more difficult and the double bend achromat is usually preferred.

5

Perturbations

Up to now we have been considering a perfect machine, and in particular perfect magnetic fields, perfectly identical magnets and a perfect alignment on the reference trajectory. In practice, one now has to look at the detrimental effect of errors in all respects. 5.1

Resonances

We now introduce a single field error at one location. The motion turn after turn in the normalised phase space (X, X  ) is represented by circles (Figure 1.11). A particle initially perfectly centred experiences a kick -x  on each turn. For simplicity, we shall take the example of a dipolar error with an integer betatron tune. For an exact integer tune, the amplitude will grow turn after turn until the particle is lost. If the tune differs slightly from the integer, after some time the kick will be out of phase with the particle oscillation and will start reducing the amplitude. The same applies to a quadrupolar

Electron beam dynamics

21

X⬘ Kick 2 Kick 1

Turn 2 Turn 1 X

Figure 1.11 Resonant excitation.

kick and a half-integer tune and similarly to higher order multipolar fields and rational tune values. Generally speaking, a resonance line is defined by a line in the tune diagram (νx , νx ) with equation mνx + nνz = p where |m|+|n| is the order of the resonance, corresponding to 2(|m|+|n|)-pole field errors and p is the harmonic number. When p is a multiple of the number of superperiods in the machine, the resonance is called systematic and is excited by the main magnetic fields of the structure (dipoles, quadrupoles, sextupoles, higher multipolar fields present in the magnets and so on). When p is not a multiple of the periodicity of the machine, the resonance is non-systematic and can only be excited by the non-identity between the superperiods (caused by magnet manufacturing tolerances, imperfect alignment and so on). Non-systematic resonances are usually much weaker than systematic ones. The effect of resonance may be limited by: • • • •

5.2

a choice of the working point (νx , νx ) away from the lowest order resonances. It is also necessary to limit the tune spread, due, for instance, to the chromaticity and energy spread of the beam; for non-systematic resonances, powering a few corrector magnets may cancel the contribution of magnetic field errors to a given harmonic of a resonance and partially restore the periodicity of the structure; considering the radiation damping which acts against the invariant growth; getting Landau damping: when the betatron tune shifts as amplitude grows, because of non-linearities, the particle goes out of synchronism with the resonance. Horizontal/vertical coupling

The initial assumption of the mid-plane symmetry of magnetic fields ensured a full decoupling of horizontal and vertical motions. Since the vertical equilibrium emittance is extremely small, the beam cross section should be a horizontal line. Practically a fraction of the horizontal motion transfers into the vertical direction. Several factors are involved: •

Betatron coupling: a tilted quadrupole bends vertically a particle horizontally off-centred. In such a case, a part of the horizontal motion is transferred into the vertical plane in such

22

L. Farvacque a way that the sum of the emittances is preserved: εx + εz = ε0 . A coupling coefficient k is defined as k = εz /εx

•

•

5.3

(52)

Powering skew quadrupole correctors can compensate this effect. Vertical dispersion: any vertical bending of the beam (resulting for instance from a tilt angle of dipole magnets) generates vertical dispersion. Consequently the synchrotron radiation emission excites a vertical betatron oscillation, as in the horizontal plane, and contributes to the vertical emittance. Coupling of the horizontal dispersion into the vertical plane: tilted quadrupoles at locations where the horizontal dispersion is non-zero also create vertical dispersion with the same consequences as above. This can also be used for correction by powering skew quadrupole correctors to try to cancel the spurious vertical dispersion. Orbit distortions, beam stability

The reference trajectory is defined assuming perfect magnetic elements. In reality, unavoidable imperfections will cause the trajectory of the beam centre of mass (closed orbit) to deviate form this perfect orbit. The main errors come from •

transverse (horizontal or vertical) misalignment of quadrupoles.

Other errors have a smaller contribution: • • • • •

errors in bending magnet length or field; bending magnet tilt; misalignment of other elements (dipoles, sextupoles, etc.); magnetic field variations in the magnetic elements (fluctuations of power supplies or geometry modification following thermal effects); parasitic external magnetic fields.

All these errors generate an angular kick on the trajectory at the location of the error. The closed orbit distortion generated by a single kick can be easily computed: for instance, in the horizontal plane √  -xkick β(s) · βkick -x(s) = (53) cos (πν − |ϕ(s) − ϕkick |) 2 · | sin πν| where βkick , ϕkick are optical functions at the kick location and  -xkick =

-(B · l) (p/e)

is the angular kick generated by the integrated field error -(B · l). For several errors, one simply adds all the orbit distortions (assuming linear optics). For time-varying perturbations one can consider that the beam stabilises on the distorted closed orbit after a few damping times (typically a few milliseconds). Therefore, the centre of mass motion can be deduced from the perturbation behaviour using the static formula up to a few hundreds of Hertz.

Electron beam dynamics

23

x⬘

Macroscopic emittance 

∆x⬘

10% 10%

x

Displaced emittance Nominal emittance 0 Center of mass invariant co

∆x

Figure 1.12 Macroscopic emittance growth.

At any point the perturbation can be measured by -x, -x  and the degradation is quantified by relating this perturbation to the equilibrium beam size and divergence σx , σx  . This is done by introducing a ‘macroscopic emittance growth’, -ε/ε = (ε−ε0 )/ε0 , envelope over a period of time of the instantaneous displaced emittances of the beam (Figure 1.12). The emittance growth has the interesting properties that • • •

it is independent of the location along the circumference of the machine; it ensures a fair balance between position and angle errors all around the machine; it is also constant along a beam line for any drift space or focusing.

2 + 2αx x  + βx 2 , the Courant–Snyder The emittance growth can be related to εco = γ xco co co co invariant of the closed orbit, possibly time-dependent:  -ε εco (54) =2 ε ε0

5.4

Perturbations induced by insertion devices

The disturbance introduced by insertion devices results from two contributions: • •

the perturbations resulting from a perfect insertion device; the effect of errors in the insertion device field.

In both cases the perturbation may be enhanced by the fact that the insertion device can be turned on or off at any time. 5.4.1

Perfect insertion device

A perfect insertion device induces the following: • •

an additional focusing, consequently destroying the machine periodicity; higher order field components possibly exciting non-systematic resonances and reducing the dynamic aperture;

24

L. Farvacque

•

a change in equilibrium emittance, in the case of a non-zero dispersion in the insertion device, or if the dispersion generated by the insertion device itself cannot be neglected.

The simplest approximation (Halbach’s formula) for the field of an insertion device with period λ is given by Bx = (kx /kz ) · B0 · sinh kx x · sinh kz z · cos ks Bz = B0 · cosh kx x · cosh kz z · cos ks

(55)

Bs = −(k/kz ) · B0 · cosh kx x · sinh kz z · sin ks where B0 is the peak magnetic field, λ the period length and kx2 + kz2 = k 2 = (2π/λ)2 . kx expresses the transverse variation of the field due to the limited pole width. It is zero for infinitely wide poles and is imaginary for standard insertion devices. The corresponding focusing strengths (see Eqn (7)) and tune shifts are then given by Kx2 =

kx2 B02 2k 2 (p/e)2

B02 kx2 L 8πk 2 (p/e)2

-νx =

(56) k 2 B02 Kz2 = z2 2k (p/e)2

B02 k2 -νz = z 2 L 8πk (p/e)2

where L is the total length of the insertion device. The focusing effect is independent of the period length λ, and in the simple case kx = 0, kz = k, it is null in the horizontal plane. This simple field approximation is valid far away from the pole surfaces but generally gives a poor approximation for realistic insertion devices. However, it describes the main effect of a perfect insertion device: a focusing effect, mainly in the vertical plane, inversely proportional to the square of the momentum. This focusing effect is noticeable on low-energy machines, but can be negligible on high-energy storage rings. The same scaling applies to higher order multipolar fields. A more general field distribution can be studied [1] in the following approximations: 1

The field integrals in both planes are vanishing over the insertion device: 

∞

−∞

2

 Bx ds = 0

∞

−∞

Bz ds = 0

(57)

The double field integrals in both planes are also vanishing over the insertion device: 

∞



s

−∞ −∞



Bx ds ds = 0



∞



s

−∞ −∞

Bz ds  ds = 0

(58)

These conditions express the basic properties of an insertion device: the field integral should not induce any angle or any displacement of the reference trajectory. We will add the additional approximation that the initial horizontal and vertical angles of the trajectory are zero or

Electron beam dynamics

25

extremely small: x  (−∞) = z (−∞) ≈ 0 Then the angles of the trajectory at the exit of the insertion device are given by: 
 ∞ ∂ 1 1  x (∞) = − <(x, z, s) ds + o (p/e)3 2(p/e)2 −∞ ∂x 
 ∞ ∂ 1 1  <(x, z, s) ds + o z (∞) = − (p/e)3 2(p/e)2 −∞ ∂z The angular kicks experienced by the particle are derived from the function
 s 2
 s 2   <(x, z, s) = Bx ds + Bz ds −∞

−∞

(59)

(60)

As usual insertion devices are periodic, the function < can be integrated over one period, resulting in a potential U given by  U (x, z) = <(x, z, s) ds (61) 1 period

The angular kick experienced by a particle over the undulator period is then ∂U 1 (x, z) 2 2(p/e) ∂x ∂U 1 -z = − (x, z) 2(p/e)2 ∂z

-x  = −

(62)

In the ideal case of the analytical field expansion (Eqn (55)), this potential expression gives again the main focusing effect. But the potential U can be obtained for any kind of insertion device, either from magnetic field computations or from magnetic measurements. 5.4.2

Field errors

In the previous paragraph we assumed that the insertion device restores at its end both: •

the beam angle:  ∞ Bx,z ds = 0 −∞

•

the beam position:  Bx,z ds ds  = 0

In practice, this is not exactly true because of imperfections. The construction errors can be minimised using shimming techniques. The difficulty is increased by the fact that these conditions must be fulfilled for any tuning of the insertion device. Residual errors can be compensated by powering correcting magnets at each end of the insertion device.

26

6

L. Farvacque

Collective effects

We can now look at what happens when we try to increase the intensity in the storage ring: at some point we can no longer consider that particles ignore each other. As particles are relativistic the space charge effect vanishes completely, but one has to take into account the fact that the beam has to be surrounded by a vacuum chamber built with conducting material. Electromagnetic fields will develop in this volume and possibly interact with the beam itself. This will lead to instability if any perturbation in the particle density grows exponentially. As the particle density in the bunch grows, the probability of interaction between particles increases. Multiple scattering between particles, called ‘intrabeam scattering’ will modify the equilibrium emittance of the beam and prevent it from reaching extremely small emittances. These effects appear in many different ways: longitudinal or transverse direction, single bunch or multibunch operation. They are responsible for the intensity (and brightness) limitation of synchrotron radiation sources.

6.1

Interaction with the vacuum chamber

The first step is to look at the electromagnetic field created by a single electric charge travelling at the velocity of light in a conductive beam pipe. We can split the problem by studying first the longitudinal field (considering a particle along the axis of a cylindrically symmetrical pipe) and then the transverse field (null as long as the particle is centred but varying with its transverse position).

6.1.1

Longitudinal wake field

The Green’s function is obtained by integrating the longitudinal electric field seen at a fixed distance (or delay τ ) behind a single punctual charge q. It corresponds to a longitudinal accelerating (or decelerating) voltage seen by a test particle following the charge q over one turn, per unit charge: 1 G (τ ) = − q




 C

E

 s s, t = + τ ds βc

(63)

This quantity completely defines the influence of the environment on the test particle. Usually it cannot be computed analytically for real vacuum chamber geometry but it can be estimated with computer codes. The total accelerating voltage (called ‘wake potential’) resulting from a real beam, knowing the particle line density λ(τ ) and the total charge of the bunch Q, can be obtained by integration:  W (τ ) = −Q ·

∞ −∞

G(τ − τ  )λ(τ  )dτ 

(64)

Going to the frequency domain, the convolution of the beam line density with the Green’s function is transformed into the product W (ω) = −I (ω) · Z (ω)

(65)

Electron beam dynamics

27

which suggests the definition of the coupling impedance Z:  Z (ω) =

∞

−∞

 I (ω) = Q

G (τ ) e−jωτ dτ

∞

−∞

λ(τ ) e−jωτ dτ

(longitudinal coupling impedance)

(66)

(beam intensity)

(67)

The wake potential W (τ ) can be considered as a perturbation to be added to the accelerating voltage provided by the RF system. In a perfectly conducting cylindrical vacuum chamber, the wake field would be null. The main contributions to the impedance come from: Resistive wall: The resistance seen by the image charge circulating in the vacuum chamber creates a longitudinal electric field component on the beam axis. High-Q resonators: The objects are typically the accelerating RF cavities, with their fundamental mode and possibly higher order mode (HOM). The wake field generated there extends over a long period so that it can affect several consecutive bunches. High-Q resonators are involved in multibunch instabilities. Broadband resonator: All the discontinuities of the cross section of the vacuum chamber can be cumulated in a broadband resonator (usually Q = 1) summarising their effect: ZBB (ω) =

Rs 1 + jQ((ω/ωr ) − (ωr /ω))

(68)

The resonance frequency ωr is related to the vacuum chamber radius. The influence of a broadband resonator is limited to very short-term: the field generated by the head of the bunch influences its tail but disappears before the next bunch. It is mainly involved in single bunch instabilities. The quantity of interest in longitudinal instabilities is in fact Z(ω)/ω or even better Z(p)/p, with p a harmonic of the revolution frequency: p = ω/ω0 . Figure 1.13 shows the main contributions to the impedance.

Z (p) Re  ||   p 

Resonator

Z (p) Im  ||   p 

Resistive wall Broad band

p

Figure 1.13 Longitudinal impedance spectrum.

p

28

L. Farvacque

6.1.2

Beam signal

It appears now that the perturbation seen by a particle is the consequence of the interaction between the beam spectrum and the coupling impedance of the machine. Therefore, we need to express in the frequency domain the signal created by a circulating unperturbed (Gaussian) beam, followed by the signal created by any perturbation of this distribution. Instability will occur if the spectrum of a perturbation interacts with the impedance so that the perturbation amplitude grows. Longitudinally the signal of a stationary Gaussian bunch is a line spectrum at multiples of the revolution frequency:

 p 2 ω02 στ2 S (ω = pω0 ) = I (69) δ(ω − pω0 ) exp − 2 p The envelope of the spectrum extends up to a frequency 1/στ . This frequency is usually similar to the resonance frequency of the broadband impedance of the vacuum chamber. The interaction occurs mainly with the inductive part of the resonator. This is why the coupling impedance is sometimes described only by the inductance value. Perturbations may be added to this distribution. They can be expressed as a combination of modes defined in the following way: • •

Mode 1 corresponds to a global oscillation of the bunch in phase and in energy around its equilibrium position at a frequency .s . Mode 2 is an oscillation of the bunch length and energy spread at frequency 2.s .

The spectrum of mode m shows sidebands at frequency m.s apart from each harmonic of the revolution frequency pω0 . Figure 1.14 shows the spectrum of the stationary distribution and of modes 1 and 2 in the absence of influence of the self-induced field. It is plotted as on a real spectrum analyser with negative frequencies folded over positive ones. For multibunch perturbations, each mode is subdivided into sub-modes differing by the phase difference between successive bunches. 6.1.3

Longitudinal instabilities

The interaction between the bunch spectrum and the real part of the coupling impedance creates power dissipation and an asymmetry of the bunch longitudinal profile. The interaction with the imaginary part creates a shift of synchrotron frequency and bunch lengthening (or shortening). Power dissipation: The interaction with the resistive part of the impedance creates losses: Equation (64) gives the energy loss for the test particle. Integrating over the whole bunch gives the energy loss for the bunch. This shows a Q2 , square of the bunch charge. The total energy loss can be expressed as -E = −kLOSS · Q2 , where k the ‘loss factor’ is  ∞ 1 kLOSS = Re(Z(ω))|λ(ω)|2 dω (70) 2π −∞ The losses depend on the bunch line density λ: a short bunch extends higher in frequency and creates more losses. The synchronous phase is shifted so that the losses are restored by the RF system.

Electron beam dynamics

29

Power spectrum

Stationary distribution

Mode 1

Mode 2

0

1

2

3

4

5 p = / 0

6

7

8

Figure 1.14 Longitudinal beam spectrum.

Potential well distortion: The imaginary part of the impedance (inductance over the lower frequency range and capacitance above the resonance frequency) creates a voltage over the bunch length that modifies the voltage created by the accelerating cavity. Consequently, the stationary distribution of particles will be modified as will the incoherent synchrotron frequency, synchronous phase and bunch length. For realistic impedances, this results in a bunch lengthening for ηC < 0 or shortening for ηC > 0, with a constant energy distribution. This will happen up to a threshold corresponding to an infinite or vanishing RF voltage slope. Above this threshold, the energy spread of the bunch will grow while the bunch lengthens. According to Equation (46), the bunch length can be reduced at zero current by reducing ηC . However, the lengthening due to the longitudinal coupling impedance gives a lower limit, function of the beam current but independent of ηC , for the minimum bunch length [2]. Coupled bunch instabilities: High-Q resonators, like higher order modes of accelerating cavities may induce wake fields which couple adjacent bunches in multibunch operation. In such cases, the impedance is narrow band while the spacing between the lines of the beam spectrum is increased by the number of bunches. It may then be possible to tune the resonators so that they do not interfere with the beam. 6.1.4

Transverse wake field

If the beam is displaced from the axis of a symmetrical vacuum chamber, the electromagnetic induced fields also have a transverse component. Among the numerous azimuthal modes, one considers only the lowest one, the dipole mode, the strength of which is proportional to the displacement - of the beam. This mode gives a deflection of the whole beam and may

30

L. Farvacque

generate coherent transverse oscillations of the beam centre of mass. Following what was done for longitudinal direction, we define a transverse Green function based on the integral of the deflecting force seen by a test particle following the reference particle, both off-centred by an offset -:
  s 1 G⊥ (τ ) = (E + βcB)⊥ s, t = + τ ds (71) q-β C βc The beam signal (‘transverse intensity’) is defined as the product of the beam transverse position by its intensity, as it can be measured on an ideal beam position monitor. The perturbation is now the deflecting force, proportional to the elongation - and integrated over one turn. This adds to the focusing provided by the quadrupoles. The transverse coupling impedance is defined as:  ∞ Z⊥ (ω) = j G⊥ (τ ) e−jωτ dτ (72) −∞

6.1.5

Transverse impedance

Impedance contributions are the same as for longitudinal direction. For simple cylindrical geometry with radius b, the transverse impedance is linked to the longitudinal impedance by the simple formula Z⊥ (ω) = 6.1.6

2c Z (ω) b2 ω

(73)

Transverse beam signal

The unperturbed beam is on-axis and consequently the signal from the stationary distribution is null. Only perturbations contribute to the transverse signal of the beam. We define perturbation modes as the following: • •

mode 0 in a global transverse oscillation of the whole bunch; mode 1 shows opposite oscillation amplitude at the head and tail of the bunch (1 node over the bunch length).

More generally mode m has m nodes over the bunch length and its spectrum has lines at frequencies (p + ν)ω0 + m.s . The symmetry point of the envelopes of all these modes is shifted from the zero frequency by a value ωξ depending on the chromaticity (Figure 1.15): ωξ = νω0

6.1.7

ξ ηC

(74)

Transverse instabilities

The interaction of the beam spectrum with the imaginary part of the impedance produces tune shifts while the interaction with the real part gives damping or anti-damping. As suggested by Eqn (73), the transverse impedance spectrum is similar to the longitudinal one.

Transverse power spectrum

Electron beam dynamics

31

Mode 0

Mode 1

0

Mode 2



Figure 1.15 Envelope of transverse spectrum.

Tune shifts: The interaction of the spectrum of mode 0 (rigid transverse oscillation of the bunch) with the imaginary part of the transverse impedance adds to the focusing effect of the main quadrupoles. This results in a coherent tune shift proportional to the bunch intensity. This can be seen mainly in single bunch operation. Head-tail instability: The wake field generated by the head of the bunch creates a deflecting force on the tail. At the same time the longitudinal oscillation within the bunch interchanges the particles between head and tail and modifies their transverse oscillation frequency through the chromaticity. This leads to a current threshold above which the coherent motion of the bunch is anti-damped. However, since a variation of chromaticity displaces the envelope of the different transverse coherent modes, it is possible to raise these thresholds by moving the most dangerous modes with respect to the real part of the impedance. 6.2

Intrabeam scattering

As the particle density in the bunches increases, the probability of collisions between particles becomes greater. If the number of collisions with small exchanges of momentum during a damping time is large, it causes a significant change in the equilibrium distribution. Without taking into account the radiation damping, this would correspond to a modification of the sharing of oscillation energies between the three planes, satisfying the relation a · σδ2 ηC + b · νx εx + c · νz εz = constant where a, b, c are constants depending on the lattice.

(75)

32

L. Farvacque

The behaviour depends on the sign of ηC . For ηC < 0, as it is for most synchrotron radiation sources, the three invariants may grow indefinitely, taking energy from the RF system. The growth rates of the invariants have to be compared with the radiation damping times. They are inversely proportional to a density factor A involving the six-dimensional volume of the bunch: A=

re2 cN 64π 2 σs σδ σx σy σx  σz β 3 γ 4

(76)

The energy dependence in β 3 γ 4 indicates that the effect mostly affects low energy machines. The theoretical minimum emittance values given in Eqns (49)–(51) suggest that one could reduce the emittance to any desired value by increasing the machine dimensions. Intrabeam scattering gives an absolute lower limitation of the emittances, at least for reasonable intensities.

7

Beam lifetime

The distribution of the particles in phase space, resulting from the equilibrium between synchrotron radiation and damping is Gaussian. However, it has to be modified to take into account the longitudinal or transverse limitations of the oscillation amplitudes. In addition, other effects may contribute to the excitation of oscillations. This leads to a rate of loss of particles expressed as the beam lifetime τ so that: 1 1 dI =− τ I dt

(77)

Different loss processes will be shown and the resulting lifetime will be the combination of all effects given by 1  1 = τi τ

(78)

i

When computing lifetimes we shall refer to the different acceptance values defined in Section 4.2: εm the transverse acceptance (x or z) and δm the momentum acceptance. 7.1

Quantum lifetime

The stationary distribution of particles is deduced from the Fokker–Planck equation. In Section 4.4, we computed the distribution for an infinitely wide aperture. A correction must be added to match the boundary condition that the particle density must be zero at the limit of aperture. The resulting lifetime for a limitation in one plane (x, z or δ) is:   r 1 rx,z,δ x,z,δ = exp − τq τx,z,δ 2

(79)

where τx,z,δ is the damping time in the considered plane and rx,z,δ is the relative aperture: rx,z

εm = = εx,z




xm , zm σx,z

2 rδ =

2 δm σδ2

Electron beam dynamics

33

This contribution to the lifetime is only significant for very small apertures. One usually considers that it can be neglected when the aperture is larger than six standard deviations of the beam size.

7.2

Coulomb scattering

This is a part of the interaction of the circulating particles with the residual gas in the vacuum chamber. We consider elastic scattering of a particle interacting with the nucleus of an atom of the residual gas. We restrict ourselves to angles large enough so that the particle gets out of the transverse acceptance of the machine. The differential cross section for Coulomb scattering is given by the Rutherford formula: dσ = d.




re Z 2γβ 2

2

1

(80)

4

sin (θ/2)

where θ is the scattering angle, Z the charge of the nucleus and re the classical electron radius: re = 2.82 · 10−15 m. This cross section has to be integrated over all scattering angles larger than the transverse angular acceptance of the machine εm /β. Usually, the angular acceptance in storage rings is smaller in the vertical plane, because of constraints in magnet and insertion device gaps. If we neglect the limitation in the horizontal plane, we get
σ = 2π

re Z γβ 2

2

βz εm

(81)

The lifetime is then obtained by combining the effect of all components of the residual gas and integrating over the circumference:

1 = τsc

2π re2 c γ 2 β 3 kT

    1 Zj2 αij βz pi  εm atomj

(82)

gas i

where αij is the number of atoms j per molecule i, pi the partial pressure of gas i, Zj the charge of atom j , k the Boltzmann constant (k = 1.38 · 10−23 J/K), T the absolute temperature and re the classical electron radius defined above. The Coulomb scattering lifetime depends strongly on the electron energy (γ −2 ) and on the transverse aperture of the machine. In case the pressure and β functions vary along the circumference, it is important to have a low pressure where β is large.

7.3

Bremsstrahlung

Here we look at inelastic scattering of the particles on the residual gas. The loss is due to the longitudinal acceptance of the machine. The differential cross section for an energy loss

34

L. Farvacque

between Eb and Eb + dEb is given by: 4re2 Z 2 dEb dσ = F (E, Eb ) 137 Eb

(83)

For high-energy electrons, the function F is given approximately by the Bethe–Heitler formula: 

2 
 183 1 Eb Eb Eb 4 F (E, Eb ) = ln 1/3 + + 1− 1− E E 9 E 3 Z

(84)

This cross section has to be integrated between the energy acceptance of the machine Em and infinity, and then integrated over the circumference of the machine. With some approximations the result is

1 4re2 c = τb 137 kT




5 4 1 − ln δm 6 3

  atomj

 Zj2 ln

183  1/3

Zj

 αij pi 

(85)

gas i

The Bremsstrahlung lifetime is roughly independent of energy. It is limited by the momentum acceptance of the machine. In similar conditions a low-energy machine is more sensitive to Coulomb scattering while a higher energy machine is dominated by Bremsstrahlung.

7.4

Touschek effect

For large intensities, the particle density in the bunch becomes large. As particles are oscillating independently inside the bunch, the probability of collisions between particles increases and may result in particle loss. The longitudinal oscillation frequency is usually much smaller than the transverse ones: in the bunch frame, the oscillation energy is larger in the transverse planes than in the longitudinal plane. A collision thus mainly transfers transverse energy into the longitudinal plane. Moreover, in the case of synchrotron radiation sources the horizontal beam size is much larger than the vertical one, so that we can restrict ourselves to the simple case of collisions transferring horizontal motion into the longitudinal direction. In the referential defined by the centre of mass of the two colliding particles, the differential cross section is given by the Möller formula: dσ¯ r2 = e4 d. 4β¯




4 sin4 θ

−

3 sin2 θ

 (86)

where σ¯ is the cross section in the centre of mass frame and β¯ the relative velocity of each particle in the centre of mass frame, supposed non-relativistic (β¯ 1).

Electron beam dynamics

35

0.35

0.3

D (v)

0.25

0.2

0.15

0.1

0.05

0 10–4

10–3

10–2 v

10–1

100

Figure 1.16 Universal function D.

This cross section is integrated for all angles resulting in a momentum deviation δ larger than the momentum acceptance of the machine. 1 N re2 c = D(v) 3 τt 8π σx σz σs β 3 γ 2 δm

(87)

where σx , σz and σs are the dimensions of the bunch: σx2 = βx εx + η2 σδ2

σz2 = βz εz

σs = bunch length

N is the number of particles in the bunch, D, the universal function defined by Eqn (88) and plotted in Figure 1.16: √
 ∞  ∞ −u  v ln u −u e −v D(v) = −3e + v e du + (3v − v ln v + 2) du 2 u u v v

(88)

v, the dimensionless parameter defined by δ 2 βx v = m2 γ εx



σx2 σx2 + σδ2 (αx η + βx η )2



The Touschek lifetime is inversely proportional to N , the number of particles in the bunch, or to the beam intensity per bunch. It is dominant in single bunch operation.

36

8

L. Farvacque

Conclusion

The design of a storage ring used as a synchrotron radiation source is guided by the aim of providing the highest brightness. This involves many parameters whose limitations are summarised here. 8.1

Emittance

Emittance reduction is an obvious way of improving the brightness of a synchrotron radiation source. This is true up to the point where the electron emittance is of the order of the single electron photon emittance (diffraction limit). 8.1.1

Horizontal emittance

A small emittance is obtained by an adequate tuning of the focusing of the storage ring. Low emittance lattice designs are well known. The emittance scales like γ 2 · θd3 , θd being the dipole angle, so that for a given energy, a smaller emittance implies a larger machine. In such a scheme, the emittance reduction is ultimately limited by intrabeam scattering (intensity dependent effect). 8.1.2

Vertical emittance

Vertical emittance is a consequence of errors: it can be reduced to less than 1% of the horizontal emittance and is now close to the diffraction limit. Optimisation may come from a good matching between electron and photon emittances by adjusting the vertical β function. 8.2

Position stability

High requirements on position stability are the consequence of the emittance reduction. This is of course more severe in the vertical plane. Vibration problems can be solved by precautions to be taken in the environment of the storage ring, use of vibration dampers and feedback systems. Low frequency drifts, mainly related to thermal effects are more difficult to control. 8.3

Intensity

The beam intensity is limited by instabilities resulting from the coupling impedance of the vacuum chamber. In multibunch operation, it usually comes from coupled bunch instabilities driven by parasitic high-Q resonators such as higher order modes of RF cavities. Various techniques have been developed to minimise this effect. In single bunch operation, the limitation comes from the broadband impedance of the vacuum chamber. 8.4

Time structure and peak intensity

Shorter bunches could improve time resolution and increase the peak intensity for a given total current. The influence of machine tuning, by reducing the momentum compaction factor, is again limited by the longitudinal vacuum chamber impedance. The other alternative is the action on the RF voltage slope, calling for larger voltage and larger frequency, possibly using harmonic cavities.

Electron beam dynamics 8.5

37

Lifetime

All means of increasing the brightness are detrimental to the lifetime: small emittances, high current, short bunches and so on. Touschek lifetime appears to be limiting all modes of operation; optimisation may come only from the enlargement of the momentum acceptance of the machine.

Bibliography Bruck, H., Accélérateurs Circulaires de Particules (Presses Universitaires de France, 1966). Sands, M., The Physics of Electron Storage Rings. An Introduction Proceedings of the International School of Physics ‘Enrico Fermi’, 1971–46. Wiedemann, H., Particle Accelerator Physics I & II (Springer, 1994). Steffen, K., ‘Basic course on accelerator optics’, CERN Accelerator School on General Accelerator Physics, Gif-sur-Yvette, 1984. Wilson, E., ‘Transverse beam dynamics’, CERN Accelerator School on General Accelerator Physics, Gif-sur-Yvette, 1984. Le Duff, J., ‘Longitudinal beam dynamics in circular accelerators’, CERN Accelerator School on General Accelerator Physics, Gif-sur-Yvette, 1984. Ropert, A., ‘Lattices and emittances’, CERN Accelerator School on Synchrotron Radiation and Free electron lasers, Grenoble, 1996. Laclare, J. L., ‘Bunched beam coherent instabilities’, CERN Accelerator School on Advanced Accelerator Physics, Oxford, 1985.

References [1] Elleaume, P., ‘A new approach to the electron beam dynamics in undulators and wigglers’, Proceedings of the EPAC conference, Berlin, 1992, pp. 661–663. [2] Limborg, C., ‘Ultimate brilliance of storage ring based synchrotron radiation facilities’, Thesis, Université Joseph Fourier, Grenoble, 1996.

2

Generalities on the synchrotron radiation Pascal Elleaume

1

Introduction

The most general characteristics of synchrotron radiation, which is the radiation generated by an ultra-relativistic electron beam travelling through a magnetic field, are derived in this chapter. It concerns the radiation field, the photon flux density, power density, brilliance, mutual intensities and polarization which can be predicted quite accurately. The derivation starts from the retarded potential which is a solution of the Maxwell equations. The presentation of the electric field in the time domain is inspired from several text books such as J. D. Jackson [1], L. Landau and E. Lifchitz [2] and R. P. Feyman [3]. The method used to derive the field in the frequency domain was pointed out to me by O. Chubar [4]. It is much simpler and more powerful than the method presented by J. D. Jackson. The definition and emphasis on the dimensionless field vector H is largely inspired from the lecture by K. J. Kim [5]. The presentation of the radiation in the near field by means of the angular representation was initiated by K. J. Kim [6]. The definition of the brilliance by means of the Wigner distribution [7] was first mentioned in literature by R. Coisson and R. Walker [8] and more extensively developed by K. J. Kim [6]. The description of the polarization in terms of the Stokes components is borrowed from Born and Wolf [9]. The treatment of the transverse coherence by means of mutual intensity and their exact computations for the Gaussian beam is inspired by the work of R. Coisson [10]. The radiation properties of a single electron or a monoenergetic filament beam are derived in Section 2. The generalization to a thick electron beam with finite size, divergence and energy spread is made in Section 3. No assumption is made with respect to the magnetic field where the electron beam propagates and only the most general results are derived. The application to the uniform field of a bending magnet is made in Chapter 4. The radiation from a periodic magnetic field is detailed in Chapter 3.

2 2.1

Filament monoenergetic electron beam Observer and electron time

As an electron propagates in a magnetic field, it is submitted to an acceleration which bends  ) be the position in its velocity. The acceleration depends on the local magnetic field. Let R(τ space of the electron at time τ (Figure 2.1). This electron produces an electromagnetic wave which is seen by a stationary observer whose position is r. The wave arrives at the observer at time t. Because the field propagates at the speed of light c, the times τ of emission of the

Synchrotron radiation, generalities

39

D n ()

ϑ() e– r

R ()

O

Figure 2.1 Electron motion and the observer.

radiation and t of receiving of the radiation by the observer are related by t =τ+

D(τ ) c

(1)

where D is the distance between the electron and the observer at the time τ when the radiation was produced by the electron. D is expressed as  )| D(τ ) = |r − R(τ

(2)

Eqn (1) is very important – it relates the observer time t and the electron time (also called retarded time) τ . Differentiating Eqn (1) with respect to the time τ , one deduces the ratio between the two times: dt  ) = 1 − n(τ ˆ )ϑ(τ dτ

(3)

where n(τ ˆ ) is a unit vector directed from the electron to the observer: n(τ ˆ )=

 ) r − R(τ  )| |r − R(τ

(4)

 ) is the electron velocity divided by the speed of light c. For an ultra-relativistic and ϑ(τ electron propagating towards the observer with a small angle θ , one can write
  1 1 dt 2 2 = 1 − ϑ cos θ = 1 − 1 − 1/γ cos θ ≈ +θ (5) dτ 2 γ2 where E = γ mc2 . E is the total electron energy and m its mass. In practical units γ = 1957E [GeV]

(6)

40

P. Elleaume

Most synchrotron sources have an electron energy from a fraction of a GeV to a few GeV: one therefore realizes that whenever the electron is emitting towards the observer within an angle small or comparable to 1/γ , the observer time t evolves a million or a billion times slower than the electron time τ . This phenomenon is known as the relativistic compression of time. As a consequence, any periodic motion of the electron with some wavelength results in a periodic electromagnetic wave, the wavelength of which is one million or one billion times smaller. This is one way to understand the extension of the spectrum of synchrotron radiation to very short wavelengths. 2.2

Liénard–Wiechert potentials

2.2.1

General case

In the Lorentz gauge, the Maxwell equations can be written (in SI units) as
<(r, t) = −

ρ(r, t) ε0

 r, t) = −
A(

j(r, t) ε0 c 2

(7)

where
<(r, t) is the Dalembertian operating on the coordinate r and time t and applied to the function <(r, t), ε0 is the vacuum permeability, ρ(r, t) and j(r, t) are the electron charge and current density at position r and time t. Equations (7) have the the following general  r, t): solutions for the scalar potential <(r, t) and the vector potential A(  1 δ(τ − t + (|r − y(τ )|/c)) <(r, t) = ρ( y, τ ) dτ d3 y + <0 (r, t) 4πε0 |r − y(τ )| (8)  1 δ(τ − t + (|r − y(τ )|/c)) 3    A(r, t) = j ( y, τ ) dτ d y + A0 (r, t) 4πε0 c2 |r − y(τ )| where <0 (r, t) and A0 (r, t) are general solutions of Eqn (7) in the absence of charge and current densities. In the following we shall neglect both <0 (r, t) and A0 (r, t) and only deal with the field generated by the charge and current densities. ρ( y , τ ) and j( y , τ ) can be expressed as  )) ρ( y , τ ) = eδ( y − R(τ

 )ρ( j( y , τ ) = cϑ(τ y, τ )

(9)

where δ( y ) is the three dimensional delta function. Substituting Eqn (9) in Eqn (8), one derives the retarded potentials (or Liénard–Wiechert potentials):    )|/c))  δ(τ − t + (|r − R(τ e e  <(r, t) = dτ =  )|  )| ret 4πε0 |r − R(τ 4πε0 |r − R(τ  (10)    )|/c))  ) e δ(τ − t + (| r − R(τ e ϑ(τ   r, t) =  ) A( ϑ(τ dτ =   )|  )|  4πε0 c |r − R(τ 4πε0 c|r − R(τ ret

where ‘exp|ret ’ means that exp is computed at the retarded time τ related to the observer time t by τ =t−

 )| |r − R(τ c

(11)

Synchrotron radiation, generalities

41

The electric and magnetic fields of the radiation are deduced from the vector and scalar potentials according to  r, t) = −∇<(r, t) − ∂ A(  r, t) E( ∂t

 r, t) = ∇ × A(  r, t) B(

(12)

Substituting Eqn (10) in Eqn (12), and making use of Eqn (4), one computes the electric and magnetic fields in the time domain:



     × dϑ/dτ  e n ˆ − ϑ e n ˆ × (( n ˆ − ϑ) )    E(r, t) =  +  2 3 2 3     4πε0 γ (1 − ϑ n) 4πε0 c (1 − ϑ n) ˆ D ˆ D ret ret (13)   ˆ )  r, t) = n(τ  r, t) B( × E(  c ret  dϑ/dτ  where the quantities n, ˆ ϑ, and the distance D are a function of the retarded time τ . It is clear from Eqn (13) that the electric field is the sum of two terms. The first one decays as 1/D 2 : it is the Coulomb field, also called the velocity field. The other term scales like 1/D and therefore propagates to much larger distances. It is also called the acceleration field because it vanishes for a constant velocity electron. The detailed computation of the  acceleration field requires the knowledge of the acceleration, dϑ/dτ , which is derived from the Lorentz force equation: γm

dϑ = eϑ × Be dτ

(14)

where Be is an external magnetic field responsible for the electron acceleration. Let dW /d᏿ be the energy radiated per unit surface, normal to the unit vector sˆ , by a single electron. Using the Poynting vector, one derives the following expression for dW/d᏿:  ∞ dW 2  r, t) × B(  r, t))ˆs dt = ε0 c (E( (15) d᏿ −∞ For a filament electron beam with current I , the power generated per unit surface dP /d᏿, is the product of the energy radiated per unit surface multiplied by I /e, the total number of electrons per second:  ∞ dP 2I  r, t) × B(  r, t))ˆs dt = ε0 c (E( (16) d᏿ e −∞ Here, we have assumed randomly spaced electrons in such a way that the power generated by the beam is simply the sum of the power generated by each electron. This assumption is discussed in Section 3.1. 2.2.2

Far field

At sufficiently large distances, one neglects the velocity field in Eqn (13), the electric field and the magnetic field are orthogonal to each other and to the direction of observation nˆ

42

P. Elleaume

which is a constant:

 × dϑ/dτ  nˆ × ((nˆ − ϑ) ) (1 − ϑ n) ˆ 3D   ˆ )  r, t) = n(τ  r, t) B( × E(  c ret

e  r, t) = E( 4πε0 c

    

ret

(17)

Making use of the identity  + n(1  nˆ − ϑ = nˆ × (nˆ × ϑ) ˆ − nˆ ϑ)

(18)

 r, t): one derives the Feymann [3] expression of the electric field E(

   n ˆ × ( n ˆ × ϑ) d e  r, t) = E(   4πε0 cD(1 − ϑ n) ˆ dτ (1 − ϑ n) ˆ ret 
2 e e d  d2  (t)) = R⊥ (τ (t)) = nˆ × nˆ × 2 R(τ 4πε0 cD dt 4πε0 cD dt 2

(19)

where (d2 /dt 2 )R⊥ is the acceleration of the electron as seen by the observer projected into the plane normal to the direction of emission n. ˆ Since the acceleration field decays as 1/D, one can express the power per unit solid angle, dP /d., produced by a filament beam of current I in the direction nˆ as 

dP dP I (n) ˆ = D2 = ε0 c d. dS e

∞

−∞

 r, t)|2 dt D 2 |E(

(20)

and substituting Eqn (17) in Eqn (20), one derives the power per unit solid angle generated in the direction nˆ with the planar polarization defined by the unit vector uˆ as e2

dP I (n, ˆ u) ˆ = d. 16π 2 ε0 c e 2.2.3



∞

−∞



2   × dϑ/dτ  nˆ × (nˆ − ϑ) uˆ (1 − ϑ n) ˆ 5

dτ

(21)

Radiated power

Performing the angle integration in Eqn (21), one derives the total power radiated by an electron beam of current I [1]: 

2   ∞  2 2  e dϑ  I 6  dϑ P = dτ (22) − ϑ × γ 6π ε0 c e dτ dτ −∞  Replacing dϑ/dτ by the expression given by Eqn (14) (Lorentz Force Equation) in Eqn (22), one obtains   I 2 2 ∞ 2 e2 c I 4 2 ∞ 1 e4 γ ϑ γ ϑ B⊥ dτ = dτ (23) P = 2 6π m2 ε0 c e 6πε0 e −∞ −∞ ρ

Synchrotron radiation, generalities

43

where B⊥ is the component of the magnetic field which is orthogonal to the velocity ϑ and ρ = (mcγ /eB⊥ ) is the radius of curvature of the trajectory under the action of the field B⊥ . For ultra-relativistic electrons, ϑ ≈ 1 and dτ ≈ ds/c where s is a coordinate measured tangentially to the trajectory. In the particular case of a uniform magnetic field, the trajectory is circular and the energy loss per electron, per turn, δE, is derived from Eqn (23) in practical units: δE [MeV] = 8.85 × 10−2

E 4 [GeV] = 2.66 × 10−2 E 3 [GeV]B⊥ [T] ρ [m]

(24)

where E is the total electron energy. The energy loss per turn grows with the third power of the electron energy. Taking the example of a 1 GeV storage ring with a 100 m circumference made with 1 T bending magnets, the electron loses half of its energy in less than 19 ms. To maintain a stable orbit, the energy loss is compensated by an acceleration of the electrons in one or several radio frequency cavities placed along the circumference. Note that at very high energies, the large energy loss due to synchrotron radiation makes the acceleration of electrons in circular machines impractical. This comes from the large number of accelerating cavities required to compensate the energy losses and/or the large circumference required to increase the radius of curvature and reduce the field in the bending magnets. For the ESRF ring, the electron energy is 6 GeV and the magnetic field in the bending magnets is 0.85 T resulting in a radius of curvature of 23.3 m and an energy loss per turn of 4.9 MeV. For a stored current of 200 mA this accounts for a radiated power of 900 kW cumulated in all bending magnets, to which one must add the power radiated in the insertion devices. 2.3 2.3.1

Radiation in the frequency domain General case

Replacing the δ function of the potentials in Eqn (10) by  ∞ 1 δ(t) = exp(iωt) dω 2π −∞

(25)

and reversing the order of the integrals over τ and ω, one derives the potentials <(r, ω) and ! A(r, ω) in the frequency domain: !  ∞ exp(iω(τ + D/c)) e dτ <(r, ω) = 4πε0 −∞ D ! (26)  ∞ e exp(iω(τ + D/c))   A(r, ω) = ϑ dτ 4πε0 c −∞ D ! where f (ω) is the Fourier transform of the function f (t) defined by ˜  ∞  ∞ 1 f (ω) = f (t) exp(iωt) dt f (t) = f (ω) exp(−iωt) dω 2π −∞ −∞ ˜ ˜

(27)

To simplify the notation, in the following, we shall write f (ω) instead of f (ω) bearing in mind that the function f is different if its variable is a time t or a frequency˜ω.

44

P. Elleaume

 r, ω) are expressed as a simple It is important to stress that the potentials <(r, ω) and A(  r, ω) and the magnetic field integral over the time τ of the electron. The electric field E( B(r, ω) in the frequency domain are deduced from Eqn (12) as  r, ω) = (−∇<(r, ω)) + iωA(  r, ω) E(

 r, ω) = ∇ × A(  r, ω) B(

(28)

 r, ω) in Eqn (28) by their expression given by Eqn (26) and Substituting <(r, ω) and A(  r, ω) making use of the identity ∇D = n(τ ˆ ) deduced from Eqn (4), one derives the fields E( and B(r, ω):

 ϑ − n(1 ˆ + (ic/ωD)) D ) exp iω τ + dτ D c −∞

  ∞  ϑ × n(1 ˆ + (ic/ωD)) D −ieω  exp iω τ + dτ B(r, ω) = 4πc2 ε0 −∞ D c

 r, ω) = ieω E( 4πcε0



∞

(29)

It is important to stress that no approximation is made and Eqn (29) applies to the most general case of a charged particle under any arbitrary motion. The motion can be induced by an arbitrary external electric field and/or magnetic field. The electron may be at rest, have a slow velocity or be ultra-relativistic and the radiation can be observed at short or long distances. In addition, Eqn (29) is easily expressed numerically since it only involves a single integral over the time of the electron. From a mathematical point of view, the convergence of the integrals is assured by the fast oscillation of the exp (iω(τ + D/c)) term when the time τ tends towards infinity. Making use of Eqn (15) and of the Parceval1 theorem, one derives the power radiated by a filament beam of current I per unit surface orthogonal to the unit vector sˆ : dP = d᏿

 0

∞

dP I dω = 4πε0 c2 d᏿ dω e



∞



 ∗   Re E(r, ω) × B (r, ω) sˆ dω

(30)

0

where B ∗ is the complex conjugate of B and Re(x) is the real part of the quantity x. Let d
(31)

where h is the Planck constant. 2.3.2

Far field radiation

If the observer is at a sufficient distance from the electron, the unit vector nˆ is independent  ) in Eqn (29) one derives the following expressions of the time τ . Replacing D by n ˆ r − nˆ R(τ

1 Note that the Parceval theorem can be shown to be a consequence of Eqn (25).

Synchrotron radiation, generalities

45

 r, ω) and B(  r, ω): for the field vectors E(

 ω  ∞  ieω n ˆ R  r, ω) = E( (ϑ − n) ˆ exp iω τ − exp i n ˆr dτ 4πcε0 D c c −∞

 ω  ∞  −ieω n ˆ R  r, ω) = B( (ϑ × n) ˆ exp iω τ − exp i n ˆr dτ 4πc2 ε0 D c c −∞

(32)

(33)

Making use of Eqn (18) and of the following identity valid for ω = 0 





nˆ R  exp iω τ − iω(1 − nˆ ϑ) c −∞ ∞



 dτ =

∞ −∞

 d(exp(iω(τ − (nˆ R/c)))) dτ = 0 dτ (34)

 r, ω): one derives an equivalent expression for the electric field E(

 ω  ∞  −ieω n ˆ R  r, ω) =  exp iω τ − E( nˆ × (nˆ × ϑ) exp i n ˆr dτ 4πcε0 D c c −∞

(35)

 r, ω), B(  r, ω) and the observation direction nˆ is clear The orthogonality between the fields E(  r, ω) and E(  r, ω) as from Eqns (33) and (35) In the following, we shall express the fields E(    r, ω) = −ie exp i ω n E( ˆ r H (n, ˆ ω) c 2cε0 D

 r, ω) × nˆ  r, ω) = − 1 E( B( c

where the dimensionless field vector H is deduced from Eqns (33) and (35) as

 ∞  ω n ˆ R  exp iω τ − H (n, ˆ ω) = dτ (nˆ − ϑ) 2π −∞ c

 ∞ ω nˆ R  exp iω τ − = nˆ × (nˆ × ϑ) dτ 2π −∞ c

(36)

(37)

We shall see later that most (if not all) physical quantities of interest can be expressed as a function of H . Another useful expression of the electric field in the far field range can be obtained if one replaces τ − (n( ˆ R − r)/c) by the observer time t in Eqn (35) and performs an integration by parts:

 ∞  e d n ˆ × ( n ˆ × ϑ)  r, ω) = E( exp(iωt) dt (38) 4πcε0 D −∞ dt 1 − nˆ ϑ  r, ω) is the Fourier transform of (e/2cε0 D) (d/dt)(nˆ × (nˆ × ϑ)/1   In other words, E( − nˆ ϑ) which represents the electric field seen by the observer in the time domain. Equation (38) is  r, ω) using the Fast Fourier Transform the basis of an efficient numerical computation of E(   at equally spaced algorithm (FFT). It requires the computation of (d/dt)(n×( ˆ n× ˆ ϑ)/1− nˆ ϑ)

46

P. Elleaume

 ), using, for example, some numerical time intervals which can be done for any velocity ϑ(τ  integration and making use of dt/dτ = 1 − nˆ ϑ. Substituting Eqn (36) in Eqns (30) and (31), and making use of the orthogonality between the electric and magnetic fields, one derives the power per unit solid angle and per unit frequency dP /d. dω as well as the spectral flux per unit solid angle d
(39)

I 2πω dP d< = α |H (n, ˆ ω)|2 (40) (n, ˆ ω) = e d. hω d. dω where α = e2 /2ε0 hc = 1/37 is the fine structure constant. d
(41)

It is important to stress that nearly all computations of synchrotron radiation found in literature are done in the far field approximation. In most cases, it is sufficiently accurate. Even though there is a lack of clear experimental confirmation, there are circumstances such as in the infrared edge radiation [11], or off-axis undulator radiation [12], where a more exact near field radiation differs significantly from the prediction of the far field approximation. 2.4

Small angle approximation

2.4.1

Generalities

Let us define the orthogonal set of axis Ox, Oz and Os. Ox and Oz are the horizontal and vertical transverse axes while Os is the longitudinal axis. In the rest of this chapter, we shall assume that both the electron velocity ϑ and the direction of observation nˆ make a small angle with√the axis Os. Furthermore, we assume an ultra-relativistic electron beam so that (1/γ ) = 1 − ϑ 2 1. Let θx and θz be the angle between nˆ and the Os axis measured in the Oxs and Ozs planes (respectively). nˆ can be expressed as


 2 + θ2 θ x z (42) nˆ = θx , θz , 1 − θx2 − θz2 ≈ θx , θz , 1 − 2 and the velocity ϑ is expressed as 

2 + ϑ2 ϑ 1 1 z x − ϑ = ϑx , ϑz , 1 − 2 − (ϑx2 + ϑz2 ) ≈ ϑx , ϑz , 1 − γ 2γ 2 2

(43)

With these definitions, the small angle approximation is equivalent to ϑx , ϑz , θx , θz ,

1 1 γ

(44)

Synchrotron radiation, generalities

47

To zero-th order of approximation, the position R can be expressed as R = (X, Z, S) ≈ (0, 0, cτ )

(45)

Since the field vector H defined in Eqn (37) is orthogonal to n, ˆ it can be approximated by a two-dimensional (2D) vector H (θx , θz , ω) ≈ (Hx , Hz , 0) expressed as 
 S  ∞ ω ω 2    H (θx , θz , ω) = (46) (1 + ι (S )) dS dS ι(S) exp i 2πγ c −∞ 2cγ 2 0 where ι is a 2D vector which depends on the longitudinal coordinate S of the electron defined by " # # " ιx (S) ϑx (S) − θx ι(S) = (47) =γ ιz (S) ϑz (S) − θz In Eqn (46), the integral over the longitudinal coordinate S has been derived from the integral over the electron time τ by means of Eqn (45). In the far field approximation, the electric  r, ω) is also a 2D vector which is derived from Eqn (36): field E(  

ie x z  z2 ω x2   E(r, ω) = E(x, z, s, ω) = H exp i s 1 + 2 + 2 , ,ω (48) 2cε0 s c 2s 2s s s A similar expression for the magnetic field B (r, ω) is derived using Eqn (36). 2.4.2

Electron trajectory

Using the longitudinal position S of the electron as a primary variable rather than the time τ , the Lorentz force expressed by Eqn (14) can be written to first order in ϑx and ϑz as dϑx e = Bz (X(S), Z(S), S) dS γ mc dϑz −e = Bx (X(S), Z(S), S) dS γ mc

dX = ϑx dS dZ = ϑz dS

(49)

where Bx (x, z, s) and Bz (x, z, s) are the horizontal and vertical components of the field at the position (x, z, s). Because the transverse velocities ϑx and ϑz of an electron are small, the oscillations of the transverse positions X(S) and Z(S) are usually also small compared to the distance over which the magnetic field changes significantly2 . Under these conditions, one may simplify Eqn (49) as  S  S  S e e   Bz (S )dS X(S) = Bz (S  ) dS  dS  ϑx (S) = γ mc −∞ γ mc −∞ −∞ (50)  S  S  S −e −e ϑz (S) = Bx (S  )dS  Z(S) = Bx (S  ) dS  dS  γ mc −∞ γ mc −∞ −∞ where Bx (S) and Bz (S) are shorthand notations for Bx (0, 0, S) and Bz (0, 0, S). 2 These conditions are fulfilled for the large majority of insertion devices being used as synchrotron sources. They may be violated for low energy electrons and/or very long undulators such as the long undulators used for free electron lasers.

48

P. Elleaume

2.4.3

Power density

The powers (dPx /d.) and (dPz /d.) radiated at large distance per unit solid angle with horizontal and vertical polarization are deduced from Eqn (21) and Eqn (47) as 

 dPx 2 4  ∞  d.  1   = e 2γ I  dP  4πε0 π e 2 5 2 −∞ (1 + ιx + ιz ) z d.

2   dιz dιx  2 2  dS 1 − ιx + ιz − 2 dS ιx ιz    dS 
2     dιz  dι x 1 − ι2z + ι2x − 2 ιx ι z dS dS 


(51) A little algebraic transformation of Eqn (51) results in the following expression for the total power per unit solid angle: dP dPx dPz e2 2γ 4 I = + = J (θx , θz ) d. d. d. 4πε0 π e

(52)

where J (θx , θz ) is given by  J (θx , θz ) =

∞

−∞

(1 + ι2 ) (dι/dS)2 − ((d/dS)ι 2 )2 dS (1 + ι 2 )5

(53)

In practical units, Eqn (52) can be expressed as # " dP W = 13.44 × 10−3 E 4 [GeV]I [Amp]J [m−1 ] d. mr 2

(54)

The power per unit surface dP /d᏿ on a screen located at a distance d from the source is derived from power per unit solid angle as dP 1 dP = 2 d᏿ d d.

(55)

This expression is valid in the far field approximation. One can derive a more accurate expression which is valid in the near field approximation by considering the angles of observation θx and θz in Eqn (47) and the distance d in Eqn (55) as functions which depend on the longitudinal coordinate S and by introducing 1/d 2 inside the integral of Eqn (51). 2.4.4

Angle integrated power

The angle integrated power horizontally and vertically polarized, Px and Pz , can be obtained by integrating the power per unit solid angle over the angles θx and θz : " #  ∞ ∞" # dPx /d. Px = dθx dθz Pz −∞ −∞ dPz /d.

(56)

Synchrotron radiation, generalities

49

Substituting Eqn (51) in Eqn (56) and making use of Eqns (47) and (49) and the following identities [13]  ∞ ∞  ∞ ∞ 1 x2 π dx dz = dx dz 2 2 n 2 2 n n−1 −∞ −∞ (1 + x + z ) −∞ −∞ (1 + x + z ) π (57) = 2(n − 1)(n − 2) one derives the angle integrated power with horizontally polarized electric field: 
 ∞  ∞ e e2 2 2 2 Px = γ I 7 Bz ds + Bx ds 4π ε0 12m2 c2 −∞ −∞

(58)

The angle integrated power Pz with electric field vertically polarized is deduced from Eqn (58) following a permutation of the Bx and Bz fields. Summing the two polarizations, one obtains the total angle integrated power:  ∞ e2 2e 2 P = (Bz2 + Bx2 ) ds (59) γ I 4π ε0 3m2 c2 −∞ In practical units, Eqn (59) becomes P [kW] = 1.266E [GeV]I [Amp] 2

2.5



∞ −∞

(Bz2 [T] + Bx2 [T] ds [m]

(60)

Angular representation

It is useful to introduce the angular representation E (θx , θz , s, ω) of the electric field !  r, ω) ≡ E(x,  E( z, s, ω) using the following definitions:  ∞ ∞  ω  ω   E (θx , θz , s, ω) = E(x, z, s, ω) exp −i (θx x + θz z) dx dz (61) 2πc −∞ −∞ c !  ∞ ∞  ω  ω   E(x, z, s, ω) = (62) E (θx , θz , s, ω) exp i (θx x + θz z) dθx dθz 2πc −∞ −∞ ! c Within the small angle approximation defined in Section 2.4, we have seen that the vector  E(x, z, s, ω) is the 2D projection of the field on the Oxz plane orthogonal to the Os axis. The angular representation can be interpreted as a projection of the field over a family of plane waves making an angle (θx , θz ) with respect to the axis Os. The transformation of the electric field through a drift space of length d can be performed using the Huygens principle:  E(x, z, s + d, ω) =

−iω 4π c



∞

−∞

   exp i(ω/c) (x − x  )2 + (z − z )2 + d 2   , z , s, ω)  dx  dz E(x −∞ (x − x  )2 + (z − z )2 + d 2 (63)



∞

The same transformation expressed in the angular representation is dramatically simpler [14]:

2 2 θ θ ω E (θx , θz , s + d, ω) = exp i d 1 + x + z E (θx , θz , s, ω) (64) c 2 2 ! !

50

P. Elleaume

Substituting Eqn (48) in Eqn (61), one derives the electric field in the angular representation:

2 2 θ ω θ ieω exp i s 1 + x + z E (θx , θz , s, ω) = 4πc2 ε0 s c 2 2 !  ∞ ∞  x z   ω ω , , ω dx dz exp i (x − sθx )2 + i (z − sθz )2 H × cs cs s s −∞ −∞ (65) Making use of the following identity  π  π exp(iLθ ) ≈ exp i δ(θ) 4 L

if L  1

2

(66)

one obtains

2 2 θ θ ie ω E (θx , θz , s, ω) = H (θx , θz , ω) exp i s 1 + x + z 2cε0 c 2 2 !

(67)

Propagating the field backwards to the source using Eqn (64), one derives the very simple expression ie  H (θx , θz , ω) E (θx , θz , 0, ω) = 2cε0 !

(68)

Eqn (68) has many important consequences. First, it gives a physical interpretation of the field vector H as the radiated field in the angular representation computed at the origin. Second, it allows the derivation of both the near and far field radiation as a function of H .  More precisely, the field E(x, z, s, ω) at any position (x, z, s) is derived from Eqns (68) and (62) in the form of the following 2D integral:  E(x, z, s, ω) =

ieω 4πε0 c2



∞



∞

−∞ −∞

H (θx , θz , ω)





θz2 θx2 ω θx x + θz z + s 1 + + dθx dθz × exp i c 2 2

(69)

 In particular, the field E(x, z, 0, ω) at the origin of the longitudinal coordinate is  E(x, z, 0, ω) =

ieω 4πε0 c2



∞

−∞



 ω  H (θx , θz , ω) exp i (θx x + θz z) dθx dθz c −∞ ∞

(70)

By field at the origin, I mean the equivalent field distribution in a plane normal to the electron beam axis such that it coincides with the field outside the undulator when it is propagated through a drift space. By this method, the possible divergence of the electric field in the vicinity of an electron is removed. Substituting B = −(1/c)E × nˆ in Eqn (31), and making use of Eqn (69), one obtains the following expression for the spectral flux per unit

Synchrotron radiation, generalities surface (d


 θz2 ω θx2  × exp i θx x + θz z + s 1 + + dθx dθz   c 2 2

51

(71)

Note that Eqn (69) only predicts the expression for the transverse electric field. The longitudinal component can be deduced from the transverse components using Maxwell’s equations. The longitudinal field component is much smaller and, in general, of little interest. We have now two different methods of computation of the near field radiation, the one summarized by Eqn (30) and the other derived in this section which is summarized by Eqn (69). From a numerical point of view, the use of the angular representation requires the computation of H over a 2D grid of angles followed by some sort of Fourier transform which can be made using a 2D FFT algorithm. This opens the possibility of efficient computation of wavefront propagation. A drift space is more easily treated in the angular representation, while a focusing lens, mirror or diffracting aperture can be approximated as a multiplication by a complex number in the normal spatial representation. The propagation through a combination of mirrors and drift spaces is made by switching from one representation to the next. The computation starts at the origin in the angular representation by means of Eqn (68). On the other hand, if one wants to compute the near field at a only a few points in space, Eqn (30) is the quickest method but it does not contain complete information of the state of the wave and, therefore, does not allow any further propagation of the field. Both methods have been used in the SRW computer code [15] which computes the radiation of an electron beam in an arbitrary magnetic field in the near field approximation and performs wavefront propagation. 2.6

Brilliance

2.6.1

Definition

Following Kim [6], we define the spectral brilliance Ꮾ of the radiation produced by a filament electron beam of current I by means of the Wigner distribution function [7]: Ꮾ( y , y , s, ω, u) ˆ





  ε0 ω2 I ∞ ∞   ξ  = E y + , s, ω uˆ ∗ 2 2π hc e −∞ −∞ ! 2



 ω  ξ  × E ∗ y − , s, ω uˆ exp −i yξ  d2 ξ  2 c !

(72)

where uˆ defines a polarization state, E is the electric field in the angular representation and ! E ∗ is its complex conjugate. The vectors y and y are, respectively, shorthand notations for ! transverse position (x, z) and transverse angle (θ , θ ). With these notations, the fields the x z

52

P. Elleaume

  y , s, ω), respectively. The specE (θx , θz , s, ω) and E(x, z, s, ω) become E ( y  , s, ω) and E( ! ! tral brilliance is also simply called brilliance and sometimes brightness or spectral radiance. In Section 2.6.3, we shall discuss its properties. Before doing so, let us derive an alternative definition of the brilliance. Inserting Eqn (61) into Eqn (72) , reversing the order of integration and making use of Eqn (25), one obtains Ꮾ( y , y , s, ω, u) ˆ





  ε0 ω 2 I ∞ ∞  ξ = E y + , s, ω uˆ ∗ 2 2π 2 ch e −∞ −∞



 ω  ξ × E ∗ y − , s, ω uˆ exp i y ξ d2 ξ 2 c

(73)

Note the symmetry of the expression of the brilliance in terms of the electric field in the spatial or angular representation. Indeed, equally, one may use Eqn (72) or (73) as a definition of the brilliance. Replacing in Eqn (72) the electric field by its expression given by Eqn (67), one derives the following expression of the brilliance in terms of the dimensionless field vector H : Ꮾ( y , y , s, ω, u) ˆ







 ω 2 I  ∞  ∞   ξ ξ  ∗ ∗  H y + , ω uˆ H y − , ω uˆ =α 2 2 2π c e −∞ −∞  ω  × exp −i y − y s ξ  d2 ξ  c

2.6.2

(74)

Mutual intensity

y , ξ, s, ω, u) ˆ as the following complex quantity: Let us define the mutual intensity ᏹ(





  ξ ξ E ∗ y − , s, ω uˆ ᏹ( y , ξ, s, ω, u) ˆ = E y + , s, ω uˆ ∗ (75) 2 2 The following expression is an alternative expression for the brilliance: ε0 ω 2 I Ꮾ( y , y , s, ω, u) ˆ = 2π 2 ech 



∞



∞

−∞ −∞





 ω c



ᏹ y, ξ, s, ω, uˆ exp i y ξ d2 ξ

(76)

Fourier inverting Eqn (76), one derives the mutual intensity as a function of the brilliance:    ω  h e ∞ ∞  ᏹ( y , ξ, s, ω, u) ˆ = Ꮾ( y , y , s, ω, u) ˆ exp −i y ξ d2 y (77) 2ε0 c I −∞ −∞ c The mutual intensity characterizes the transverse coherence of the photon beam. In particular it determines the level of modulation of the fringes that one observes in the interference by two narrow slits in a Young experiment. We shall discuss it in more detail when dealing with the radiation from a thick electron beam. Its close relation to the brilliance is already a clear indication that large transverse coherence usually means high brilliance. In the far field range,

Synchrotron radiation, generalities

53

one can substitute Eqn (36) in Eqn (75) and express the mutual intensity as a function of the dimensionless field vector H :  s, ω, u) ᏹ( y , ξ, ˆ e2 ≈ 2 2 2 4c ε0 s 2.6.3



H







  y  − ξ /2 y + ξ/2 ω y  ξ H ∗ , ω uˆ exp i , ω uˆ ∗ s s c s

(78)

Properties of the brilliance

It is easy to show from Eqn (72) that the brilliance is a real quantity (zero imaginary part). Integrating the brilliance over the spatial coordinates y and the angle coordinate y one derives  ∞ ∞ 2 cε0 I     Ꮾ(ω, y, y , s, u) ˆ d2 y = 2 y , s, ω) E ( h e ! −∞ −∞ d<  d< (n, ˆ ω, u) ˆ (79) = 2  ( y , ω, u) ˆ ≡ d y d. 

∞



∞

−∞ −∞

Ꮾ(ω, y, y , s, u) ˆ d2 y = 2

=

cε0 I h e

 2   y , s, ω E(

d< d< ( y , s, ω, u) ˆ ≡ ( y , s, ω, u) ˆ 2 d y d᏿

(80)

In view of Eqn (79) and Eqn (8), Ꮾ appears to be the density of photons per unit surface and solid angle. From Eqn (74), one derives the transformation of the brilliance in a drift space: ˆ ˆ = Ꮾ( y − L y  , y , s, ω, u) Ꮾ( y , y , s + L, ω, u) Assuming a thin lens of focal length F , the transformation of the electric field is
 ω y2   E( y , s, ω) → E( y , s, ω) exp i c F

(81)

(82)

and the brilliance transforms as Ꮾ( y , y , s, ω, u) ˆ → Ꮾ( y , y + y/F, s, ω, u) ˆ

(83)

Indeed, both Eqns (81) and (83) can be expressed in the more general form Ꮾ( y1 , y1 , s1 , ω, u) ˆ = Ꮾ( y2 , y2 , s2 , ω, u) ˆ

whenever " # " # y2 y = M 1 y2 y1

(84)

(85)

where M is a matrix which transforms the transverse coordinates between the planes defined by s = s1 and s = s2 . In other words, the brilliance is invariant with respect to any linear transformation of the phase space.

54

P. Elleaume

Let us define o = (0, 0); the on-axis brilliance Ꮾ( o, o, s, ω, u) ˆ can be derived from Eqn (74): Ꮾ( o, o, s, ω, u) ˆ

 ω 2 I  ∞  ∞ (H (θx , θz , ω)uˆ ∗ )(H ∗ (−θx , −θz , ω)u) ˆ dθx dθz =α π c e −∞ −∞

The total flux < can be written in a very similar way   I ∞ ∞  <(ω, u) ˆ =α ˆ dθx dθz (H (θx , θz , ω)uˆ ∗ )(H ∗ (θx , θz , ω)u) e −∞ −∞

(86)

(87)

From Eqns (86) and (87), one can show that the on-axis brilliance satisfies the following inequality o, o, s, ω, u)| ˆ ≤ |Ꮾ(

4 <(ω, u) ˆ λ2

(88)

where λ is the wavelength associated with the frequency ω. By changing the origin of the transverse coordinates and angles, this inequality is also true for the brilliance Ꮾ( y , y, s, ω, u) ˆ at any point in space. The equality in Eqn (88) is satisfied for the following conditions Ꮾ( o, o, s, ω, u) ˆ =±

4 <(ω, u) ˆ λ2

∗ if H (θx , θz , ω) = ±H (−θx , θz , ω)

(89)

To summarize, the brilliance defined by means of the Wigner distribution function has many properties that one would expect from a distribution function of photons in phase space. It gives the angular flux per unit solid angle (angular flux per unit surface) when integrated over the spatial (angular) variables and it is invariant over any linear transformation of geometrical optics. It is nevertheless important to stress a major drawback of this definition of brilliance, namely, that it can be negative3 and therefore cannot be fully interpreted as a density of photons in phase space. There are two simple ways to resolve this paradox: 1 2

One can conclude that the Wigner distribution function is simply not the phase space distribution and one must find something else! One can note that a phase space distribution of photons cannot exist as a physically observable quantity in the sense of quantum mechanics. The reason is that one cannot measure simultaneously the position and angle (momentum) of a particle. Therefore it makes no sense to look for a photon density in phase space and one should be quite happy with the Wigner distribution which fullfills so well the job of a photon density except for being, sometimes, negative!

I leave the reader to adopt whichever point of view he prefers even though I do prefer number 2. I have not found any safer or more elegant alternative than the Wigner function to approximate the distribution of photons in phase space. In the following, I shall keep to this. This discussion may sound strange to a novice. Despite a number of dicussions that I have

3 Indeed, we shall see in Chapter 3, that the brilliance of the radiation from a planar undulator is negative on the even harmonics.

Synchrotron radiation, generalities

55

had in the past with various experts, I have been unable to obtain any acceptable consensus on the questions. To avoid the difficulty, some people derive their own definition of brilliance which is usually less powerful than the Wigner distribution. Those definitions usually have the right scaling but easily differ by a factor 2, 3, π etc. from one author to another. 2.6.4

Application to a Gaussian beam

To obtain a further understanding of the properties of the brilliance defined in terms of the Wigner distribution, let us study a hypothetical case where the dimensionless field vector H is real and Gaussian with a polarization u: ˆ

2 2 θ θ H = uˆ exp − x2 − z 2 (90) 4Qz 4Qx  Substituting H given by Eqn (90) in Eqn (74) and performing the integration, one derives the brilliance:

 ω 2 θz2 x2 z2 θx2 (91) − − Ꮾ= <(ω, u) ˆ exp − 2 − 2Qx2 2Qz2 πc 2Qx  2Qz2 with Qx and Qz defined by Qx Qx  = Qz Qz =

λ ω = πc 4π

(92)

Substituting Eqn (90) in Eqn (70), one derives the Gaussian dependence of the electric field in the spatial representation: 
eωQx  Qz x2 z2  E( y , 0, ω) = uˆ exp − 2 − 4ε0 c2 4Qx 4Qz2

(93)

Let us now study the transmission of such a radiation field through a filter having a transmission which depends on the vertical coordinate z according to


z2 T (z) = exp − 2 4Q

 (94)

From Eqns (93) and (94), the field E  of the transmitted beam can be written as 
eωQx  Qz x2 z2   E ( y , 0, ω) = uˆ T (z) exp − 2 − 4ε0 c2 4Qx 4Qz2

(95)

Substituting Eqn (95) into Eqn (73), one determines the brilliance Ꮾ of the transmitted beam. One easily shows that Ꮾ is related to the flux < of the transmitted beam by a relation similar to Eqn (91):

 ω 2 2 2 2 2 θ θ x z (96) Ꮾ = < (ω, u) ˆ exp − x2 − z  2 − − πc 2Qx2 2Qz2 2Qx  2Qz

56

P. Elleaume

in which the horizontal rms size and divergence are unchanged after passing through the mask while the vertical rms beam size Qz , divergence Qz  and the spectral flux < are modified according to Qz =

Qz R

Qz  = RQz

< =

1 < R

(97)

where R is given by  R=




1+

Q QZ

2 >1

(98)

As expected the vertical beam size, the total flux and the brilliance have decreased by R after passing through the mask. The vertical angular spread expanded leaving the product of the vertical size times the vertical angular spread unchanged and given by Eqn (92). Let us now study the brilliance of the radiation with a field vector H expressed as

θ2 θ2 H = uθ ˆ z exp − x2 − z 2 4Qx  4Qz

(99)

Substituting H given by Eqn (99) in Eqn (74) and carrying out the integrations one determines the brilliance:



 ω 2 θz2 θz2 θx2 z2 x2 z2 Ꮾ= <(ω, u) ˆ −1 + 2 + 2 exp − 2 − − − πc Qz 2Qx2 2Qz2 Qz 2Qx  2Qz2 (100) where Qx and Qz are also defined by Eqn (92). The on-axis brilliance is negative and becomes positive away from the axis. The Gaussian case studied in this section is of particular interest because one easily obtains a complete expression of the brilliance at any position (x, z) and angle (θx , θz ). When dealing with the radiation from a real insertion device, it is difficult to obtain a simple analytical expression of the brilliance and one must leave it in an integral form.

2.7

Photon beam size and divergence

Let f ( y , y ) be any function of the transverse position and angle. We define the average f  as ∞ ∞ f  =

y , y )Ꮾ( y , y , s, ω, u) ˆ d2 y d2 y −∞ −∞ f ( ∞ ∞ y , y , s, ω, u) ˆ d2 y d2 y −∞ −∞ Ꮾ(

(101)

Synchrotron radiation, generalities In particular, the horizontal rms beam size Qx2 and divergence Qx2 , are ∞ ∞ 2 y , y , s, ω, u) ˆ dx dz −∞ −∞ x Ꮾ( 2 2 Qx = x  =  ∞  ∞ y , y , s, ω, u) ˆ d2 y d2 y −∞ −∞ Ꮾ( 2  ∞  ∞ 2    −∞ −∞ x E(x, z, s, ω) dx dz =   2  ∞ ∞  E(x, z, s, ω)   dx dz −∞ −∞  ∞  ∞ 2 y , y , s, ω, u) ˆ dx dz −∞ x Ꮾ( 2 2  Qx  = x  = −∞ ∞ ∞ ˆ d2 y d2 y y , y , s, ω, u) −∞ −∞ Ꮾ( 2  ∞  ∞ 2    H (θ , θ , ω) θ  dθx dθz x z −∞ −∞ x  =   2  ∞ ∞   H (θ , θ , ω)   dθx dθz x z −∞ −∞

57

(102)

(103)

A similar expression can be derived for the vertical photon beam size and divergence. In Eqns (102) and (103), we have assumed that the photon beam satisfies x = x   = 0, which is obtained by a proper change of origin of the x axis. As expected, the rms sizes depends on s while the rms divergence is invariant. The rms beam size at the source (s = 0) can be expressed as  2 ∞ 2 ∞  (θx , θz , ω) exp(iω/c(θx x + θz z))dθx dθz  dx dz x H  −∞ −∞ −∞ −∞ Qx2 =   2   ∞ ∞  ∞ ∞  −∞ −∞  −∞ −∞ H (θx , θz , ω) exp(iω/c(θx x + θz z))dθx dθz  dx dz ∞ ∞

(104) It is left as a mathematical exercise to show that the size and divergence of Eqns (103) and (104) satisfy the following inequality [9]: Qx Qx  ≥

c λ = 2ω 4π

(105)

and the equality is satisfied if and only if H (θx , θz , ω) is equal to a real Gaussian function of θx and θz within a complex multiplicative constant. Eqn (105) is a well known result of diffraction theory; it is the Heisenberg uncertainty principle applied to photons. Note that Qx depends on s; therefore, the equality in Eqn (105) is only reached for a Gaussian beam at the position s where its size is minimum (waist). Because of the linearity of Eqn (101), the transformation of the average position by any sequence of drift spaces and thin lenses is obtained from the same matrix M as in Eqn (85) " # " # " #" # x2  x1  a b x1  =M = (106) x2  x1  c d x1  From Eqn (106), one derives the second moments:   2  2ab a x2 x2  x2 x   = ac ad + bc 2 x2 x2  c2 2cd

following matrix equation for the transformation of the   b2 x1 x1  bd  x1 x1  x1 x1  d2

(107)

58

P. Elleaume

Let us define the centred coordinates x = x − x and x  = x  − x  : their second moments ˜ are related to the second moment of x˜ and x  as follows x 2  = x 2  − x2 x x   = xx   − xx   x 2  = x 2  − x  2 (108) ˜ ˜˜ ˜  Making use of Eqns (106) and (107), one easily shows that x2 , x x   and x 2  are trans˜ established ˜ ˜relations ˜ formed with the same matrix relation as that of Eqn (107). Similar are for the vertical moments. Equation (107) provides a very simple method to compute the sizes and divergences along the beamline once the three second moments are known at the source. The main limitation is that it does not take into account the aberrations such as those induced by curved mirrors, for example. This could be partly corrected by propagating the third and fourth order moments. However, the advantage of simplicity vanishes. Note that along a drift space, the moment x x   vanishes at the position of the waist where x 2  is minimum. Let us ˜ ˜ Ᏹx define the following˜quantity Ᏹ2x = x 2 x 2  − x x  2

(109) ˜ ˜ ˜˜ It is easy to show that Ᏹx is invariant under any linear transformation of the phase space (x, x  ) having a determinant equal to 1. In the particular case where x x   = 0, it reduces to ˜ ˜from Eqn (105), one the product of the rms beam size multiplied by the rms divergence and has Ᏹx ≥ λ/4π. For these reasons, we shall call it the rms emittance of the photon beam. However, one should keep in mind that, as for the brilliance, the photon beam emittance is not a physically measurable quantity in the sense of quantum mechanics. 2.8

 Importance of the field vector H

We have seen that most spectral quantities of interest – electric field, brilliance, flux per unit surface or angle, size and divergence can be computed from the dimensionless field vector H defined in Eqn (37). H encapsulates the information on the magnetic field in which the electron beam generates the radiation. The complete study of any radiation source can be made by computing the vector H as a function of the direction (θx , θz ) and the frequency ω. Using Eqns (46) and (47), one derives the following dependence of H on the electron energy γ : 
ω  ω  H = γ 2 f γ θx , γ θz , 2 (110) γ γ where f(a, b, c) is a 2D vector function of three variables. Increasing the electron energy by a factor 2 shifts the spectrum to higher energies by 4 and shrinks the divergence of the radiation by 2. Replacing the field vector H given by Eqn (110) in the expression of the brilliance Eqn (74), one finds that the brilliance at the scaled angles (γ θx , γ θz ), scaled position (γ x, γ z) and scaled frequency ω/γ 2 grow like γ 4 . The total flux at the scaled frequency (obtained, for example, by integrating the angular spectral flux over all angles) is independent of the electron energy.

3

Thick electron beam

So far, we have considered either a single electron or a filament electron beam with random longitudinal positions between each electron. We shall now detail the characteristics of the radiation from a thick beam with finite energy spread.

Synchrotron radiation, generalities 3.1

59

Longitudinal distribution of the electron beam

For a filament electron beam, the brilliance and spectral flux have been derived by summing the contribution from each electron. Let us now reconsider this approximation in more detail. Let us first assume a monoenergetic filament beam, that is, a beam where all the electrons have the same energy and travel along the same trajectory. Let ti be the time at which an electron i passes at a fixed point of the trajectory. The vector H produced by such a collection of electrons can be written as

 H = H0 exp(iω(ti − t0 )) (111) i

where H0 is the H vector generated by a reference electron labelled 0. From Eqn (111), one derives     |H |2 = |H0 |2  1+ exp(iω(ti − tj )) i=j

i

= |H0 |2 (Ne + Ne (Ne − 1)exp(iω(t1 − t2 )))

(112)

where Ne is the number of electrons in the bunch and · is the average over all the electrons. Let ρe (t) be the electron distribution with time t and ρe (t1 , t2 ) be the two-particle distribution. In the absence of correlations ρe (t1 , t2 ) = ρe (t1 )ρe (t2 ) and one writes  ∞ ∞ exp(iω(t1 − t2 )) = exp(iω(t1 − t2 ))ρe (t1 , t2 ) dt1 dt2 −∞ −∞

  = 

∞

−∞

2  exp(iωt)ρe (t) dt 

(113)

In a storage ring the distribution of the longitudinal position of the electrons in a bunch is essentially uncorrelated with a Gaussian with an rms length σs ; therefore, Eqns (112) and (113) become

2 σ 2  ω 2 2 |H | = |H0 | Ne + Ne (Ne − 1) exp − 2 s (114) c Assuming a typical number of electrons Ne = 1 × 1010 , then for all wavelengths satisfying λ < 1.3σs , the term proportional to Ne2 (sometimes called coherent radiation) in Eqn (114) is dominated by the one proportional to Ne (incoherent radiation). The rms length σs is of the order of a few millimetres or centimetres; therefore, angular spectral flux in the infrared, visible and X-ray parts of the spectrum grows proportionally to the number of electrons, while at wavelengths longer than a few millimetres, it grows proportionally to Ne2 . Large powers can, therefore, be generated by the electron beam in the millimetre range. The precise computation of this power in this wavelength range is difficult because the radiation is partly reflected, attenuated and guided by all parts of the vacuum chamber. In some circumstances, this radiation can interact with the electron beam of the ring and produce some instability. There exists a way of generating some short wavelength radiation with intensity growing proportionally to Ne2 . To do so, one injects some periodic modulation in the longitudinal

60

P. Elleaume

density of an electron bunch. This takes place in a free electron laser (see Chapter 8) or by having a laser beam resonantly interacting with the electron motion in an undulator. In such experiments, the laser creates a periodic micro-bunching of the longitudinal density which results in a coherent emission at this wavelength and its harmonics. Coherent narrow band emission has been achieved at energies as high as 6 eV [16]. A detailed study of the properties of this radiation is out of the scope of this chapter. In the following, we assume randomly spaced electrons and a sufficiently short wavelength and we compute the flux and brilliance of an electron beam by summing over the contribution from each electron. 3.2

Transverse distribution of the electron beam

For the moment, let us assume a monoenergetic electron beam. Summing over the brilliance ¯ of the photon beam produced by an of each electron of the beam, one derives the brilliance Ꮾ electron beam as the convolution of the single electron brilliance Ꮾ defined in Section 2.6.1 with the electron beam distribution (defined in Chapter 1): ¯ ( Ꮾ y , y , s, ω, u) ˆ 

=

∞



∞



∞



∞

−∞ −∞ −∞ −∞

 y − Y  , s, ω, u)ρ  Y  , s) d2 Y d2 Y  Ꮾ( y − Y, ˆ e (Y,

(115)

 Y  , s) is the electron beam density at the longitudinal coordinate s. Y = (X, Z) where ρe (Y,   and Y = (ϑx , ϑz ) are the transverse position and angle. The convolution expressed by Eqn (115) is valid if one is allowed to simply add the brilliance from each electron (see previous section) and if the various electrons are submitted to the same acceleration whatever their angle or transverse position in the bunch. This last condition is equivalent to neglecting any magnetic field gradient. It is realized in the very large majority of insertion devices being used as synchrotron source. One consequence of Eqn (115) is that the angular spectral flux ¯ ¯ ᏿) (d


∞



∞

−∞ −∞

¯  d< ˆ e (Y  ) d2 Y  ( y − Y  , ω, u)ρ d.

 ∞ ∞ ¯ dP¯  dP    ( y)= ( y − Y )ρe (Y  ) d2 Y  d. −∞ −∞ d.

¯ d< ( y , s, ω, u) ˆ = d᏿



∞



∞

−∞ −∞

d<  s, ω, u)ρ  s) d2 Y  ( y − Y, ˆ e (Y, d᏿

 ∞ ∞ ¯ dP¯ dP  s)ρe (Y,  s) d2 Y ( y , s) = ( y − Y, d᏿ d ᏿ −∞ −∞

 s) are the partial electron beam densities: where ρe (Y  ) and ρe (Y,  ∞ ∞  ∞ ∞   2  Y , s) d Y  s) =  Y  , s) d2 Y  ρe (Y ) = ρe (Y, ρe (Y, ρe (Y, −∞ −∞

−∞ −∞

(116)

(117)

(118)

Note that, from Eqn (116), one may want to write ρe (Y  , s) rather than ρe (Y  ). The s-independence of ρe (Y  ) results from the invariance of the angle Y  under propagation

Synchrotron radiation, generalities

61

in a drift space. At a sufficient distance from the source, the power per unit surface is approximated by the power per unit solid angle divided by the square of the distance to the source. The power per unit solid angle from a filament electron provides a significant variation only for angles of the order of 1/γ . This angle is usually much greater than the divergence of the electron beam. Consequently, one usually forgets the convolution when computing the power per unit surface (or per unit angle) in a beamline. Note that this simplification does not apply to the spectral flux per unit solid angle or per unit surface for sources like undulators or very high energy wigglers and one must perform the convolution. In many situations, one may avoid the computation of the full convolution if one is simply interested in the rms size or divergence of the photon beam. Indeed, it is a consequence of Eqns (116) and (117) that the ¯ x, Q ¯ z and rms divergences Q ¯ x , Q ¯ z of the radiation generated by the thick beam rms sizes Q are the quadratic sum of the filament beam contributions Qx , Qz , Qx  , Qz and the electron beam contributions σx , σz , σx  , σz : ¯ x2 = Qx2 + σx2 Q

¯ z2 = Qz2 + σz2 Q

¯ x2 = Qx2 + σx2 Q

¯ z2 = Qz2 + σz2 Q

(119)

Note that both the electron and photon beam sizes depend on the longitudinal position s and the photon beam sizes and divergences depend on the frequency ω of the radiation. The photon beam sizes can also be linearly transformed into linear optical elements by means of Eqns (106) and (107). Assuming that the middle of the straight section where the insertion device is installed is a symmetry point of the ring lattice (see Chapter 1), the electron beam density is approximated by

2 /β + β ϑ 2 2 /β + β ϑ 2 ) (Z − sϑ ) 1 (X − sϑ z z z x x x z x  Y  , s) = ρe (Y, exp − − (2π)2 εx εz εx εz (120) where βx and βz are the so-called horizontal and vertical beta functions of the lattice in the middle of the straight section, and εx and εz are the corresponding electron emittances. The electron beam sizes and divergences at a distance s from the middle of the straight section are derived from Eqn (120) as σx2 = εx (βx + s 2 /βx ) σx2 = εx /βx

σz2 = εz (βz + s 2 /βz )

σz2 = εz /βz

(121)

One can generalize Eqn (109) to define the emittance Ᏹ¯ x of the photon beam produced by a thick beam of electrons as ¯ 2x = x 2  x 2  − xx  2 Ᏹ

(122)

in which each term x 2 , x 2  and xx   are the sum of a contribution from the electron beam and from the radiation produced by a filament electron beam. One can show that Ᏹ¯ x is always smaller than the sum Ᏹx +εx of the emittance of the radiation produced by a filament electron beam plus the electron beam emittance. One easily shows that the equality Ᏹ¯ x = Ᏹx + εx is satisfied if the quantity xx   for the electron beam and for the photon beam produced by a filament electron beam are both equal to zero for the same longitudinal coordinate s and if the ratio of the beam size to the divergence are equal (matching condition): βx =

σx Qx =  σx Qx 

(123)

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P. Elleaume

3.3

Electron energy spread

As discussed in Chapter 1, all electrons of the beam do not have the same energy. The energy γ of an electron deviates slightly from the average energy γ0 . Let δ = (γ − γ0 )/γ0 be the relative energy deviation. The electron beam density ρe depends also on δ and we shall  Y  , s, δ). The single electron brilliance depends on the electron energy as write it as ρe (Y, ¯ discussed in Section 2.8. Therefore, the expression given by Eqn (115) of the brilliance Ꮾ can be generalized as
  ω    ¯ ( y − Y )(1 + δ), ( y − Y )(1 + δ), s, Ꮾ y , y , s, ω, u) ˆ = Ꮾ ( , uˆ (1 + δ)2  Y  , s, δ) d2 Y d2 Y  dδ × (1 + δ)4 ρe (Y,

(124)

where Ꮾ is the brilliance produced by a filament electron beam. For bending magnet and wiggler radiation, the brilliance Ꮾ varies little over the frequency shift ω(1 − (1/(1 + δ)2 )) ∼ 2δω induced by the energy deviation δ which is typically in the 1 × 10−3 range. Under such conditions Eqn (124) reduces to Eqn (115) with  ∞     Y  , s, δ) dδ ρe (Y, Y , s) = ρe (Y, (125) −∞

It simply means that the influence of the energy spread to the brilliance only takes place through the change of beam size and divergence that it brings. For undulator radiation, the spectrum is made up of a series of narrow peaks and the electron energy spread combined with the fast variation of the brilliance with the frequency causes an additional reduction of the brilliance. 3.4

Polarization

A full description of the polarization characteristics of a photon beam produced by a bunch of electrons can be done within the modern formalism of density matrix or the old fashioned Stokes parameters. The density matrix is written in terms of four real coefficients which are in direct relation with the four Stokes parameters. In this section we shall recall the definition of the Stokes parameters and summarize a few properties. First, let us consider an experimental set-up which integrates all the radiation over some frequency bandwidth -ω and some surface ᏿. Let < be the integrated flux with polarization vector u. ˆ It can be expressed as a sum of spectral flux per unit surface d
Following Born and Wolf [9], one defines the four Stokes parameters s0 , s1 , s2 and s3 as s0 = <(uˆ x ) + <(uˆ z ) s1 = <(uˆ x ) − <(uˆ z ) s2 = <(uˆ 45 ) − <(uˆ 135 ) s3 = <(uˆ r ) − <(uˆ 1 )

(127)

Synchrotron radiation, generalities

63

The unit vectors uˆ x = (1, 0) √ and uˆ z = (0, 1) are parallel √ to the Ox and Oz axis. The unit vectors uˆ 45 = (uˆ x + uˆ z )/( 2) and uˆ 135 = (uˆ x − uˆ z )/( 2) correspond to two orthogonal linear polarizations making √angles of 45 and 135 degrees √ with respect to the Ox axis. The unit vectors uˆ r = (uˆ x + iuˆ z )/( 2) and uˆ 1 = (uˆ x − iuˆ z )/( 2) correspond to the two right and left handed orthogonal circular polarizations. s0 is the total number of photons collected over all polarizations. s0 is always positive while s1 , s2 and s3 are positive or negative and represent the difference of flux between two orthogonal states of polarization. This description is complete in the sense that the flux <(u) ˆ in an arbitrary polarization state defined by the unit vector uˆ = αx uˆ x + αz uˆ z can be expressed as a function of the Stokes components by <(u) ˆ = 21 (s0 + (αx αx∗ − αz αz∗ )s1 + 2 Re(αx αz∗ )s2 − 2 Im(αx αz∗ )s3 )

(128)

where Re(x) and Im(x) stand for the real and imaginary parts of the quantity x. One frequently normalizes the s1 , s2 and s3 Stokes components with respect to the total intensity s0 and defines the polarization rates I1 , I2 and I3 as I1 =

s1 s0

I2 =

s2 s0

I3 =

s3 s0

(129)

I1 , I2 and I3 take values between −1 and 1. They are, respectively, the normal linear, inclined linear and circular polarization rates. Taking the simple case of a plane wave described by  the flux <(u) a transverse field E, ˆ can be expressed as <(u) ˆ ≈ |E u| ˆ 2 and one easily shows that the Stokes parameters satisfy the following condition: s02 = s12 + s22 + s32

(130)

Summing the flux contributions from each electron, over the surface S and over the frequency bandwidth -ω results in the following inequality s02 ≥ s12 + s22 + s32

(131)

The breaking of the equality can be understood as the presence of some depolarization. In the extreme case of a fully depolarized radiation s1 , s2 and s3 are all equal to zero, and from Eqn (128), one predicts equal flux on any state of polarization. It is important to stress that the depolarization originates from several sources: the spread of the electron in phase space, the finite surface of integration of the radiation beam and the finite frequency bandwidth. As a matter of fact, the wiggler radiation observed away from the axis of the electron beam can be largely depolarized while the undulator radiation observed on-axis on an odd harmonic of the spectrum presents negligible depolarization. 3.5 3.5.1

Coherence General case

One distinguishes between the longitudinal and the transverse coherence. The longitudinal coherence can be qualitatively understood as the longitudinal distance over which the field is oscillating with a defined phase and frequency. The longitudinal coherence is at most equal to the duration of the radiation pulse produced by a single electron. It can be shorter if the

64

P. Elleaume

electrons emit at different frequencies. The longitudinal coherence length lc is also related to the wavelength of the radiation and the width of the spectral distribution -λ by the relation lc =

λ2 -λ

(132)

Note that the use of a monochromator reduces the bandwidth -λ and accordingly increases the coherence length lc . The transverse coherence of a photon beam is related to the phase preservation of the field in the transverse direction. It is quantitatively defined as the distance between two narrow slits which results in the extinction of the interference fringes observed in Young’s experiment. The detailed analysis of the transverse coherent length requires the use of the mutual intensity. Similarly to the brilliance, one expresses the mutual intensity ᏹ¯ of a photon beam as the following convolution:





   ξ ξ ¯ ( Ei y + , s, ω uˆ ∗ Ei∗ y − , s, ω uˆ ᏹ y , ξ, s, ω, u) ˆ = 2 2 i (133)  ∞ ∞  s) d2 Y = ᏹ(Y − y, ξ, s, ω, u)ρ ˆ e (Y, −∞ −∞

where Ei is the electric field created by the i-th electron, ᏹ is the mutual intensity produced y , s) is the density of the electron beam. A general by the filament electron beam and ρe ( expression of the mutual intensity as a function of the dimensionless field vector H and electron beam density ρe can be derived by substituting Eqns (74) and (77) in Eqn (133). The final expression involves a large number of integrals. However, in the far field approximation, one can replace Eqn (78) in Eqn (133) and obtain a simpler expression: ¯ ( ᏹ y , ξ, s, ω, u) ˆ e2 ≈ 2 2 2 4c ε0 s



∞



∞

−∞ −∞

ω Y ξ × exp i c s



H







 + ξ/2  Y + ξ/2 Y ∗ ∗ H , ω uˆ , ω uˆ s s



 s) d2 Y y − Y, ρe (

(134)

 s) should be understood y − Y, Note that in all equations following Eqn (133), the quantity ρe ( as the electron beam density as if it had propagated inside the drift space of the beamline together with the photon beam without being deviated by the bending magnet or quadrupole field of the storage using lattice. The contrast of the interference fringes produced by two narrow slits in Young’s experiment is the product of three contributions. One is the finite longitudinal coherence length; it can be increased to 1 by reducing the spectral bandwidth using a monochromator. The second comes from the difference of flux per unit surface incident on the slits; it can be increased to 1 by properly centring the pair of slits on the photon beam. The third contribution is deduced from the mutual intensity. Let us define the contrast C as the ratio between the sum and the difference of the maximum and minimum signal. C is related to the mutual intensity by ¯ (  s, ω, u)| |ᏹ y , ξ, ˆ  y, s, ω, u) C(ξ, ˆ = ¯ ( ¯ ( ᏹ y + ξ/2, 0, s, ω, u)( ˆ ᏹ y − ξ/2, 0, s, ω, u)) ˆ

(135)

Synchrotron radiation, generalities

65

where ( y , s) is the position of the mid-point between the two slits and ξ is the lateral displacement between the slits. 3.5.2

Application to a Gaussian beam

Let us consider the particular case where the brilliance (and therefore the mutual intensity) can be factorized as a product of contributions in the vertical and horizontal planes: ¯ ( ¯ z (z, ξz , s, ω, u) ¯ x (x, ξx , s, ω, u)  s, ω, u) ˆ ᏹ ˆ ᏹ y , ξ, ˆ =ᏹ ¯ ( ¯ x (x, θx , s, ω, u) ¯ z (z, θz , s, ω, u) Ꮾ y , y , s, ω, u) ˆ =Ꮾ ˆ Ꮾ ˆ

(136)

We assume that the slits are parallel to the Oz axis in such a way that the distance between the slits can be written as ξ = (ξ, 0). The contrast of the interference fringes depends on ¯ x (x, ξx , s, ω, u) ¯ x (x, θx , s, ω, u). ᏹ ˆ and therefore on Ꮾ ˆ To simplify the notation, we abbreviate ¯ (x, ξ, ω) and Ꮾ ¯ (x, θ, ω) omitting the longitudinal coordinate s of the slit and the them as ᏹ polarization u. ˆ Equation (77) becomes  ∞   ¯ (x, ξ, ω) ≈ ¯ (x, θ, ω) exp −i ω θ ξ dθ (137) ᏹ Ꮾ c −∞ Let us now assume that both the electron beam distribution and the brilliance produced by ¯ (x, θ, ω) at the the filament electron beam are Gaussian; using Eqn (115), the brilliance Ꮾ slit location which results from a 2D convolution is also Gaussian and can be expressed in the form 
bθ 2 + 2axθ + gx 2 ¯ (138) Ꮾ(x, θ, ω) ≈ exp − 2Ᏹ where the coefficients a, b, c, Ᏹ and the rms size Q and divergence Q  are related to each other by x 2  = Q 2 = bᏱ θ 2  = Q 2 = g Ᏹ xθ  = −a Ᏹ g=

(139)

(1 + a 2 ) b

One can easily derive from Eqn (139) that Ᏹ2 = x 2 θ 2  − xθ2 ; therefore, Ᏹ can be identified with the emittance of the photon beam. Note that Eqn (138) does not make any assumption concerning the longitudinal position of the slit with respect to the waist of the radiation beam. Substituting Eqn (138) in Eqn (137), one derives the expressions for the mutual intensity and fringe contrast
ω a  ω 2 ξ 2 Ᏹ  2 x ¯ (x, ξ, ω) = ᏹ ¯ (0, 0, ω) exp − −i xξ − (140) ᏹ 2bᏱ c b c 2b
 ¯ (x, ξ, ω)| ᏹ ξ2 C= = exp − 2 (141) 2¯ (x − ξ/2, 0, ω) ¯ (x + ξ/2, 0, ω)ᏹ ᏹ

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P. Elleaume

where - is given by √

bᏱ Q - = 2 = 2 (2(ω/c)Ᏹ)2 − 1 (Ᏹ/(λ/4π))2 − 1

(142)

The contrast of the fringes decreases as the distance ξ between the slits increases. - is the critical distance between the slits which makes the contrast drop to exp (−1/2). Therefore, - is the transverse horizontal coherence length. It is the horizontal distance over which the radiation field loses its phase coherence due to the mixing introduced by the many electrons constituting the electron bunch. There are a number of important remarks that can be made from Eqn (142). •

• • •

The coherence length - is a local quantity which depends on the longitudinal position of the slit through the rms size Q. The ratio -/Q is invariant under propagation. This is the basis of the definition of the coherent flux as the flux intercepted by an aperture centred on the photon beam and equal to the transverse coherence length -. The emittance of the photon beam can be deduced from Eqn (142) following a measurement of both the transverse coherence - and the rms size Q. For a filament electron beam, the emittance satisfies Ᏹ = λ/4π and - is infinite. There is perfect phase correlation over the whole space because the beam is fully, transversely coherent. For a large emittance Ᏹ  λ/4π , and an observer located at a large distance from the source, one can approximate Q ∼ σ ∼ Dσ  with Ᏹ ∼ σ σ  and one derives the well known expression of the coherence length: -=

•

λD 2πσ

where σ and σ  are the rms size and divergence of the electron beam at the source. A similar coherence length can be defined in the vertical plane.

List of symbols General c ε0 α

(143)

Speed of light Vacuum permeability Fine structure constant

Electron m e h I R = (X, Z, S) τ

Mass of the electron Charge of the electron Planck constant Electron beam current Position of the electron Electron time

Synchrotron radiation, generalities γ ϑ = (ϑx , ϑz , ϑs ) σx , σz σx  , σz σγ /γ

Energy of an electron divided by mc2 Electron velocity normalized to the speed of light Horizontal and vertical rms electron beam sizes Horizontal and vertical rms electron beam divergences Rms electron relative energy spread

Radiation Observer time Position of the observer Distance between the observer and the electron Wavelength of the radiation  Frequency of the radiation nˆ = θx , θz , 1 − θx2 − θz2 Unit vector directed from the electron towards the observer   A(r, t), A(r, ω) Vector potential of the radiation in the time and frequency domain   E(r, t), E (r, ω) Electric field of the radiation in the time and ! frequency domain   B(r, t), B (r, ω) Magnetic field of the radiation in the time and ! frequency domain uˆ Complex unit vector describing the polarization of the radiation  H (n, ˆ ω) Dimensionless field vector of the emission in the direction nˆ <(r, t), <(r, ω) Scalar potential of the radiation in the time and frequency domain d< Spectral flux per unit solid angle d.

t r = (x, z, s) D λ ω  

d< d᏿

Spectral flux per unit surface

dP d.

Power per unit solid angle

dPx d.

Power per unit solid angle horizontally polarized

dPz d.

Power per unit solid angle vertically polarized

P Px Pz ᏹ Ꮾ Qx , Qz Qx  , Qz

Power of the radiation Horizontally polarized power Vertically polarized power Mutual intensity of the radiation Brilliance or brightness of the radiation Horizontal and vertical rms photon beam sizes Horizontal and vertical rms photon beam divergences

67

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P. Elleaume

References [1] Jackson, J. D., Classical Electrodynamics, Chapter 14 (John Wiley & Sons, Inc., New York, 1962). [2] Landau, L. and E. Lifchitz, Classical Theory of Fields, Chapter 8 and 9 (Pergamon, Oxford). [3] Feynman, R. P., The Feynman Lectures on Physics, Chapters 28 and 32 (Addison-Wesley Pub. Co., Reading, Mass., 1963). [4] Chubar, O., Sci. Instrum. 66(2), 1872–4 (1995). [5] Kim, K. J., Characteristics of Synchrotron Radiation, AIP Conference Proceedings 184, vol. 1, p. 567 (American Institute of Physics, New York, 1989). [6] Kim, K. J., Nucl. Instr. Meth. A246, 71 (1986). [7] Wigner, W., Phys. Rev. 40, 749 (1932). [8] Coisson, R. and R. P. Walker, SPIE Proceedings 582, 24, 1986. [9] Born, M. and E. Wolf, Principles of Optics, Chapter 10 and Appendix 8, 6th edn (Pergamon Press, New York, 1980). [10] Coisson, R., Applied Optics 34(5), 904 (1995). [11] Bosch, R. A. and O. V. Chubar, ‘Long-wavelength edge radiation in an electron storage ring’, Synchrotron Radiation Instrumentation, Tenth US National Conference, American Institute of Physics Conference Proceedings 417, E. Fontes (ed.) pp. 35–41 (1997). [12] Walker, R. P., Nucl. Instr. Meth. A267, 537 (1988). [13] Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, Inc., New York, 1964). [14] Goodman, J. W., Introduction to Fourier Optics (Mac Graw Hill, New York, 1968). [15] Chubar, O. and P. Elleaume, ‘Accurate and efficient computation of synchrotron radiation in the near field region’, Presented at the EPAC98 Conference, Stockholm, 22–26 June 1998. The SRW code is available from ‘http://www.esrf.fr/machine/groups/insertion devices/Codes/software.html’ [16] Girard, B. et al., Phys. Rev. Lett. 53, 2405 (1984); Ortega, J. M. et al., Nucl. Inst. Meth. A237, 268–72 (1985).

3

Undulator radiation Pascal Elleaume

1

Introduction

The large majority of insertion devices being used as synchrotron sources generate a magnetic field which is periodic along the electron beam path. If, in addition, the integral of the magnetic field over a single period vanishes, the velocity of the electron is also a periodic function of s. Insertion devices for which the magnetic field satisfies both these properties are generically called undulators. The simplest case is the planar undulator which presents a sinusoidal field parallel to a single plane. It is the most popular undulator, the characteristics of its radiation will be discussed in Section 3. The ellipsoidal undulator produces a field the transverse components (orthogonal to the main electron beam axis) of which are both sinusoidal with the same period but with a different amplitude and phase. The characteristics of its radiation will be discussed in Section 4. The radiation from the planar and the ellipsoidal undulators have many features in common which are a consequence of the periodicity of the magnetic field. Therefore, to avoid duplication, we shall derive in Section 2 the general properties of the radiation produced by electrons in a periodic magnetic field. Finally, Section 5 presents some modifications of the radiation properties when the magnetic field is almost but not completely periodic. These include tapered undulators and phase errors. The development in this chapter makes extensive use of material in Chapter 2. Therefore, we advise the reader interested in fully understanding the subjects to read Chapter 2 before hand. Due to its utmost importance, the characteristics of undulator radiation have been derived more or less independently at various degrees of approximation by a large number of people. The oldest paper (to my knowledge) reporting the angular spectral flux produced by a planar undulator is by Alferov et al. [1] and for the case of a helical undulator by B. Kincaid [2].

2 2.1

Periodic field Generalities

We use the following orthogonal set of axes: Ox (horizontal), Oz (vertical) and Os (longitudinal). The magnetic field of the undulator is B = (Bx , Bz , Bs ) where all components Bx , Bz and Bs are a function of the coordinates (x, z, s). The electron beam is assumed to travel along the longitudinal axis Os with some small deviations in both the horizontal and vertical directions.

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P. Elleaume

2.2

Electric field produced by a single electron

Let us consider an electron of charge e, mass m and energy γ mc2 travelling through an undulator. We have seen in Chapter 2 that at a sufficiently large distance s from the source,  the electric field E(x, z, s, ω) at the position (x, z, s) and frequency ω can be expressed as  

z2 x z  ie ω x2 H exp i s 1 + 2 + 2 , ,ω 2cε0 s c 2s 2s s s

 E(x, z, s, ω) =

(1)

where c is the speed of light, ε0 is the vacuum permeability. H is a dimensionless field vector which can be written as #  ∞" ω ϑx (s) − θx  H (θx , θz , ω) = 2πc −∞ ϑz (s) − θz
  s  ω 2  2 2  2  × exp i 1 + γ (ϑx (s ) − θx ) + γ (ϑz (s ) − θz ) ds ds 2cγ 2 0 (2) where ϑx (s) and ϑz (s) are the horizontal and vertical electron velocities at the longitudinal coordinate s normalized to the speed of light, and γ is the electron energy normalized to the rest energy mc2 . In the derivation of Eqns (1) and (2), we have assumed an observation point located close to the longitudinal axis Os: (x/s), (z/s) 1, and an electron making a small angle with respect to the longitudinal axis: ϑx , ϑz 1. As discussed in Chapter 2, one can express the velocity (ϑx , ϑz ) as a function of the transverse horizontal and vertical field components (Bx , Bz ) according to e ϑx (s) = γ mc



s



−∞

Bz (s ) ds

−e ϑz (s) = γ mc



s

−∞

Bx (s  ) ds 

(3)

We further assume that the transverse field integrals over one period satisfy  0

λ0

Bx (s  ) ds  =



λ0

Bz (s  )ds  = 0

0

In this chapter, the fields Bx , Bz and the velocities ϑx , ϑz are assumed to be periodic function of s with period λ0 . As a consequence, one may split the integral in Eqn (2) into an integral over each period of the magnetic field. By placing the origin O in the centre of the undulator one derives the following expression for the field vector H :   θz , ω)  H (θx , θz , ω) = h(θ,

(N −1)/2


exp 2iπq

q=−(N −1)/2

sin(πN (ω/ω1 )) = N H (θx , θz , ω) N sin(π(ω/ω1 ))





ω  ω1 (4)

 x , θz , ω) is derived from Eqn (2) by where N is the number of periods of the undulator. h(θ replacing the integral in the range (−∞, ∞) by an integral over a single period (−λ0 /2, λ0 /2) of the magnetic field. ω1 is a resonant frequency for which the phase advance of the radiation

Undulator radiation

71

emitted from one period to the next is equal to 2π. ω1 depends on the direction of observation (θx , θz ) and can be expressed as ω1 (θx , θz ) =  λ 0 0

=

4πcγ 2 (1 + γ 2 (ϑx (s) − θx )2 + γ 2 (ϑz (s) − θz )2 ) ds

4πcγ 2 λ0 (1 + (Kx2 /2) + (Kz2 /2) + γ 2 θx2 + γ 2 θz2 )

(5)

where the dimensionless deflection parameters Kx and Kz are defined as  2γ 2

Kx =

λ0





λ0

0

Kz =

ϑx2 ds

2γ 2 λ0



λ0 0

ϑz2 ds

(6)

It is clear from Eqns (6) and (3) that the deflection parameters Kx and Kz are independent of the electron energy. They only depend on the magnetic field. We shall see in the following sections that the resonant frequency ω1 plays a very important role and it is useful to derive the associated resonant wavelength λ1 : 2π c λ0 λ1 = = 2γ 2 ω1



K2 K2 1 + x + z + γ 2 θx2 + γ 2 θz2 2 2

(7)

Figure 3.1 presents the variation of sin(πN (ω/ω1 ))/(N sin(π ω/ω1 )) which appears in Eqn (4) as a function of the normalized frequency ω/ω1 for N = 40. As N tends to infinity, the function tends to 0 for all frequencies except those satisfying ω = nω1 where n is an integer. It is clear from Figure 3.1 that the spectrum of such radiation is made up of a series of peaks. The frequency of each peak is such that ω = nω1 where n is an integer. In the following,

1.0

0.5

0.0

–0.5

–1.0 0

1

2 / 1

3

4

Figure 3.1 Graph of sin(π N(ω/ω1 ))/(N sin(π(ω/ω1 ))) as a function of ω/ω1 for N = 40 periods.

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P. Elleaume

n will be called the harmonic number. In the limit of a large number of periods N, one can proceed with the following approximations: ∞

 sin(πN (ω/ω1 )) sin(π N ((ω/ω1 ) − n)) ≈ (−1)n(N −1) N sin(π(ω/ω1 )) πN ((ω/ω1 ) − n)

(8)

n=1

and H (θx , θz , ω) ≈ N

∞ 

(−1)n(N −1) hn (θx , θz )

n=1

sin(π N ((ω/ω1 ) − n)) π N ((ω/ω1 ) − n)

(9)

 x , θz , nω1 ) represents the contribution to the electric field on the n-th where hn (θx , θz ) = h(θ harmonic of the spectrum in the direction (θx , θz ). More precisely, hn (θx , θz ) is derived from Eqn (2): # ϑx (s) − θx −λ0 /2 ϑz (s) − θz
  2iπn s (1 + γ 2 (ϑx (s  ) − θx )2 + γ 2 (ϑz (s  ) − θz )2 )  × exp ds ds λ0 0 1 + (Kx2 /2) + (Kz2 /2) + γ 2 θx2 + γ 2 θz2

n hn (θx , θz ) = λ1



λ0 /2

"

(10) where λ1 is the resonant wavelength given by Eqn (7), Kx and Kz are the deflection parameters defined by Eqn (6) and (ϑx , ϑz ) are derived from the transverse magnetic field using Eqn (3). The important point is that for a periodic magnetic field the information contained in the field vector H (θx , θz , ω) can be reduced to a set of discrete vector coefficients hn (θx , θz ) with n = 1, 2, 3 . . . which only depends on the direction (θx , θz ). All characteristics of the radiation from an undulator can be derived from hn (θx , θz ). In particular, a general expression  for the electric field E(x, z, s, ω) produced by an electron at the position (x, z, s) in the near or far field is derived from Eqn (69) of Chapter 2 and from Eqn (9) as  ∞ ∞ ∞ ieωN  n(N −1) n (θx , θz ) sin(π N ((ω/ω1 ) − n))  h E(x, z, s, ω) = (−1) 4πε0 c2 π N ((ω/ω1 ) − n) −∞ −∞ n=1



θz2 ω θx2 × exp i θx x + θz z + s 1 + + dθx dθz c 2 2 (11) in which the resonant frequency ω1 depends on the angle (θx , θz ) according to Eqn (5). In the far field approximation, the angle integration in Eqn (11) can be performed: 

eN ω x2 z2  E(x, z, s, ω) ≈ exp i s 1 + 2 + 2 2cε0 s c 2s 2s (12) ∞  x z  sin(π N (ω/ω (x/s, z/s) − n))  1 n(N −1)  hn × , (−1) s s π N (ω/ω1 (x/s, z/s) − n) n=1

where hn (θx , θz ) is given by Eqn (10).

Undulator radiation 2.3

73

Angular spectral flux

For a filament electron beam, the angular spectral flux (d
n=1

≈

∞  d
d.

(13)

ˆ (θx , θz , ω, u)

where (d
(14)

where ω1 is the resonant frequency which depends on the angle of observation (θx , θz ). The angular spectral flux at the resonant frequncy ω = nω1 is therefore written as I d
(15)

It is proportional to the square of the number of periods (Figure 3.2). Clearly, the angular spectral flux of the n-th harmonic expressed by Eqn (14) is a product of two terms: the ‘lobe function’ |hn (θx , θz )uˆ ∗ |2 which contains the contribution of the magnetic field and the ‘interference function’ (sin(πN((ω/ω1 ) − n))/πN ((ω/ω1 ) − n))2 which depends on the angle θx2 + θz2 through ω1 and is the same for any type of undulator field. Since the number of period N is assumed to be large, the variations of the interference function vs (θx , θz ) are usually much more rapid than those of hn (θX , θZ ). From Eqns (4) and (14), one predicts that the radiation at any frequency ω is emitted over some narrow cone of emission centred on the average direction of the electron velocity. Each cone corresponds to a specific harmonic. The higher the harmonic number n, the larger the cone angle. This feature of the radiation is particularly spectacular when observed in the visible range of the spectrum. This can be achieved with a sufficiently low electron energy (0.1–1 GeV) in such a way that ω1 or some low order harmonic of ω1 falls in the visible or near infrared. Each colour of the visible range is emitted on a different cone. As a result, the pattern of the radiation observed on a screen placed at normal incidence to the radiation at some distance from the undulator consists of a series of circular rainbows of increasing diameter. Each rainbow corresponds to a specific harmonic, the higher the harmonic number, the larger the diameter of the rainbow. The intensity emitted along the circumference of one rainbow varies proportionally to the lobe function |hn (θx , θz )u| ˆ 2 . As discussed in Chapter 2, the angular spectral flux produced by a thick electron beam results from the convolution

74

P. Elleaume

1.6

Angular spectral flux

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

2

4 / 1

6

8

Figure 3.2 Graph of the angular spectral flux d
of the filament beam emission with the density distribution of the electrons in angle. The sharpness of the rings of emission observed on a screen can be blurred due to the electron beam sizes and divergences. Some plots of these rings of emission together with their blurring by the electron beam will be shown for the particular case of the planar undulator in Section 3.4. 2.4

Angle integrated spectral flux

Let us now consider the angle integrated flux <(ω, u) ˆ defined as  <(ω, u) ˆ =

∞



∞

−∞ −∞

∞

 d< (θx , θz , ω, u) ˆ dθx dθz =
(16)

n=1

where
∞



∞

d

(17)

Undulator radiation where the function F (x) is a function defined by  1 ∞ sin2 (x) F (x) = dx π x x2

75

(18)

and the resonant frequency ω1 (θx , θz ) is defined by Eqn (5). The angle θn corresponds to the direction in which the radiation of the n-th harmonics is observed with the frequency ω. It is derived from the relation ω = nω1 (θn , 0)

(19)

In particular, for the on-axis resonant frequency ω = ωn = nω1 (0, 0), the spectral flux is

Kz2 1 Kx2 π IN
2 2 K 1 N K 1 
Photon beam divergence

In this section we shall study the divergence of the monochromatic radiation. We shall discuss the conditions which are to be met to generate the smallest divergence possible. We shall only consider the case of the filament electron beam. The divergence of the general case is obtained by quadratically summing the divergence produced by a filament electron beam and the divergence of the electron beam. The general expression for the rms divergence is derived from Eqn (14). The minimum divergence is expected when the cone of emission of the radiation has an infinitely small aperture angle. In such a situation, the frequency ω is close to the on-axis resonant frequency nω1 (0, 0). In the limit of small angles (θx , θz ) 1/γ and large number of periods N  1, Eqn (9) for the field vector can be approximated as H (θx , θz , ω) = N

∞ 

(−1)n(N −1) hn (0, 0)

n=1

where θ 2 + θz2 L V = πn x + πN λ1 (0) 2




sin(V) V

(23)



λ1 (0) −n λ

(24)

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P. Elleaume

= n = n (1–1/Nn) = n (1+ 0.4/Nn)

1.0

(Sin (Γ)/Γ)2

0.8 0.6 0.4 0.2 0.0 –4

–3

–2

–1 (L /)1/2

0

Figure 3.3 Graph of (sin(V)/ V)2 as a function of the angle θ = frequencies. ωn is an abbreviation for nω1 (0, 0).

1

2

 θx2 + θz2 for three different

λ1 (0) is the resonant wavelength on-axis obtained by susbstituting (θx , θz ) = (0, 0) in Eqn (7). Note that Eqn (23) presents a circular symmetry in the variables(θx , θz ). Figure 3.3 presents

the variations of (sin(V)/ V)2 as a function of the angle θ = θx2 + θz2 for three different frequencies. Among the three curves, the minimum divergence is reached for a frequency slightly lower than the on-axis resonant frequency ωn = nω1 (0, 0). However, it also corresponds to a reduced angle integrated spectral flux. On the other hand, the angular spectral flux at the frequency ωn (1 − (1/nN)) which corresponds to a maximum of the angle integrated spectral flux (see Section 2.4) produces a larger divergence. In other words, there is a compromise between the flux and small size of the divergence. The splitting observed at high energies can be explained as follows. At frequencies ω higher than ωn , ω satisfies ω = nω1 (θ, 0) for a cone angle θ = θn = 0. The radiation is emitted in the surface of a cone and the two peaks of Figure 3.3 correspond to the opposite side of the cone. On the other hand, if ω < ωn , no resonance condition on harmonic n can be satisfied and the angular flux drops to zero over a frequency range of the order of ωn /nN. A fit of (sin(V)/ V)2 for ω = ωn with a Gaussian profile exp(−(θ 2 /2Q 2 )) gives an angular spread Q  of the central cone equal to   λn λn  Q = 0.69 ≈ (25) L 2L where, as before, λn is the wavelength associated with the frequency ωn and L = N λ0 is the undulator length. Substituting Eqn (7) in Eqn (25) results in 
 Kz2 1 Kx2 1  Q = 1+ + (26) 2 2 2γ nN For a large number of periods, Q  is much narrower than the (1/γ ) angle of emission typical of the bending magnet or wiggler radiation. This is due to the constructive interference of the

Undulator radiation

77

radiation from one period to the next. The longer the undulator and the higher the harmonic number the smaller the divergence. Note that a more natural way to define the divergence is to use the rms divergence: ∞ ∞ 2 θx (d
Photon beam sizes

In this section, we derive the expression of the flux per unit surface at the source in the middle of the undulator. By flux in the middle of the undulator, we mean the flux derived from the electric field from which the electric field at any location downstream of the undulator can be derived using the Huygens principle. Experimentally, it cannot be measured directly. It is deduced from an imaging experiment in which a lens or focusing mirror is placed between the source and an observation plane in such a way that the plane of observation and the transverse plane in the middle of the undulator are conjugate with respect to the lens. As in the previous section we assume a filament electron beam. The more general case of a thick electron beam can be derived following a convolution with the electron beam size. The spectral flux per unit surface in the middle of the undulator on harmonic n is derived from Eqn (11) and Eqn (71) of Chapter 2:  ω 2 d
1

(28) Similarly to the angular spectral flux, for the frequencies ω close to the on-axis resonant frequency, ωn = nω1 (0, 0), one may approximate hn (θx , θz ) = hn (0, 0) and the integral in Eqn √ (28) presents a circular symmetry. It, therefore, depends only on the radial coordinate r = x 2 + z2 . Let us write it as d
78

P. Elleaume

1.2

= n = n (1–1/nN ) = n (1+ 0.4/nN)

1.0 0.8 0.6 0.4 0.2 0.0 –10

–5

0 2πr/(2L)1/2

5

10

Figure 3.4 Spectral flux per unit surface in the middle of the undulator for three frequencies close to the on-axis resonant frequency ωn = nω1 (0, 0).

where ω1 (θ) is an abbreviation for ω1 (θ, 0) and J0 (x) is the Bessel function of order 0. Similarly to the angular spectral flux, the variations of the spectral flux per unit surface d
2.740  λn L 4π

(30)

where λn is the wavelength associated with the frequency ωn = nω1 (0, 0). One can show from Eqn (11) that the size Q is minimum for a position s = 0 of the plane located exactly in the middle of the undulator. Combining the minimum size given by Eqn (30) and the divergence given by Eqn (25), one derives the emittance ε of the photon beam: ε = Q  Q ≈ 1.89

λn 4π

(31)

It would have been more useful to derive the photon beam rms emittance from the second moments of position x and angle θx , but for the reason already discussed in the previous section, it is unpractical. As a result, the value of the emittance depends on the definition used for the size and divergence. For example, if one defines the size as the full width at half maximum (fwhm) divided by 2.35, Q, Q  and ε become  λn λn 3.00   Q = 0.53 λn L ε = 1.59 Q= (32) L 4π 4π

Undulator radiation

79

So far, all beam sizes and divergences were derived for the frequency ω = ωn . Q, Q  and ε vary with the frequency ω. A more complete numerical investigation shows that one obtains a minimum of the emittance for a value of ω close to but not exactly equal to ωn . Both the minimum value of the emittance and the frequency at which it takes place depends on the definition chosen for the size and divergence. It is important to stress again that abandoning the definition of emittance in terms of the second order moment for reasons detailed in Section 2.5 has certain consequences. The invariance of the emittance under propagation in drift space and aberration free focusing elements is likely to be (slightly ?) violated. The inequality ε ≥ λ/4π which is always true for the rms emittance is not even guaranteed and we are fortunate that Eqns (31) and (32) satisfy this condition. For reasons of simplicity, in the rest of this chapter, we shall use the following definitions for the divergence, size and emittance  

Q =

λ 2L

√ Q=

2λL 2π

ε=

λ 2π

(33)

Indeed, these are approximations and should not be considered as fundamental results.

3

Planar undulator

In this section, the general properties of the radiation produced by an electron beam propagating inside a periodic magnetic field, as derived in Section 2, are applied to the particular and most useful case of the sinusoidal planar undulator. 3.1

Electron trajectory

We consider a vertical sinusoidal field B defined as

  s  ˆ B = 0, −B sin 2π ,0 λ0

(34)

Note that in order to satisfy Maxwell’s Equations this expression of the magnetic field can only apply over a line defined here by the vector position r = (0, 0, s). It cannot be valid if the transverse position (X, Z) of the electron is different from zero. Nevertheless, we assume that the electron beam size extends over sufficiently short distances in such a way that the deviation of the real field from Eqn (34) can be neglected. For ultra-relativistic electrons, Eqn (3) can be solved to first order in 1/γ :  


1 s K ϑ = (ϑx , ϑz , ϑs ) = , 0, 1 + o cos 2π (35) γ λ0 γ where K is a dimensionless parameter also called the deflection parameter. o(1/γ ) is a quantity of order smaller than 1/γ . The effect of an intial small transverse velocity (ϑx,0 , ϑz,0 ) at the entrance of the undulator has been removed following a rotation of the axis Ox, Oz and Os. The deflection parameter K is proportional to the product of the peak field times the undulator period and is given by K=

ˆ 0 eBλ 2π mc2

(36)

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P. Elleaume

or, in practical units, K = 0.0934Bˆ [T] λ0 [mm]

(37)

K appears to be the amplitude of oscillations of the horizontal velocity in units of 1/γ . Since ϑ 2 = 1 − 1/γ 2 , one derives the following expressions for the longitudinal velocity ϑs :  ϑs =




 s 1 s 1 1 K2 K2 2 2 ≈1− +o 1 − 2 − 2 cos 2π − cos 2π 2 2 λ0 γ0 γ2 γ γ 2γ 2γ (38)

or ϑs = ϑ¯s + δϑs where ϑ¯s is the average velocity independent of s, and δϑs is the oscillating part: 

 K2 K2 s 1 ¯ ϑ¯s 1+ ϑs = 1 − cos 4π δϑs = λ0 2 4γ 2 2γ 2

(39)

(40)

Making use of

 ds 1 = cϑs = c 1 + o dτ γ

(41)

one derives the horizontal (vertical) position X(Z) as a function of the longitudinal coordinate s and Z(s): 


Kλ0 1 1 s X= +o Z=o (42) sin 2π λ0 γ γ 2π γ Equation (42) defines an electron trajectory which is essentially a sine wave in the horizontal plane which has the same phase as the magnetic field. Its amplitude is equal to Kλ0 /2πγ . For a typical undulator with a K value of 2 and a period of 40 mm, the amplitudes of the oscillations are 8.1, 2.2 and 0.9 µm for electron energies of 0.8, 2 and 7 GeV (respectively). The amplitude of the motion is well within the electron beam size in the undulator which is usually of the order of a few hundred microns. 3.2

Radiation field in the time domain

At a large distance and small angle from the source, the electric field produced by a single electron is transverse to the direction of observation nˆ ≈ (θx , θz , 1 − θx2 /2 − θz2 /2). Substituting Eqn (35) into Eqn (19) of Chapter 2, one derives the following expression for the transverse components (Ex , Ez ) of the field seen by an observer at time t " # Ex (θx , θz , t) Ez (θx , θz , t) " #
e d 2γ K cos(2π(s/λ0 )) − γ θx = −γ θz 4π ε0 cD dt 1 + (K cos(2π(s/λ0 )) − γ θx )2 + (γ θz )2 (43)

Undulator radiation

81

where D is the distance to the source and the expression for the longitudinal position s of the electron as a function of the observer time t is deduced from Eqn (3) of Chapter 2: ds dt cϑ 2cγ 2 ≈c = = dt dτ 1 − nˆ 1 + (K cos(2πs/λ0 ) − γ θx )2 + (γ θz )2

(44)

In a direction of observation located in the plane of the motion (θz = 0), the vertical component of the electric field vector vanishes. Figure 3.5 presents the horizontal component Ex of the field together with the angular spectral flux observed on-axis of the electron (θx = θz = 0) for a planar undulator made up of 5 periods of 42 mm and a deflection parameter K equal to 0.5, 1 and 2. The periodicity of the magnetic field implies the periodicity of the electric field in the time domain and therefore a spectrum made up of a series of harmonics. For small values of K, the field seen by the observer in the time domain is nearly sinusoidal and the spectrum is dominated by the fundamental harmonics. As K increases, the distance between two successive peaks increases resulting in a lower frequency of the fundamental peak. At the same time, the shape of the electric field pulse becomes narrower resulting in a distortion of the sine wave. This distortion explains the presence of higher harmonics in the spectrum. For a sufficiently large K value, the shape of each pulse of electric field tends towards that produced by an electron in a bending magnet whose field is equal to the undulator peak field. In this case, the envelope curve joining the top of each harmonic of the spectrum tends towards the angular spectral flux of the corresponding bending magnet multiplied by (2N )2 where N is the number of periods. 3.3

Radiation field in the frequency domain

Let us apply the general results obtained in Section 2 concerning the radiation produced by an electron in a periodic magnetic field. Substituting Eqn (35) in Eqn (10), one derives the expression for hn (θx , θz ) for a planar undulator: # K/γ cos(2π(s/λ0 )) − θx −θz 0

 s −2γ θx K sin(2π(s/λ0 )) + (K 2 /4) sin(4π(s/λ0 )) × exp 2iπn + ds λ0 2π(1 + (K 2 /2) + γ 2 (θx2 + θz2 ))

n hn (θx , θz ) = λ1



λ0

"

(45) where λ1 is the fundamental resonant wavelength associated with the frequency ω1 and is derived from Eqn (7) as
 K2 2π c λ0 2 2 2 2 1+ (46) + γ θx + γ θz = λ1 = ω1 2γ 2 2 Making use of the following identities [3] cos(z cos(θ)) = J0 (z) + 2

∞ 

(−1)k J2k (z) cos(2kθ )

k=1

sin(z cos(θ)) = 2

∞  k=0

(−1)k J2k+1 (z) cos((2k + 1)θ )

(47)

K = 0.5 800 4 × 1015 Photons/sec/0.1%/mrad2

600 400 kV

200 0 –200 –400 –600 –800 0.0

3 2 1 0

0.2

0.4

0.6 nm

0.8

1.0

0

10

20

30 keV

40

50

60

0

10

20

30 keV

40

50

60

0

10

20

30 keV

40

50

60

K=1 1.5 8 × 1015 Photons/sec/0.1%/mrad2

1.0

MV

0.5 0.0 –0.5 –1.0 –1.5

6 4 2 0

0.0

0.2

0.4

0.6 0.8 nm

1.0

1.2

1.4

K=2 7 × 1015 Photons/sec/0.1%/mrad2

2

MV

1 0 –1 –2

6 5 4 3 2 1 0

0.0

0.5

1.0

1.5 nm

2.0

2.5

Figure 3.5 Horizontal component of the electric field seen by the observer on-axis for a single electron (left-hand side) and the associated angular spectral flux computed for a filament electron beam (right-hand side). The computation assumes an electron current of 200 mA, an energy of 6 GeV and an undulator made of 5 periods of 42 mm with a deflection parameter K = 0.5, 1 and 2. For a small K, the electric field is nearly sinusoidal resulting in a spectrum dominated by the fundamental with a small contribution from harmonics 3 and 5. For a large K the electric field is made up of a series of narrow peaks resulting in a large number of harmonics in the spectrum.

Undulator radiation

83

K=2  x = 0.3

MV

1 0

–1 –2 0.0

0.5

1.0

1.5

2.0

2.5

nm

 z = 0 6 × 10 Photons/sec/0.1%/mrad2

2

15

4 2 0 0

10

20

30 40 keV

50

60

Figure 3.6 Horizontal component of the electric field produced by a single electron in the time domain and its associated angular spectral flux in a direction γ θx = 0.3, γ θz = 0. The computation assumes a 200 mA filament electron beam of 6 GeV and an undulator made of 5 periods of 42 mm with a deflection parameter K = 2. The peaks of the electric field are not equidistant. As a result the even harmonics are present.

where Jn (x) is the Bessel Function of the variable x and order n. One derives the following expression for the field vector on-axis

" 2  2 # nK nK nK − J(n−1)/2 J(n+1)/2 hn (0, 0) = γ uˆ x 1 + K 2 /2 4 + 2K 2 4 + 2K 2 (48) if n = 1, 3, 5, . . . hn (0, 0) = 0

if n = 2, 4, 6, . . .

where uˆ x is the unit vector parallel to the horizontal axis. As a consequence, the emission on-axis on the even harmonics vanishes. This can equivalently be understood from Eqn (43) or from Figure 3.5. The radiation field in the frequency domain presents no even harmonics because the pulses of positive and negative electric fields are equidistant in the time domain. This is not true if one observes the radiation off-axis in the horizontal plane. It is illustrated in Figure 3.6, which presents the electric field and angular spectral flux for γ θx = 0.3 and γ θz = 0 for a planar undulator with K = 2. Due to the finite horizontal angle of observation, the adjacent positive and negative peaks of the electric field in the time domain are not equidistant resulting in the presence of even harmonics in the frequency domain. 3.4

Angular spectral flux

The angular spectral flux produced by a filament electron beam in a planar undulator is given by Eqn (14) where hn (θx , θz ) is expressed by Eqn (45). Figure 3.7 presents the result of a computation of the angular spectral flux produced by a filament beam as a function of the horizontal and vertical angles γ θx and γ θz for the frequencies ω = ω1 (0, 0) and ω = 1.3ω1 (0, 0). The undulator has a deflection parameter K = 1.5 and contains 50 periods. As discussed in Section 2.3, the ring structure is a consequence of the periodicity of the field and is numerically generated by the interference function (sin(π N ((ω/ω1 ) − n))/ (πN((ω/ω1 ) − n)))2 . In the left-hand figure, the central spot corresponds to harmonic n = 1.

84

P. Elleaume 2

2

= 1 (0, 0)

1

 z

 z

1

= 1.3 1 (0, 0)

0

0

–1

–1

–2

–2 –2

–1

0  x

1

2

–2

–1

0  x

1

2

Figure 3.7 Angular spectral flux produced by a filament beam as a function of the horizontal and vertical angles γ θx and γ θz for two frequencies ω. The undulator is made up of 50 periods with a deflection parameter K = 1.5. Each ring is associated to a single harmonic of the spectrum. This ring pattern is typical of the undulator radiation.

Two rings are visible, they correspond to harmonics n = 2 and n = 3. To increase the visibility of the rings the image has been over-exposed by a factor 2. In the right-hand figure harmonic n = 1 is not visible because the selected frequency is too high and the rings also correspond to harmonics 2 and 3. The variation of intensity observed along the circumference of each ring is proportional to the function |hn (θx , θz )|2 and is a signature of the magnetic field of the undulator. It can be shown from Eqn (45) that for odd values of n, the maximum value of |hn (θx , θz )| as a function of θx and θz is reached on-axis (θx = θz = 0). The angular spectral flux on-axis is obtained by substituting Eqn (48) in Eqn (14): d
if n = 1, 3, 5, . . . (49)

with n2 K 2 Fn (K) = (1 + (K 2 /2))2





J(n−1)/2

nK 2 4 + 2K 2




− J(n+1)/2

nK 2 4 + 2K 2

2 (50)

The term |uˆ x uˆ ∗ |2 in Eqn (49) simply means that the radiation is fully horizontally polarized. In practical units, the angular spectral flux on-axis on harmonic n can be written as d
(51)

Figure 3.8 presents a plot of Fn (K) as a function of the deflection parameter K for several harmonic numbers.

Undulator radiation

0.5

n=7

n=5

n=3

85

n =1

Fn (K)

0.4 0.3 0.2 0.1 0.0 0

1

2

3

4

5

K

Figure 3.8 The function Fn (K) for different values of n. K is the deflection parameter.

2

2

= 1 (0, 0)

1

0

 z

 z

1

= 1.3 1 (0, 0)

–1

0

–1

–2

–2 –2

–1

0  x

1

2

–2

–1

0  x

1

2

Figure 3.9 Angular spectral flux produced by a 6 GeV electron beam with horizontal (vertical) rms divergence of 10(4) µrad. The frequencies and the undulator parameters are those of Figure 3.7.

The central bright spot seen on the left-hand side of Figure 3.7 is the central cone of the first √ harmonic. Its rms divergence is Q  ≈ λ/2L (see Section 2.5) where λ is the wavelength of the radiation and L is the length of the undulator. As discussed in Chapter 2, for a thick electron beam, the angular spectral flux is the convolution of the angular spectral flux produced by a filament beam by the angular distribution of the electron beam. Figure 3.9 presents the result of such a convolution made for the same conditions as in Figure 3.7 assuming an electron energy of 6 GeV and an rms horizontal (vertical) divergence of 10(4) µrad (applicable to an ESRF high beta undulator section). Clearly, both the central cone and the rings are enlarged. The rings of the high harmonic numbers become barely visible above the background.

86

P. Elleaume

As a result of the convolution, the horizontal and vertical divergences Qx and Qz of the photon beam at the frequency ω ≈ ωn = nω1 (0, 0) are Qx2 ≈

λ + σx2 2L

Qz2 ≈

λ + σz2 2L

(52)

where σx (σz ) are the rms horizontal (vertical) divergences of the electron beam. For odd harmonics, the on-axis angular spectral flux can therefore be approximated as the product of the angular spectral flux produced by a filament electron beam multiplied by 1 (1 + Lσz2 /2λ)

(1 + Lσx2 /2λ)

(53)

Undulator patterns similar to those of Figure 3.9 can be observed experimentally by placing a monochromator followed by a screen at a distance D from the undulator. In such a case a further broadening of the ring pattern is observed due to the electron beam size. In such an experiment, the sizes Qx and Qz of the central cone observed on the screen are
 λ 2 D2 2 + Qx ≈ εx βx + D βx 2L (54)
 λ 2 D2 2 + Qz ≈ εz βz + D βz 2L where εx and εz are the horizontal and vertical emittances of the beam and βx and βz , are the beta functions of the lattice at the source (see Chapter 1). In Eqn (54) we have neglected the electron energy spread and assumed the beta function to be at a minimum in the middle of the undulator and the distance D to be large compared to the undulator length L. These conditions are met in the large majority of undulators in use in the various synchrotron sources in the world. Note that the electron beam contribution to the size of the central cone is either dominated by the electron beam angular divergence if β < D or by the electron beam size if β > D. The β function which minimizes the size of the central cone and therefore maximizes the spectral flux per unit surface is equal to the distance D. 3.5

Polarization and lobe structure

The angular spectral flux in a direction (θx , θz ) is proportional to the lobe function |hn (θx , θz )uˆ ∗ |2 where uˆ is the unit vector defining the polarization. The function hn (θx , θz ) derived from Eqn (45) can be shown to be purely real. Therefore, the lobe functions |hn√ uˆ r |2 √ 2 and |hn uˆ l | associated with the right hand (uˆ r = (i, 1)/ 2) and left hand (uˆ 1 = (1, i)/ 2) circular polarizations are equal. The circular polarization rate is equal to zero in any direction. In other words, the radiation from a planar undulator is, for any direction of observation, linearly polarized. The plane of polarization depends on the direction of emission and on the harmonic number. It is clear from Eqn (45), that in a direction located in the plane of the motion (θz = 0), the vertical electric field component is always zero. For a more detailed presentation of the direction of polarization as a fuction of the direction of observation see [4]. The numerical computation of the lobe function can be made using a numerical integration of Eqn (45). Figure 3.10 presents a contour plot of |h1 u| ˆ 2 , |h2 u| ˆ 2 and |h3 u| ˆ 2 as a function of γ θx and γ θz for a deflection parameter K = 2. The plain curve corresponds to an electric field in the horizontal plane uˆ = uˆ x = (1, 0) and the dashed curve corresponds to an electric

Undulator radiation

87

Harmonic #1 3 2

 z

1 0 –1 –2 –3 –3

–2

–1

0  x

1

2

3

0  x

1

2

3

0  x

1

2

3

Harmonic #2 3 2

 z

1 0 –1 –2 –3 –3

–2

–1

Harmonic #3 3 2

 z

1 0 –1 –2 –3 –3

–2

–1

Figure 3.10 Contour plot of the lobe functions |hn uˆ x |2 (plain curves) and |hn uˆ z |2 (dashed curves) as a function of the normalized horizontal and vertical angles γ θx and γ θz for the harmonics n = 1, 2 and 3. The deflection parameter K = 2.

field in the vertical plane uˆ = uˆ z = (0, 1). The radiation with vertical electric field |hn uˆ z |2 is typically 5–10 times lower than that of the horizontal field |hn uˆ x |2 . To improve the visibility of the vertical polarization, a different step of contour has been used for each polarization. The lobes present some maxima and minima. The minima of the lobe are precisely equal to

88

P. Elleaume

zero and correspond to a change of sign of the vector hn . The lobe of emission on-axis is zero for all even harmonics 2, 4, 6, . . . , while it reaches a maximum for the odd harmonics 1,3,5, . . . . The higher the harmonic number the larger the number of lobes. In addition, the maxima of the lobes of the horizontal polarization are always in the horizontal plane while no radiation with vertical polarization is produced in this plane. Finally, the maxima of the vertical (horizontal) polarization always corresponds to a zero emission in the orthogonal horizontal (vertical) polarization.

3.6

Spectrum through a slit

We have seen in Section 2.3 that the spectrum produced by a filament electron beam in a given direction is made up of a series of harmonic peaks the frequency of which are multiples of the fundamental frequncy ω1 . We have also seen that for a planar undulator, ω1 is given by ω1 =

2cγ 2 λ0 1 + (K 2 /2) + γ 2 θx2 + γ 2 θz2 

(55)

where (θx , θz ) is the angle betweeen the electron and the observer. For a single electron and an observation at a single point in space, the width of the peaks is equal to 1/nN where n is the harmonic number and N is the number of periods. Extremely narrow peaks can, therefore, be expected when observing the radiation from a many period undulator on a high harmonic number. For example, the radiation produced by a 100 period undulator would have a spectral width below 0.1% on harmonic 11. In real situations there are a number of limitations which prevent the reaching of such a narrow spectral line. These will be discussed in this section. It can be shown from Eqn (55) that, for a given point of observation, all electrons of a thick beam do not have the same resonant frequency ω1 . This can occur through the angular divergence of the electron beam which spreads both the horizontal and vertical angle of observation θx and θz . A spread of the transverse position of the electron beam also induces a spread of ω1 because an observer located sufficiently close to the source sees the radiation produced by the various electrons under a different angle (θx , θz ). On-axis, the frequency ω1 is proportional to the square of the electron energy. As a result, the electron energy spread is also responsible for a spread of ω1 . Finally, an observer interested in collecting a large flux usually integrates the spectral flux over some finite aperture and the value of ω1 varies from one point to the next inside this aperture. To illustrate this, Figure 3.11 presents the spectrum of an undulator of the ESRF computed using the SRW [5] computer code. The electron beam energy is 6.03 GeV and the current is 200 mA. The horizontal (vertical) emittances are 4 nm (0.04 nm) and the horizontal (vertical) beta functions are 35 m (2.5 m). The undulator presents a period of 35 mm, a length of 5 m and a deflection parameter K of 2.34. The aperture of observation is placed at a 30 m distance from the undulator on-axis of the electron beam. Its horizontal (vertical) width is 1 mm (0.2 mm). The main peaks of Figure 3.11 correspond to harmonics n = 1, 3 and 5. Note that some radiation is emitted on the even harmonics. If one simultaneously shrinks the emittance of the electron beam and the width of the aperture, the harmonics n = 1, 3 and 5 would narrow with a higher peak intensity and their shape would tend towards (sin(π N ((ω/ω1 ) − n))/ (πN((ω/ω1 ) − n)))2 expected for a filament electron beam while even harmonics would disappear completely. Note that the local minimum in the low energy tail of harmonic 5 originates from the fact that at some photon energy, the radiation is emitted on a ring which

Undulator radiation

89

Photons/sec/0.1%

400 × 1012

300

200

100

0 2

4

6 8 10 Photon energy [keV]

12

14

Figure 3.11 Spectrum through a slit produced by an undulator of the ESRF.

falls partly in a minimum of the lobe structure defined by |hn u| ˆ 2 resulting in a reduced emission. There is an important difference between the spectral broadening induced by the electron energy spread and that induced by the electron emittance and slit width. This can be understood from Eqn (55). The electron relative energy spread σγ /γ is Gaussian (see Chapter 1) and it introduces a Gaussian spread σω /ω in the spectrum of each harmonic: σγ σω =2 ω γ

(56)

On the other hand, the spread in the angle θx and θz shifts the resonant frequency ω1 to the lower energy side only. As a consequence, the peak of an odd harmonic observed on-axis of the electron beam always presents a steep slope on the high energy side which is determined by nN and by σγ /γ and a reduced slope on the low energy side which is determined by the electron emittance, beta function and size of the observation aperture. For medium energy synchrotron sources in the 1–2 GeV range with small emittances, the broadening of the peak is usually dominated by the electron energy spread while for the high energy sources in the 6–8 GeV range both emittance and energy spread play a significant role in the broadening of the peaks. In most third generation sources the rms electron energy spread is around 0.1% resulting in a minimum fwhm of the harmonic peak of 0.5%. Adding the natural width 1/nN and the broadening induced by the emittance and the slits, one hardly observes a peak with fwhm smaller than 1% without using any additional monochromator. Undulator field errors can also broaden the peaks (see Section 5).

3.7

Angle integrated flux

The angle integrated spectral flux has been studied in Section 2.4 in the general case of a periodic magnetic field. We have seen in Sections 2.5 and 2.6 that the divergence and beam size of the radiation is the smallest around the frequencies ωn = nω1 (0, 0). These

90

P. Elleaume

frequencies are, therefore, the most useful from a user point of view. Let us derive the angle integrated spectral flux
(57)

where Qn (K) is expressed as 2



nK 2 nK 2 nK 2 K 2 Fn (K) Qn (K) = 1 + − J = J (n−1)/2 (n+1)/2 4 + 2K 2 4 + 2K 2 2 n 1 + (K 2 /2) if n = 1, 3, 5, . . . Qn (K) = 0

if n = 2, 4, 6, . . . (58)

In practical units, the flux at the frequency ωn with an electric field in the horizontal plane can be written as
(59)

Figure 3.12 presents the variation of Qn (K) as a function of K for the harmonics n = 1, 3, 5 and 7. Qn (K) is maximum for harmonic n = 1 and decreases with increasing harmonic number n. Q1 (K) tends towards 1 for large K values. As a result the flux on any harmonic is smaller than π αN (I /e) which only depends on the number of periods of the undulator and the ring current. Figure 3.13 presents the angle integrated spectral flux computed in the whole spectrum for several values of the deflection parameter K. The peaks of the harmonics n = 1, 3, 5, . . . are clearly visible even though they are not as sharp as they would be if observed though a narrow slit. At low values of the deflection parameter K, the spectrum is dominated by the fundamental while at high values of K, the spectrum tends toward that of a bending magnet with some oscillations superimposed. Note that all odd harmonics present

1.0 0.8

Qn (K )

n=1 0.6

n=3

n=5

0.4 n=7

0.2 0.0 0

1

2

3 K

Figure 3.12 Qn vs K for several values of n.

4

5

Undulator radiation

Photons/sec/0.1%

200 × 1012

91

K = 0.5

150

100

50

0 0

5

10

15

20

25

30

35

600 × 1012 K =1 Photons/sec/0.1%

500 400 300 200 100 0 0

5

10

15

20

25

30 K =2

1.0 × 1015 Photons/sec/0.1%

35

0.8 0.6 0.4 0.2

0

5

10

15

20

25

30

35

Figure 3.13 Angle-integrated spectra produced by a 200 mA electron current of 6 GeV in a 42 mm period undulator with 40 periods and with a deflection parameter is K of 0.5, 1 and 2.

an abrupt discontinuity on the high energy side. The precise shape of this discontinuity is determined by the product nN and by the electron energy spread (see Section 3.6). The total variation of flux over the discontinuities corresponds to the on-axis spectral flux given by Eqn (57).

92

P. Elleaume

3.8

Brilliance

From Eqn (45), one derives the following properties of hn (θx , θz ) hn (−θx , −θz ) = hn (θx , θz )

if n = 1, 3, 5, . . .

hn (−θx , −θz ) = −hn (θx , θz )

if n = 2, 4, 6, . . .

(60)

From Eqns (60) and (89) of Chapter 2, one derives the following relations between the on-axis o, o, 0, ω, u) ˆ and the angle integrated flux
if n = 1, 3, 5, . . . (61) if n = 2, 4, 6, . . .

These relations are valid for any frequency ω and polarisation u. ˆ The brilliance is negative on the even harmonics! As discussed in Section 2.6 of Chapter 1, a negative brilliance should not be considered as a fundamental difficulty. It simply means that the brilliance cannot be simply assimilated to a distribution function especially when dealing with the purely diffractive radiation field of a single electron emission. In all cases, the most useful radiation is that produced on the odd harmonics where the on-axis brilliance is positive. Its maximum (vs ω) coincides with the maximum of the angle-integrated spectral flux
4 I Qn (K)
(62)

where λ is the wavelength associated with the frequency ωn . The brilliance of the radiation produced by a thick beam is the convolution of the brilliance produced by a filament electron beam and the electron beam distribution. The following expression is the usual estimate of the maximum brilliance [6] on-axis for an odd harmonic: Ꮾn ( o, o, 0, ωn , uˆ x ) ≈

ˆ
(63)

where Qx , Qz , Qx  and Qz are the rms photon beam sizes and divergences in the middle of the undulator which are approximated as a function of the rms electron beam sizes σx , σz , σx  and σz by λ 2L λL Qx2 ≈ σx2 + 8π 2

Qx2 ≈ σx2 +

λ 2L λL Qz2 ≈ σz2 + 8π 2

Qz2 ≈ σz2 +

(64)

Note that with this expression of the photon beam size and divergence, the brilliance defined in Eqn (63) coincides with the expected result in both limits of infinitely large and infinitely small electron beam emittance. One should nevertheless only consider Eqn (63) as an approximation. The exact result is obtained by convoluting Eqn (74) of Chapter 2 by the distribution of the electron beam in the transverse phase space.

Undulator radiation 3.9

93

Tunability of the spectrum

We have seen that the smallest photon beam emittance and, therefore, the highest brilliance is reached if one collects the radiation on-axis of the electron beam at a frequency corresponding to an odd harmonic ωn given by ωn =

2ncγ 2 λ0 (1 + K 2 /2)

(65)

The frequency ωn can be tuned either in a discontinuous manner by jumping from one harmonic to the next inside the series n = 1, 3, 5, . . . or continuously by changing the value of the deflection parameter K. The change of K is achieved by changing the peak field of the undulator as predicted by Eqn (36). The peak undulator field is reduced by opening the magnetic gap (permanent magnet undulator) or reducing the current (electromagnet undulator). However, by design, there is a maximum value of the field. For a permanent magnet device it corresponds to the minimum magnetic gap authorized by the vacuum chamber and/ or electron beam. For an electromagnet undulator, it corresponds to the maximum current available from the power supply compatible with a reasonable temperature rise of the copper in the conductor windings. Let Kmax be the maximum value of K. Kmax is an essential property of an undulator because it determines its tunability. It is clear from Eqn (65) that to bridge the gap left between harmonics 1 and 3, one needs Kmax to be larger than 2. In practice, to obtain a sufficient overlap of the brilliance at the junction between harmonics 1 and 3, one usually selects Kmax ≈ 2.2. Similarly, to ensure the tunability between harmonics 3 and 5, one needs Kmax > 1.16, but in practice one would use Kmax = 1.3–1.4. The tunability between higher harmonics requires an even lower value of Kmax and is therefore easier to reach. Finally, one should mention the possible choice of Kmax around 1.5 for which one would bridge the gap between harmonics 1 and 3 by using harmonic 2, even though harmonic 2 gives a reduced flux and brilliance. This approach is most useful for large electron emittances and usually considered unacceptable for experiments requiring high brilliance performed on a low emittance source. To illustrate this discussion, Figure 3.14 presents the brilliance of a 5 m long undulator with a 42 mm period installed on a high beta section of the ESRF. The electron energy is 6 GeV, the current is 200 mA, the emittances are 4 nm (0.04 nm) in the horizontal (vertical) plane and the beta function is 35 m (2.5 m) in the horizontal (vertical) plane. The photon energy is changed by varying the field of the device and the resulting brilliance is plotted as a function of the photon energy for the harmonics 1, 3, 5 and 7. This undulator presents a Kmax of 2.4 resulting in a good overlap of the photon energy range covered by the fundamental and the third harmonics. 3.10

Power density

Substituting the dimensionless electron velocity ϑ given by Eqn (35) into the general expression of the power per unit solid angle dP /d. produced by a filament electron beam (see Eqn (51) of Chapter 2), one derives dP 4π 2 e2 2γ 4 I (θx , θz ) = NK2 2 d. 4πε0 π e λ0 
  λ0 /2
1 s (γ θx − K cos(2π(s/λ0 )))2 2 sin 2π ds × −4 3 λ0 d5 −λ0 /2 d

(66)

94

P. Elleaume

Photons/sec/0.1%/mm2/mrad2

1020 Harm # 5

Harm # 1 Harm # 3 1019

Harm # 7

1018

1017

2

3

1

4

5

6 7 8 9

10 Photon energy (keV)

2

3

4

5

6 7 8 9

100

Figure 3.14 Brilliance produced by a 5 m long undulator of a 42 mm period installed on an ESRF high beta straight section. Kmax is 2.4, resulting in a good overlap of the photon energy range covered by the fundamental and the third harmonics.

with d given by
2
s d = 1 + γ θx − K cos 2π + (γ θz )2 λ0

(67)

In particular, the power per unit solid angle on-axis is dP 21γ 2 (0, 0) = P G(K) d. 16πK

(68)

where P is the angle-integrated power whose expression is given by Eqn (59) of Chapter 2 and the function G(K) is defined as 16K 7



π

−π

(1 − K 2 cos2 (x)) K 6 + (24/7)K 4 + 4K 2 + (16/7) 2 sin (x)dx = K = G(K) (1 + K 2 )7/2 (1 + K 2 cos2 (x))5 (69)

In practical units the angle-integrated power and the power per solid angle on-axis are expressed as # " dP W = 10.84Bˆ [T ]E 4 [GeV]I [Amp]N G(K) (0, 0) d. mr 2 (70) 2 4 P [kW ] = 0.633Bˆ [T ]E [GeV]I [Amp]L [m]

Undulator radiation

95

1.0

G (K)

0.8 0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

K

Figure 3.15 Behaviour of the function G(K).

1.0 K = 0.3

fK ( X, 0)

0.8 0.6 0.4 K=1 0.2 K = 10 0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

 x /K

Figure 3.16 Plot of fK (γ θx , 0) as a function of the normalized horizontal angle γ θx /K.

where Bˆ is the undulator peak field and L = N λ0 is the undulator length. The function G(K) is shown in Figure 3.15. It is clear that for K > 1, G(K) is close to 1 and the on-axis power density simply scales proportionally to the number of periods and the peak field. dP /d. is precisely equal to 2N times the power density produced in a bending magnet with a field ˆ equal to B. Following K. J. Kim [7], one defines the function fK (γ θx , γ θz ) by dP dP (θx , θz ) = (0, 0)fK (γ θx , γ θz ) d. d.

(71)

Figures 3.16 and 3.17 present the variations of fK as a function of the normalized angles γ θx /K and γ θz . fK is maximum and equal to 1, on-axis. In the vertical plane it decays

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P. Elleaume 1.0

fK (0,  Z)

0.8 0.6 K = 10

K=1 0.4 0.2

0.0

K = 0.3 0.2

0.4

0.6

 Z

0.8

1.0

1.2

1.4

Figure 3.17 Plot of fK (0, γ θz ) as a function of the normalized horizontal angle γ θz .

with a profile similar to that of a bending magnet. In the horizontal plane, it extends over the range ±K/γ . This corresponds to the angle range spanned by the electron in the course of its oscillations in the undulator field. The solid angle . in which the power is emitted can be estimated from Eqn (68): .=

4

P 16πK = dP /d. 21γ 2

(72)

Ellipsoidal undulator

After the planar undulator, the next most popular undulator is the ellipsoidal undulator. It is characterized by sinusoidal horizontal and vertical field components with identical periods. Note that the planar undulator studied in the previous chapter is a particular case of an ellipsoidal undulator without any horizontal field component. The helical undulator is a particular case of the ellipsoidal undulator in which both components have identical peak fields and their phase difference is π/2. The essential use of ellipsoidal undulators is in the production of intense and brilliant radiation with an arbitrary elliptical polarization. 4.1

Electron trajectory

The trajectory of an electron travelling inside the field of an ellipsoidal undulator is derived in a manner similar to that of a planar undulator (Section 3.1). One expresses the magnetic field B on the axis of an ellipsoidal undulator as
 

s s  ˆ ˆ B = Bx sin 2π ,0 (73) + ϕ , −Bz sin 2π λ0 λ0 where Bˆ x and Bˆ z are the horizontal and vertical peak fields, λ0 is the undulator spatial period and ϕ is the phase between the vertical and horizontal sinusoidal fields. The Lorentz force

Undulator radiation

97

can be solved to first order in 1/γ to give the dimensionless expression ϑ of the electron velocity: ϑ =




  


Kx Kz 1 s s , + ϕ ,1 + o cos 2π cos 2π γ λ0 γ λ0 γ

(74)

where c is the speed of light, γ mc2 is the total electron energy and m is the mass of the electron. Kx and Kz are the horizontal and vertical dimensionless deflection parameters related to the undulator peak fields by the relations Kx =

eBˆ z λ0 2π mc2

Kz =

eBˆ x λ0 2πmc2

(75)

For a planar undulator, Kx = K and Kz = 0. As for the planar undulator, we have assumed that the average velocity of the electron beam is constant and parallel to the longitudinal Os axis. The longitudinal component of the velocity ϑs at second order in 1/γ is derived from Eqn (35) as

1 + Kx2 /2 + Kz2 /2 ϑs = 1 − 2γ 2






 Kz2 Kx2 s s 1 + + cos 4π cos 4π + 2ϕ + o 2 2 4γ λ0 4γ λ0 γ2 (76)

and the position X and Z of the electron at the longitudinal coordinate s as 

Kx λ0 1 s X= +o sin 2π γ 2π γ λ0



1 Kz λ0 s +ϕ +o Z= sin 2π 2πγ λ0 γ

(77)

The trajectory is therefore a sort of helix which projects into the transverse plane as an ellipse. In the specific case of a purely helical field defined by Kx = Kz = K and ϕ = π/2, the longitudinal velocity ϑs is a constant, given by ϑs = 1 −

1 + K2 2γ 2

(78)

Conversely, a constant longitudinal velocity implies a pure helical field. 4.2

Electric field in the frequency domain

The ellipsoidal undulator is a particular case of a periodic magnetic field. Therefore, as discussed in Section 2.2, the electric field in the frequency domain can be deduced from Eqns (11) and (12) where the field vector hn (θx , θz ) is specific to the ellipsoidal undulator. Substituting Eqn (35) in Eqn (10), one expresses hn (θx , θz ) as 

 
Kx s  − θx  cos 2π n λ0  λ0   γ
 exp (inψ(s)) ds hn (θx , θz ) = λ1 0  Kz cos 2π s + ϕ − θ  z γ λ0

(79)

98

P. Elleaume

where ψ(s) is given by s ψ(s) = 2π + [−2γ θx Kx sin(2π(s/λ0 )) − 2γ θz Kz sin(2π(s/λ0 ) + ϕ) λ0 *

+ (Kx2 /4) sin(4π(s/λ0 )) + (Kz2 /4) sin(4π(s/λ0 ) + 2ϕ)] [1 + Kx2 /2 + Kz2 /2 + γ 2 (θx2 + θz2 )]

(80)

Making use of Eqn (47), one derives the field vector hn (0, 0) on-axis hn (0, 0) = γ hn (0, 0) = 0

n[Jn+1/2 (nd) − J(n−1)/2 (nd)] (Kx uˆ x + Kz uˆ z eiϕ ) 1 + Kx2 /2 + Kz2 /2

if n = 1, 3, 5, . . .

if n = 2, 4, 6, . . . (81)

with d=

[Kx4 + Kz4 + 2Kx2 Kz2 cos(2ϕ)]1/2 4 + 2Kx2 + 2Kz2

(82)

where Jn (x) is the Bessel function of the variable x and order n, uˆ x = (1, 0) and uˆ z = (0, 1) are transverse unit vectors parallel to the Ox and Oz axis. For a pure helical undulator, all hn (0, 0) are equal to zero except for n = 1: h1 (0, 0) =

γK (uˆ x + iuˆ z ) 1 + K2

(83)

There is no emission on-axis of a helical undulator on any harmonic number higher than one. This property is a consequence of the constant longitudinal velocity of the electron. 4.3

Angular spectral flux

As discussed in Section 2.3, the radiation is emitted over a series of cones, each cone corresponding to one harmonic of the spectrum. The intensity of the radiation at the surface of a cone is proportional to |hn (θx , θz )|2 . The frequency ωn of the radiation emitted in the cone associated with harmonic n is deduced from Eqn (5): ωn (θx , θz ) = nω1 (θx , θz ) =

4πncγ 2 λ0 (1 + (Kx2 /2) + (Kz2 /2) + γ 2 θx2 + γ 2 θz2 )

(84)

Substituting Eqn (81) in Eqn (15) , one derives the spectral flux on-axis produced by a filament electron beam as d< I ˆ = α N 2 γ 2 Fn (85) (0, 0, ωn , u) d. e with Fn =

n2 |(Kx uˆ x + Kz uˆ z eiϕ )uˆ ∗ |2  J(n−1)/2 (nd) − J(n+1)/2 (nd) (1 + Kx2 /2 + Kz2 /2)2

Fn = 0

2

if n = 1, 3, 5, . . .

if n = 2, 4, . . . (86)

Undulator radiation

99

with d given by Eqn (82). In practical units, the angular spectral flux at the frequency ωn is given by d< (0, 0, ωn , u) ˆ [Photons/sec/0.1%/mrad 2 ] = 1.744 × 1014 N 2 E 2 [GeV]I [Amp]Fn d. (87) One of the most important differences in the radiation between the planar and the ellipsoidal undulators appears in the polarization. For a mono-energetic filament electron beam with observation on-axis, the linear polarization rates I1 and I2 (associated with the Stokes parameters s1 and s2 ) and the circular polarization rate I3 (associated with s3 ) on any odd harmonic can be expressed as I1 =

ρ2 − 1 ρ2 + 1

I2 =

2ρ cos(ϕ) ρ2 + 1

I3 =

2ρ sin(ϕ) ρ2 + 1

(88)

where ρ = Bˆ x /Bˆ z is the ratio of the horizontal and vertical peak fields. In the particular case of a pure helical undulator, the radiation on-axis only contains harmonic 1 and is fully circularly polarized: F1 =

2K 2 (1 + K 2 )2

Fn = 0

if n > 1

(89)

Compared to a planar undulator, the absence of harmonics higher than 1 on-axis of a helical undulator results in poor tunability. To enlarge the tunability while keeping 100% circular polarization, one must increase K. In many circumstances, one can tolerate unequal deflection parameters Kx = Kz which generate higher harmonics than 1, dramatically increasing the tunability. The price is a reduction of the circular polarization rate. One faces a trade-off between circular polarization rate and tunability. With a thick electron beam some depolarization is introduced. For an observer located on-axis of the electron beam, it is usually low and Eqn (88) remains accurate. Let us consider the particular case of the ESRF. Collecting the radiation produced by an undulator located on a high beta straight section within a slit whose aperture is equal to the fwhm of the central cone, one observes I12 + I22 + I32 ≥ 0.99 for all odd harmonics (and most even harmonics) of order n < 7. This is probably true for all third generation synchrotron sources. 4.4

Angle integrated spectral flux

Integrating the angular spectral flux over all angles at the frequency ωn = ωn (1 − 1/nN) slightly lower than the resonant frequency on-axis ωn = nω1 (0, 0), one derives the angleintegrated flux:

Kx2 + Kz2 Fn I 
K 2 + Kz2


Fn (91) n

100

P. Elleaume

Equation (88) for the polarization rates applies with reasonable precision to the angleintegrated flux
I 2K 2 e (1 + K 2 )

(92)

Let <1,P and <1,H be the angle-integrated flux produced by a planar and a helical undulator with the same deflection parameter K. Their ratio is 
#2 "
1 K2 K2 <1,P (93) = − J0 J1 4 + 2K 2 <1,H 2 4 + 2K 2 which tends toward 0.5 (0.24) for small (large) K. The helical undulator produces more flux than the planar undulator. This can be partly understood from the fact that on-axis, the radiation of the planar undulator is spread over many harmonics while that of the helical undulator is only emitted in the fundamental harmonic. The planar undulator is nevertheless more frequently used due to its larger tunability and simpler engineering. 4.5

Brilliance

One can show from Eqns (79) and (80) that the symmetry relations of hn (θx , θx ) set by Eqn (60) are also fulfilled for any ellipsoidal undulator. Therefore, all conclusions of Section 3.8 concerning the brilliance of the planar undulator apply to the ellipsoidal undulator after replacing the angle-integrated flux
Power density

The power per unit solid angle produced by an ellipsoidal undulator is obtained by substituting the expression for the dimensionless transverve velocity ϑ of Eqn (35) into Eqn (51) of Chapter 2. The resulting integral is easily computed numerically.  In the particular case of the helical undulator it only depends on the azimuthal angle θ = 4π 2 e2 2γ 4 I dP (θ) = NK2 2 d. 4πε0 π e λ0 with



λ0 /2

−λ0 /2



θx2 + θz2 :

1 (γ θ )2 sin2 (2π(s/λ0 )) ds −4 d3 d5


s d = 1 + K + (γ θ) − 2γ θK cos 2π λ0 2

2

(94)

(95)

On-axis, the power density is K2 dP e2 2γ 2 I 4π 2 = N d. 4π ε0 π e λ0 (1 + K 2 )3 or in practical units " # W N dP K2 4 = 530.6E [GeV]I [Amp] dθX dθZ mr 2 λ0 [mm] (1 + K 2 )3

(96)

(97)

Undulator radiation

101

Power density [kW/mrad2]

80

60

K=2

Energy = 6 GeV Current = 200 mA Period = 50 mm N. periods = 100 K=1

40

20

0 0.0

K = 0.5

0.5

1.0

1.5 

2.0

2.5

3.0

Figure 3.18 Power per unit solid angle produced by a helical undulator for several values of the deflection parameter K. The power density presents a circular symmetry around the electron axis and for sufficienly large K it peaks on a circle defined by γ θ ≈ K.

Figure 3.18 presents the power per unit solid angle for three helical undulators with K = 0.5, 1 and 2. For sufficiently large K, it reaches a maximum away from the electron beam axis in a direction approximately equal to K/γ . This behaviour can be understood as follows. The emission of the power is made in the direction of propagation of the electron within 1/γ . For a purely helical field this direction follows a cone of half angle K/γ . Therefore, the radiation is expected to accumulate around this cone, the thickness of the cone being of the order of 1/γ . A similar behaviour is expected for an ellipsoidal undulator. Here the cone of emission of the power has an ellipsoidal cross section. For large K values, the minimum of the power density on-axis of an ellispdoidal undulator can produce an important reduction of the heatload incident on the optical components of the beamline. The heatload reduction can reach one or several orders of magnitude at very low photon energies corresponding to very large K values. The reduced heatload, together with the high brilliance and high circular polarization are the two most important justifications for the use of ellipsoidal undulators instead of planar undulators.

5 5.1

Field errors Introduction

We have seen in Section 3.2 that the electric field in the time domain produced by a planar undulator is made of a sequence of equidistant pulses. The equal distance between the pulses results in a constructive interference of the radiation in the frequency domain. Small non-periodic magnetic field errors slightly destroy this equidistance, which results in an incomplete constructive interference of the radiation in the frequency domain. As a result, the brilliance is reduced, the high harmonic number being the most sensitive. A similar effect takes place on any undulator (planar, ellipsoidal, . . .) but is of most importance for planar undulators whose spectrum contains many harmonics.

102

P. Elleaume

To address this question quantitatively, one can replace Eqn (4) by the following more general expression: H (θx , θz , ω) ≈

∞ 

hn (θx , θz )

n=1

q=(N −1)/2 

exp (iϕq )

(98)

q=−(N −1)/2

where hn (θx , θz ) is given by an equation similar to Eqn (10) in which the integral is made over one of the most representative periods. φq is the phase of the radiation produced by the q-th period of the undulator which is given by 
 qλ0 2π qλ0 2 2 φq ≈ + (ϑx (s) − θx ) + (ϑz (s) − θz ) ds (99) λ 2γ 2 0 The fluctuations of hn (θx , θz ) (when computed on one or the other period of the field) have much less consequences on H than the fluctuations of φq . Consequently, for small errors, the angular spectral flux presents almost the same lobes of emission and cone pattern as an ideal undulator. However, the angular width of the cone and the angular spectral flux in the cone are modified due to the phase errors. The angular spectral flux on an odd harmonic n can be derived from Eqn (98) as 2 d
q=(N −1)/2 

q  =(N−1)/2 

exp (i(φq − φq  ))

(100)

q=−(N −1)/2 q  =−(N−1)/2

In the case of planar and ellipsoidal undulators, the main effect of field errors is to reduce the peak angular spectral flux on the odd harmonics. Such field errors can also generate some non-zero angular spectral flux on-axis on the even harmonics. In the following, we shall review the two most important cases of errors namely the tapered undulator and the random phase errors. 5.2

Tapered undulator

In a tapered undulator the field or the period is continuously changing from one extremity to the other. As a result, the time distance between two successive pulses increases linearly (at first order) from one extremity of the sequence of pulses to the other. Alternatively, one may consider a tapered undulator as an undulator where the resonant frequency ω1 varies from one extremity to the other. Let -ω1 be its total variation. If -ω1 /ω1 1, the discrete sum in Eqn (98) can be replaced by an integral which can be analytically solved to give

  ∞  ω nN -ω1 n(N−1)   H (θx , θz , ω) ≈ N (101) hn (θx , θz )(−1) hc N −n , ω1 2 ω1 n=1

where hc (ν, ρ) is the line-shape function defined by [8]
 π ν2  √ √ √ √ (C(ν/ ρ + ρ) − C(ν/ ρ − ρ)) hc (ν, ρ) = exp −i 2 ρ √ √ √ √  × +i(S(ν/ ρ + ρ) − S(ν/ ρ − ρ)) (102)

Undulator radiation 1.0


= 1/2n N ∆ 1/ 1
=0
=1
=2
= 10

0.8

hc2 (,
)

103

0.6

0.4

0.2

0.0 –6

–4

–2

0  = N ( / 1− n)

2

4

6

Figure 3.19 Plot of the line-shape function h2c (ν, ρ) as a function of ν for ρ = 0, 1, 2 and 10.

C(x) and S(x) are the Fresnel Integrals:  x  x π  π  S(x) = t 2 dt t 2 dt cos sin C(x) = 2 2 0 0

(103)

ρ = (nN/2)(-ω1 /ω1 ) is the dimensionless taper parameter. As expected, in the absence of taper, ρ = 0, the line-shape function becomes hc (ν, 0) = sin(π ν)/π ν and Eqn (101) reduces to Eqn (9). Figure 3.19 presents a plot of h2c (ν, ρ) as a function of ν for taper parameters ρ = 0, 1, 2 and 10. Clearly, for a given device characterized by the relative frequency variation -ω1 /ω1 , the modification induced to the spectrum increases with the number of periods N and the harmonic number n. High harmonics and/or long undulators are extremely sensitive to any small value of -ω1 /ω1 . Since the angular spectral flux is proportional to h2c (ν, ρ), the spectrum produced by an electron beam in a tapered undulator consists of a series of broad peaks. The tapered undulator is of interest to those experiments which require a flat spectrum over a sufficiently reduced photon energy range. It can be considered as an intermediate device somewhere between a perfect undulator and a wiggler with a flat spectrum. Note that tapering of the field results in an increase of both the divergence and source size of the radiation and therefore results in reduced brilliance. The variation of the resonant frequency -ω1 can either be obtained by a change of the period or a change of the peak field (or both). For a period taper, ρ = (nN/2)(-λ0 /λ0 ) where -λ0 is the total variation of the spatial period from one end to the other. For a field taper, ρ = (nN/4)-(Kx2 + Kz2 )/(1 + Kx2 /2 + Kz2 /2). In the particular case (and most useful) of a pure permanent magnet planar undulator presenting a magnetic gap difference from one end to the other equal to -g, the taper parameter can be expressed as ρ = π(nN/2)(K 2 /1 + (K 2 /2))(-g/λ0 ). For a permanent magnet undulator, the gap taper can be performed mechanically in a reversible manner using two motors (one for the gap and one for the taper). The incremental cost of tapering is low and one can view the tapering

104

P. Elleaume

capability as an added value to be used only in a few specific experiments. At the ESRF, the large majority of undulators are equipped with remotely controlled gap and taper. 5.3

Random phase errors

An important category of field errrors comes from small random fluctuation of the periods and/or magnetic fields from one period to the next. These errors are always present to some extent in the permanent magnet undulators due to the fluctuation of the magnetization from one magnet block to the next and due to the positioning errors of each block. These field errors are almost unpredictable and are deduced from the magnetic field measurement. Nevertheless, for a very large number of periods, a statistical approach of the effect of these errors can be performed. The phase errors modify the spectrum in two ways. First, it attenuates and spreads the shape of the peak. Then it shifts the frequency of the top of the peak to a higher or lower frequency. The frequency shift can be interpreted as an average reorganization of the phase around a new sequence regularly spaced with a distance slightly different from 2π . Let us define the rms phase error σ as the fluctuations of the phases away from this new regular sequence. Assuming the fluctuations of φq − φp are independent of the difference q − p, and assuming a Gaussian distribution of φq − φp , one has exp(i(φq − φq  )) = exp(−σ 2 )

(104)

where x is the average of x over many periods. Substituting Eqn (104) in Eqn (100), one deduces that the angular spectral flux is reduced by exp(−σ 2 ). A good correlation between the peak angular spectral flux and the rms phase error σ has been observed at the ESRF and ELETTRA [9]. The reduction of the angular spectral flux is independent of the number of periods and occurs only through the phase errors σ . Table 5.1 presents the reduction of the angular spectral flux as a function of the harmonic number and of the rms phase errors of the fundamental. For a given magnetic field configuration σ is proportional to the harmonic number n. The phase error of 6 (1) degrees is a typical value reached before (after) spectrum shimming (see Chapter 5). Note that for a given phase error the attenuation of the angular spectral flux is independent of the number of periods. If one does not take into account the frequency shift and compute the angular spectral flux at a fixed frequency corresponding to an ideal undulator, one observes statistically a larger reduction of the angular spectral flux which grows with the number of periods. However, this method of deriving the reduction of the angular spectral flux is of little interest since the frequency shift induced by phase errors can always be compensated by a small change of the field.

Table 3.1 Reduction of the angular spectral flux as a function of the harmonic number and rms phase error σ Harmonic #

σ = 6◦

σ = 1◦

1 5 9 13

0.99 0.76 0.41 0.16

1 0.99 0.98 0.95

Undulator radiation

105

Let us now provide an estimation of the rms error σ as a function of the magnetic field errors. Let δx (s) 1 and δz (s) 1 be the angle errors associated with the field errors δBz (s) and δBx (s): e δx (s) = γ mc



s

−∞

δBz (s)ds

−e δz (s) = γ mc



s

−∞

δBx (s) ds

(105)

For a planar undulator with a deflection parameter K with observation in a direction (θx , θz ), the phase fluctuation σ is deduced from Eqns (35) and (99):  nW − 2θx δx (s) + δx2 (s) + Kδx (s) cos(2πs/λ0 ) − 2θz δz (s) + δz2 (s) σ ≈ (1 + (K 2 /2) + γ 2 θx2 + γ 2 θz2 )

(106)

where for any function f of the longitudinal coordinate s, f (s) and W (f ) are defined as + f (s) =

,

(p+1)λ0

f (s)ds

W (f ) = f 2 (s) − f (s)

2

(107)

pλ0

The phase error is minimum on-axis of the undulator (θx , θz ) = (0, 0). On-axis, the dominant term is Kδx (s) cos(2πs/λ0 ). The phase errors are essentially driven by the component of field errors in the same plane as the nominal field. The reduction of the angular spectral flux and the broadening of the harmonic line of the spectrum induced by the phase errors is, in nature, very similar to the reduced efficiency of the X-ray diffraction in the direction corresponding to large Miller indices. The reduced diffraction is induced by thermal motion of the atoms with respect to their position in a perfect lattice. It is taken into account when computing the intensity of the diffraction efficiency by means of the Debye–Waller attenuation factor which is analogous to the exp(−σ 2 ) coefficient derived in this section. Similarly to X-ray diffraction, one also observes that for large phase errors, the angular spectral flux is spread away from the centre frequency ωn and tends to produce a background whose relative intensity increases with the photon energy.

List of symbols General c ε0 α

Speed of light Vacuum permeability Fine structure constant

Electron m e h I R = (X, Z, S) τ

Mass of the electron Charge of the electron Planck constant Electron beam current Position of the electron Electron time

106

P. Elleaume

γ ϑ = (ϑx , ϑz , ϑs ) σx , σz σx , σz σγ /γ

Energy of an electron divided by mc2 Electron velocity normalized to the speed of light Horizontal and vertical rms electron beam sizes Horizontal and vertical rms electron beam divergence Rms electron relative energy spread

Undulator K Kx Kz λ0 N L

Deflection parameter Deflection parameter of the vertical field (horizontal deflection) Deflection parameter of the horizontal field (vertical deflection) Undulator period Number of periods Undulator length

Radiation t r = (x, z, s) D λ ω λn ωn n uˆ <
Observer time Position of the observer Distance between the observer and the electron Wavelength of the radiation Frequency of the radiation Wavelength of the radiation on the n-th harmonic Frequency of the radiation on the n-th harmonic Harmonic number Complex unit vector describing the polarization of the radiation Angle integrated spectral flux Angle integrated spectral flux on the n-th harmonic of the spectrum

d< d.

Spectral flux per unit solid angle

d
Spectral flux per unit solid angle on the n-th harmonic of the spectrum

d< d᏿

Spectral flux per unit surface

dP d.

Power per unit solid angle

dPx d.

Power per unit solid angle horizontally polarized

dPz d.

Power per unit solid angle vertically polarized

P Px Pz Ꮾ

Power of the radiation Horizontally polarized power Vertically polarized power Brilliance or brightness of the radiation on the n-th harmonic

Undulator radiation Ꮾn

Qx , Qz Qx , Qz

107

Brilliance or brightness of the radiation on the n-th harmonic Horizontal and vertical rms photon beam size Horizontal and vertical rms photon beam divergence

References [1] Alferov, D. F., Yu A. Bashmakov and E. G. Bessonov, Sov. Phys. Tech. Phys. 18, 1336 (1974). [2] Kincaid, B. M., J. Appl. Phys. 48, 2684 (1977). [3] Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, Inc., New York). [4] Kitamura, H., Jap. J. of Appl. Phys. 19(4), L185–L188 (1980). [5] Chubar, O. and P. Elleaume, ‘Accurate And Efficient Computation of Synchrotron Radiation in the Near Field Region’, Presented at the EPAC98 Conference, Stockholm, 22–26 June 1998. The SRW code is available from ‘http://www.esrf.fr/machine/groups/insertion devices/Codes/software.html’ [6] Kim, K. J., Nucl. Instr. Meth. A246, 71 (1986). [7] Kim, K. J., Nucl. Instr. Meth. A246, 67 (1986). [8] Shih, C. C. and M. Z. Caponi, Phys. Rev. A 26(1), 438–50, (1982). [9] Diviacco, B., Proc. 1993 US Particle Accelerator Conference, p. 1590.

4

Bending magnet and wiggler radiation Richard P. Walker

1

Introduction

In this chapter, we first examine the main properties of the synchrotron radiation emitted in standard dipole magnets, including the spectral and angular intensity of photon flux, the angular distribution of total power and the polarization properties. Section 2 then considers the radiation properties of wigglers. Wigglers are magnetic devices consisting of a sequence of dipoles with alternating polarity and are very similar to the undulators discussed in Chapter 3 except that they operate in a regime where interference effects for the most part can be neglected, and their purpose is to increase the intensity of the high frequency part of the synchrotron spectrum. The study of using such structures goes back many years, and went on in parallel to that of the undulator. The main criterion for determining whether the emission takes place in a single harmonic, or in many harmonics, was already outlined in Motz’s article [1]. Purcell later discussed the number of harmonics in the spectrum and the mechanisms by which the interference structure becomes ‘washed out’ to produce a smooth bending-magnet like spectrum [2]. The first device to be built for this application was an electromagnet consisting of a single high field strength pole with half-strength poles of opposite polarity on either side in order to produce a single bump of the electron trajectory. It was termed a ‘wavelength shifter’ because of its harder spectrum compared to the bending magnet radiation [3]; the term is still sometimes applied to devices with a single high field pole. The first ‘multipole wiggler’ devices with many poles were also electromagnetic. For example, a 1.8 T, 5-pole device was installed in SPEAR in 1979 [4], followed by a similar device in ADONE [5]. A review article covering the early phase of development is given in [6]. Subsequently, the modern generation of permanent magnet devices with many poles was initiated with the 55-pole device installed in SPEAR in 1983 [7]. Higher fields were also pursued using superconducting technology, the first examples being a 3.5 T multipole device installed on VEPP3 in 1979 [8], and a 5 T wavelength shifter on the SRS in 1982 [9]. Nowadays the use of wigglers is not only to provide higher flux compared to bending magnet sources but also to produce radiation with different polarization characteristics, particularly circularly polarized radiation; this topic is treated in Section 4. Finally, in Section 5, the geometric properties of both bending magnet and wiggler sources are discussed.

Bending magnet and wiggler radiation

2 2.1

109

Bending magnet radiation Qualitative description

The most important qualities of the radiation emitted by relativistic electrons in bending magnets can be understood quite simply in terms of the relative motion of the electrons and the emitted photons. If an electron travelling with velocity (ϑ) close to that of the speed of light (c) emits radiation at an angle θ with respect to its instantaneous direction of motion, then the distance between the radiation wavefronts from two points separated by distance -s is given by: -s  = (-s/β) − -s cos θ , where β = ϑ/c. Expanding θ for small angles and noting that 1/β ≈ 1 + 1/2γ 2 , where γ is the usual relativistic factor (γ = 1/(1 − β 2 )1/2 ), it follows that the ratio between the time taken for the electron to travel the distance -s and the time duration of the resulting radiation impulse is given by -t (radiation) 1 ≈ -t (electron) 2




1 + θ2 γ2

 (1)

In most practical cases γ is very large; in practical units γ = 1957E [GeV]. Thus, when the direction of motion is pointing towards the observer within a small angle ∼1/γ there is a very strong time compression effect of the order 1/2γ 2 . An alternative method to arrive at the same result was given in Chapter 2 (Eqn (5)). The apparent acceleration of the electron seen by the observer, which according to Eqn (19) of Chapter 2 is proportional to the electric field of the received radiation, is therefore enhanced by a large factor. This fact has an implication both for the frequency spectrum of the received radiation as well as the angular distribution. Due to this process the radiation received in a given direction is effectively emitted only over a small arc of the trajectory ∼2ρ/γ , where ρ is the radius of curvature of the trajectory, and hence over a time ∼2ρ/cγ . However, due to the relative motion of the electrons and emitted photons the observer receives a pulse shortened by a factor ∼1/2γ 2 and therefore of duration -t ∼ ρ/cγ 3 . Using general arguments about Fourier transforms it can be seen that a pulse of this duration will contain frequency components up to about ωtyp where ωtyp

1 ∼ ∼ -t


 c γ3 ρ

It can be seen therefore that the frequency is increased by a large factor γ 3 compared to the angular frequency of rotation of the electron ω0 = c/ρ. For example, a 1 GeV electron in a 1 T field has a radius of curvature of 3.3 m and hence an angular frequency of 90 MHz, corresponding to a wavelength of 20 m. The radiation frequency however is much larger, 6.7 × 1017 Hz with a corresponding wavelength of 28 Å. The great difference between the typical radiation frequency and the frequency of orbital motion was first appreciated by Schwinger [10] who explained why synchrotron radiation was able to be seen in the visible region of the spectrum rather than in the microwave region where it had first been expected. Equation (1) also shows that the time compression depends strongly on the angle of observation in the vertical plane, implying therefore that the majority of the radiation is confined within a small vertical angle of the order ∼ 1/γ . In the horizontal plane, on the other hand, the deflection angle in the bending magnet is usually much larger than this, and so the emission is effectively independent of horizontal angle.

110

R. P. Walker

2.2

Detailed analysis

2.2.1

Spectral/angular distribution

The spectral and angular characteristics of bending magnet radiation have been discussed by several authors, for example [10–15]. We start our analysis from the general expression for the photon flux in the direction of unit vector nˆ per unit solid angle and unit relative frequency bandwidth for an electron in arbitrary motion, as derived in Chapter 2 (Eqn 40): d< I = α |H (n, ˆ ω)|2 d. dω/ω e where α is the fine structure constant (e2 /4πε0
c = 1/137), I the beam current and e the electronic charge. The main term, which is related directly to the radiation amplitude, is given by

   ∞  ω n ˆ · R  exp iω τ − H (n, ˆ ω) = dτ nˆ × (nˆ × ϑ) c 2π −∞ where R is the position of the electron with respect to the origin and ϑ its velocity. To analyse the case of bending magnet radiation we define the electron motion to be in the x–s plane and the observation direction to be orthogonal to the electron acceleration at some time τ = 0 and making an angle θ with respect to the plane of motion, as shown in Figure 4.1. Since we are only concerned with small angles and times around τ = 0 the following approximations can be used: nˆ = (0, θ, 1 − θ 2 /2) ϑ = β(−ω0 τ, 0, 1 − ω02 τ 2 /2) Hence to order 1/γ ,  ≈ (ω0 τ, θ, 0) nˆ × (nˆ × ϑ) z

x n







s

R

Electron trajectory

Figure 4.1 Geometry for the analysis of synchrotron radiation.

Bending magnet and wiggler radiation

111

Taking the origin to be the electron position at τ = 0 we can expand the expression for the electron position at other times as follows: R ≈ (−c2 τ 2 /2ρ, 0, βcτ − c3 τ 3 /6ρ 2 ) The argument of the exponential term can then be approximated as follows:

 # "
c2 3 ω 1 nˆ R 2 +θ τ + 2τ ≈ ω τ− c 2 γ2 3ρ resulting in the following expressions for the radiation amplitudes:  #. - "
 c2 τ 3 1 ω c ∞ ω 2 dτ Hx = τ exp i +θ τ + 2π ρ −∞ 2 γ2 3ρ 2  #. - "
 ∞ c2 τ 3 1 ω ω 2 Hz = dτ θ exp i +θ τ + 2π −∞ 2 γ2 3ρ 2 Changing the variable to −1/2
cτ 1 2 x= +θ ρ γ2 and introducing the parameter 3/2
ωρ 1 2 +θ ξ= 3c γ 2 allows the integrals to be written as follows:  ∞

. ω ρ 1 3ξ x3 2 +θ x exp i x+ dx Hx = 2π c γ 2 2 3 −∞ 1/2  ∞

. x3 ω ρ 1 3ξ 2 Hz = θ x + dx + θ exp i 2π c γ 2 2 3 −∞ The integrals above can be identified as the Airy function and its derivative, however we use here the more common formulation in terms of modified Bessel functions, which are defined as follows: # "
 ∞ 1 1 3 3 dx = √ K2/3 (ξ ) x sin ξ x + x 2 3 3 0 # "
 ∞ 1 1 3 cos ξ x + x 3 dx = √ K1/3 (ξ ) 2 3 3 0 Expanding the exponential term in the radiation amplitude and using the above relations then leads to the following: √ ω 3γ Hx = i (1 + γ 2 θ 2 )K2/3 (ξ ) ωc 2π (2) √ ω 3γ 2 2 1/2 Hz = γ θ(1 + γ θ ) K1/3 (ξ ) ωc 2π

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R. P. Walker

where we have introduced the so-called ‘critical frequency’ (the significance of which will become clear later on) defined by ωc =

3cγ 3 2ρ

We have, therefore, finally, " # d< 3α I 2 2 X2 2 2 2 2 = γ y (1 + X ) K2/3 (ξ ) + K (ξ ) d. dω/ω 4π 2 e 1 + X 2 1/3

(3)

with y = ω/ωc , X = γ θ , ξ = y(1 + X 2 )3/2 /2, and where the two terms represent the horizontally and vertically polarized intensities, respectively. In practical units [photons/s/mrad2 /0.1%bandwidth] we have " # X2 d< 2 2 2 2 2 13 2 K (ξ ) = 1.327 × 10 E [GeV]I [A]y (1 + X ) K2/3 (ξ ) + 1 + X 2 1/3 d. dω/ω (4) The corresponding critical wavelength (λc ) and photon energy (εc ) are given in practical units as follows: λc [Å] =

18.6 , B [T ]E 2 [GeV]

εc [keV] = 0.665B [T]E 2 [GeV]

(5)

Figure 4.2 shows both polarization components as a function of vertical angle for various photon energies; in each case the intensity is normalized to the peak value on-axis. On-axis the radiation is linearly polarized in the horizontal plane. As the angle θ increases the vertical polarization component increases, reaching a maximum at a certain angle. The angular divergence of both components changes significantly with photon energy, becoming much wider with reducing energy. At photon energies close to the critical energy the horizontal component is reasonably well approximated by a Gaussian distribution with the following standard deviation [13], as shown by the dotted lines in Figure 4.2: σR

0.565 ≈ γ




ω ωc

−0.425 (6)

On-axis, X = 0, only the horizontal polarization exists, and Eqn (4) is sometimes written in the following way:  d<  = 1.327 × 1013 E 2 [GeV]I [A]H2 (y) d. dω/ω θ =0 2 (y/2) [13]. Figure 4.3 shows the function H (y) which peaks at with H2 (y) = y 2 K2/3 2 ω/ωc = 0.83 with a value of 1.47 and decreases rapidly for frequencies above the critical frequency.

Bending magnet and wiggler radiation

113

1.0

Relative intensity

0.8

Ⲑc = 0.01 0.1

0.6

1.0

0.4 3 0.2

0.0 0

1

2

3



4

5

6

7

8

Figure 4.2 Angular distribution of horizontally (solid lines) and vertically (dashed lines) polarized radiation in the vertical plane for various photon energies. Dotted line from Eqn (6).

101

G1, H2

10

H2 G1

10–1

10–2

10–3 10–3

10–2

10–1 Ⲑc

10

101

Figure 4.3 The functions H2 (y) and G1 (y).

2.2.2

Polarization

The 90◦ phase shift between the horizontal and vertical components, evidenced by the fact that Hx is imaginary while Hz is real in Eqn (2), gives rise to a circularly polarized component out of the orbit plane. Using Eqn (40) of Chapter 2 the intensities polarized in the principal

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R. P. Walker

directions (see Eqn 124 of Chapter 2) can be written as follows: I I Ix = α |Hx |2 , Iz = α |Hz |2 e e 2 I |Hx + Hz | I |Hx − Hz |2 Ix + Iz Ix + I z I45◦ = α = , I135◦ = α = (7) e 2 2 e 2 2 √ √ Ix + I z ∓ 2 I x I z I |Hx + iHz |2 I |Hx − iHz |2 Ix + I z ± 2 I x I z Ir = α = = , Il = α 2 2 e 2 2 e where I refers to the spectral and angular intensity (d
S1 = Ix − Iz  S3 = ±2 Ix Iz

S2 = 0

It should be noted that the above result refers to bending magnet radiation and is not true in general for other types of sources, such as insertion devices. Figure 4.4 shows the variation of Stokes parameters and polarization rates (see Chapter 2, Eqn (127)) as a function of vertical angle at the critical frequency. It can be seen that as the angle increases the radiation becomes progressively more circularly polarized, but because the total intensity also diminishes, there exists a maximum value of the S3 component √ at some angle, which depends on the radiation frequency. Note, in the above, that the 2 Ix Iz term retains the sign of the Hz term, which depends on the sign of θ . The direction of circular polarization, therefore, reverses above and below the orbit plane. It should also be noted that the expressions are functions of the vertical angle. Integrating the intensities over a finite angular acceptance leads in general to an unpolarized component (Section 3.4 of Chapter 2). Experiments requiring circularly polarized radiation can therefore be carried out on bending magnet sources by moving the acceptance off-axis in the vertical plane. An alternative 1.0

S0, S1, S3, P1, P3 (rel. units)

S0 0.8

P3

S1

0.6 S3 0.4

P1

0.2

0.0 0.0

0.5

1.0 

1.5

2.0

Figure 4.4 Variation of Stokes parameters and polarization rates with vertical angle, at the critical photon energy.

Bending magnet and wiggler radiation

115

possibility is to introduce a variable deflection angle on the electron beam at the source location. In this way circularly polarized radiation can be generated on-axis, with a helicity that depends on the sign of the electron beam deflection angle. Such a scheme was implemented recently at SRRC (Taiwan) to permit a dynamic switching of the polarization state [16]. 2.2.3

Flux integrated over vertical angles

In some experimental situations the vertical acceptance angle may be sufficient to collect all of the radiation in the vertical plane, and so it is convenient to have an expression for the total flux integrated over the vertical angle. Eqn (3) can be integrated, yielding the following expression for the photon flux per relative bandwidth per unit horizontal angle: √  ∞ d< 3 I = K5/3 (y  ) dy  αγ y e y dθx dω/ω 2π

(8)

In practical units [photons/s/0.1%bandwidth/mrad horizontal angle], we have d< = 2.457 × 1013 E [GeV]I [A]G1 (y) dθx dω/ω

(9)

∞ where G1 (y) = y y K5/3 (y  ) dy  [13]. Figure 4.3 includes the function G1 (y) which has its maximum value (0.92) below the critical frequency at ω/ωc = 0.29, while Figure 4.5 shows the ‘Universal curve’ for the integrated flux, so called because expressed in this way it is valid for rings of any energy, current and bending magnet field strength. Also shown are the partial values obtained by numerically integrating the angular distribution over various angular acceptances. The increasingly large acceptance needed to collect the majority of the flux at low photon energies is clearly seen.

Flux (ph/s/mrad/0.1%bw/GeV/A)

1014

1013 5/

2/ 1/

1012

1011 10–3

10–2

10–1 Ⲑc

10

101

Figure 4.5 The ‘Universal curve’ of synchrotron radiation, and the fraction of flux obtained within different vertical acceptance angles.

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R. P. Walker

2.2.4

Total photon flux

A quantity of interest to the designers of the vacuum system that must contain the electron beam is the total number of photons emitted, since this determines the gas load due to photodesorption that must be pumped in order to obtain a sufficiently good vacuum pressure, and hence beam lifetime. Noting that dω/ω = dy/y and integrating Eqn (8) we obtain √   d< 3 I ∞ ∞ = K5/3 (y  ) dy  αγ dθx 2π e 0 y The integral in question has a value 5π/3 and hence 5 d< I = √ αγ = 1.286 × 1017 E [GeV]I [A] ph/s/mrad horizontal dθx e 2 3 That is, it depends only on the ring energy. Alternatively, in terms of the flux per metre of bending magnet circumference, the result depends only on the magnetic field strength: 5 αγ I d< = 0.386 × 1020 B [T]I [A] ph/s/m = √ ρ e ds 2 3 It is interesting to note that although the number of photons emitted is usually large, this is due to the large number of circulating electrons. Dividing the above expressions by the number of electrons passing a given point per second (I /e) gives the number of emitted photons per electron: 5 d< = √ αγ = 20.6E [GeV] ph/rad/electron dθx 2 3 5 αγ d< = √ = 6.18B [T] ph/m/electron ds 2 3 ρ Thus in a typical bending magnet only about 10 photons are emitted per electron per pass. 2.2.5

Radiation power

The spectral/angular distribution of the radiation flux, Eqn (3), can be converted into power units by means of the following relations: photons/s/relative bandwidth =

W/eV W/unit bandwidth = h/2π e

Of more general use is the expression for the angular distribution of radiation power which can be obtained by integrating Eqn (3) over all frequency, or alternatively from the general expression Eqn 51 of Chapter 2. The result is the following: " # 5γ 2 θ 2 7e2 γ 5 I 1 dP 1+ = d. 64π εo ρ e (1 + γ 2 θ 2 )5/2 7(1 + γ 2 θ 2 ) where as usual the first (second) terms refer to the horizontal (vertical) polarizations, respectively. Figure 4.6 shows the angular distribution of power density in the vertical plane. In practical units " # 5γ 2 θ 2 1 dP 2 4 1+ [W/mrad ] = 5.420E [GeV]B [T]I [A] d. 7(1 + γ 2 θ 2 ) (1 + γ 2 θ 2 )5/2

Bending magnet and wiggler radiation

117

1.0

Relative intensity

0.8 Ptot 0.6 P 0.4

0.2 P 0.0 0.0

0.2

0.4

0.6

0.8 

1.0

1.2

1.4

1.6

Figure 4.6 Angular distribution of total power (solid line) and the horizontally and vertically polarized components (dashed lines) in the vertical plane. Dotted line from Eqn (10).

The distribution is well approximated by a Gaussian with standard deviation [13] σP ≈

0.608 γ

(10)

as shown by the dotted line in Figure 4.6. On-axis, the peak power density is therefore given by dP [W/mrad2 ] = 5.420E 4 [GeV]B [T]I [A] d. By comparing this with Eqn 70 of Chapter 3, it can be seen that this is the same result as that for the power density for one half period of a planar undulator or wiggler when K > 1. Integrating over the vertical angle gives the linear power density: dP e2 γ 4 I ; = 6π εo ρ e dθx

dP [W/mrad horizontal] = 4.221E 3 [GeV]B [T]I [A] dθx

By integrating the horizontal and vertical components separately it can be shown that 7/8th of the radiation power is polarized horizontally and 1/8th vertically. The total power emitted in synchrotron radiation due to all of the bending magnets (assuming constant bending field) is, therefore, P =

e2 γ 4 I = 2.652 × 104 E 3 [GeV]B [T]I [A] 3εo ρ e

The linear power density can also be obtained by integrating Eqn (8), using the fact that √ √ ∞ 1 0 G1 (y)dy = 8π/9 3. It is interesting to note also that since 0 G1 (y) dy = 4π/9 3

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R. P. Walker

Table 4.1 Main parameters of synchrotron radiation facilities of three typical energies Ring

E [GeV]

B [T]

εc [keV]

I [A]

P [kW]

dP /dθx [W/mrad]

dP /d. [W/mrad2 ]

SuperACO ELETTRA ESRF

0.8 2.0 6.0

1.57 1.2 0.85

0.67 3.2 20.3

0.4 0.3 0.2

8.5 76 974

1.4 12.2 155

1.4 31.2 1194

it follows that half of the linear power density, and hence half of the total power, lies below the critical frequency (or wavelength), and half above it. Table 4.1 compares the various parameters of interest for synchrotron radiation facilities of three typical energies. The rapid increase in total power, linear power density and especially peak power density with machine energy is evident. 2.3

Practical calculation of bending magnet radiation

As shown above, the calculation of synchrotron radiation properties involves the evaluation of the modified Bessel functions of fractional order, K1/3 and K2/3 , as well as the integral of the function K5/3 . Over the years various series and asymptotic forms have been presented in the literature for these functions. Here, we mention only the convenient and rapidly converging series approximations developed by Kostroun, valid for any fractional order ν and argument x [17]:  / ∞ e−x  −x cosh(rh) Kν (x) = h e cosh(νrh) + 2 r=1 /   ∞ ∞ e−x  −x cosh(rh) cosh(νrh) + Kν (η) dη = h e 2 cosh(rh) x r=1

In the above h is a small interval, a suitable value for which is 0.5. The series can be evaluated until the rth term is less than a certain value (e.g. 10−6 ).

3 3.1

Wiggler radiation The wiggler as a high K undulator

The majority of wiggler devices contain many emitting source points, with an on-axis magnetic field distribution and hence electron trajectory that to a good approximation is sinusoidal, similar to that considered in the case of undulators (Chapter 3): Bz = B0 cos ks ϑx = (K/γ ) sin ks

(11)

where K = eB0 λ0 /2πmc. The main difference is the larger field amplitude (B0 ) and the longer period (λ0 ) in the wiggler case. For example, a typical undulator might have a period of 50 mm and a field of 0.5 T, leading to a K value about 2.5. On the other hand a wiggler would usually have a field in the range 1.5–2 T and a period of 125–200 mm, leading to a K value

Bending magnet and wiggler radiation

119

in the range 20–40. The large difference in K values leads to a significant difference in the spectrum of the radiation produced by the device. It has already been shown that the number of harmonics in the spectrum increases significantly as the K value increases (see, e.g. Figures 3.5 and 3.13). The number can be estimated by comparing the equation for the wavelength of the harmonics, Eqn (46) of Chapter 3, with that for the critical wavelength of the radiation corresponding to the peak field amplitude B0 (Eqn (5)), which gives rise to the following expression for the harmonic number corresponding to a given wavelength: n=

(3/4)K(1 + K 2 /2) (λ/λc )

We can get an idea of the number of harmonics in the spectrum by calculating the harmonic number corresponding to the critical wavelength, λ = λc . Table 4.2 shows that the number increases very rapidly with the value of K. Since the harmonics are equally spaced in frequency (ω1 , 2ω1 , 3ω1 , 4ω1 , . . .) they become relatively closer together at high frequency (-ω/ω = 1/n) making it easier for them to become smoothed out, as we will see below. A device with a large K value can therefore exhibit both strong interference effects typical of an undulator in the region of the fundamental and first few harmonics and a smooth spectrum similar to that of a bending magnet at high frequency. The same device could therefore be used both in an ‘undulator mode’ at low frequency, requiring a tuning of the magnetic field to adjust the spectral peaks to the required frequency, as well as in a ‘wiggler mode’, where, because of the smooth spectrum, different frequencies can be selected without adjusting the magnetic field strength. Roughly speaking, to produce a smooth spectrum generally requires that there are a sufficient number of harmonics (K ≥ 5) and that the frequency is not too close to the fundamental (n ≥ 10). The effects that can contribute to a smoothing of the spectrum are as follows: • •

•

Wavelength acceptance: If the monochromator bandpass exceeds the relative spacing of the harmonics (-ω/ω = 1/n) then the harmonics are smoothed out. Angular acceptance: Because of the dependence of the undulator wavelength on angle, if the angular acceptance is sufficiently large the (n + 1)th harmonic off-axis can overlap with the nth harmonic on-axis. It is easy to show that this occurs when γ 2 θ 2 ≥ (1 + K 2 /2)/n. Electron beam divergence: The effect of a non-zero electron angle is to shift the radiation pattern by the same angle, implying therefore that the radiation is viewed off-axis.

Table 4.2 Harmonic numbers giving a wavelength corresponding to the critical wavelength (λ = λc ) as a function of K K

n

1 2 3 5 10 20

1 5 12 51 383 3015

120

• •

R. P. Walker The effect of the beam divergence as a whole is, therefore, similar to that of the angular acceptance above. Electron beam energy spread: Since the frequency of the undulator harmonics depends on the square of the electron beam energy, the energy spread in the beam is sufficient to smooth out the harmonics if: -E/E ≥ 1/2n. Magnetic field errors: Random errors in the magnetic field distribution lead to an imperfect constructive interference, particularly for the higher harmonics, which can also contribute to the smoothing out of the spectrum.

In most cases the most dominant effect is the angular acceptance, followed by the effects of the electron beam and magnetic field errors; the bandpass of the monochromator is usually too small to have a significant effect. Similar considerations can also be applied to the off-axis spectrum, as well as the angular distribution, the only difference being that the harmonics become more closely spaced in angle, off-axis, and so become more easily smoothed out by the effects of angular acceptance and electron beam divergence.

3.2

Spectral and angular distributions

Figure 4.7 shows an example of a spectrum for a device with K = 5 calculated using the general expression for undulator emission presented in the previous chapter, and implemented in the program URGENT [18]. It can be seen that at sufficiently high photon energies the interference structure becomes smoothed out and that the intensity is equal to 2N times that of a bending magnet – shown by the solid line in the figure, which was calculated using Eqn (4). The smoothing effect in this case is due to integration over an angular acceptance of ±0.2 mrad. Another indication of the correspondence between the two methods of calculation

Flux (ph/s/0.1%bw)

1015

1014

1013

1012 101

102 103 Photon energy (eV)

104

Figure 4.7 Spectral flux of a high K device calculated using the exact undulator radiation method and with the wiggler model (smooth curve). E = 2 GeV, K = 5, N = 10, λ0 = 0.1 m, acceptance angles ±0.2 mrad.

Bending magnet and wiggler radiation

121

2.4 2.0 Relative brilliance

Ⲑco = 0.1 1.6 1.2 1.0

0.8 5.0 0.4 0.0 0.0

0.2

0.4

0.6  x /K

0.8

1.0

1.2

Figure 4.8 Angular distributions in the horizontal plane calculated with the undulator radiation method and with the wiggler model (smooth curves). E = 2 GeV, N = 10, λ0 = 0.1 m, acceptance angle ±0.13mrad; K = 11 (upper), 5 (middle), 2.8 (lower).

is shown by the fact that even undulator radiation can be successfully modelled as a sequence of bending magnets, provided the correct phase relationship is included between them [19]. For large angles off-axis in the horizontal plane (θx ) it follows from Eqn (11) that the field at the tangent point (θx = ϑx ) differs from the peak value, which therefore alters the critical energy of the emitted radiation from that point:  
B εc γ θx 2 = = 1− (12) B0 εco K Thus, in general, a wiggler spectrum can be calculated as that for a bending magnet, but with a critical energy that varies with the angle of observation in the horizontal plane. Figure 4.8 shows an attempt to verify this model for the angular distribution of wiggler radiation by comparison with an exact undulator radiation calculation. Apart from some discrepancies near zero angle, due to the strong influence of the on-axis harmonic, and also at the largest angles, the variation in critical energy with angle given by Eqn (12) can be seen to be validated. In particular, the reduction of the intensity of the highest photon energies off-axis can be clearly seen. 3.3

Polarization

Considering the emission from a wiggler to be the sum of the intensities from a series of dipoles of alternating polarity, it follows from the analysis of Section 2.2.1 that the difference between the two polarities (ignoring the difference in phase) is related to the fact that the x co-ordinate of the electron trajectory reverses, and hence also the x component of ϑ,  nˆ × (nˆ × ϑ),and the electric field vector H . From Eqn (7) it follows that the right- and left-handed intensities (Ir , Il ) become reversed, or in other words that the S3 component

122

R. P. Walker 1.0

S0, S1, S4, P1, P4 (rel. units)

S0 P4

0.8 S1 0.6

0.4 P1 0.2

0.0 0.0

S4

0.5

1.0  z

1.5

2.0

Figure 4.9 Variation of Stokes parameters and polarization rates with vertical angle for a standard wiggler at the critical photon energy.

changes sign, for a given off-axis angle. Now, summing the intensities, and hence also the Stokes parameters, for the two dipole polarities at a fixed angle leads to the cancellation of the S3 component, and hence the appearance of an unpolarized component: S0 = Ix + Iz

S1 = Ix − Iz

S2 = S 3 = 0

S4 = 2Iz

Figure 4.9 shows the variation of polarization components as a function of vertical angle in this case. The analysis above ignored interference effects; however, a more detailed analysis (Section 3.5 of Chapter 3) confirms the absence of a circularly polarized component in a planar wiggler, apart from a possible small contribution from the field termination: if the wiggler field is symmetric about its centre and so has an uneven number of full poles, this can result in an incomplete cancellation between the end-poles and the extra full pole. 3.4

Effect of higher harmonic field components

In many practical cases the field is not perfectly sinusoidal but often contains a significant third harmonic making the field more peaked under the poles: B = B1 cos ks + B3 cos 3ks

(13)

The effect can be quite important if radiation is to be utilized at large horizontal angles, since the critical energy reduces more rapidly with angle, and the maximum emission angle is also reduced compared to the purely sinusoidal case. No simple expression for the critical energy as a function of angle (i.e. the equivalent of Eqn (12)) can be given in this case; the tangent

Bending magnet and wiggler radiation

123

1.0 B3 = 0

0.8

c Ⲑc0

0.6 B3/B0 = 0.15 0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

 x /K

Figure 4.10 Influence of the third harmonic field component on the variation of critical energy with observation angle in the horizontal plane.

point must be determined by numerically solving the equation
 e B1 sin ks B3 sin 3 ks + = θx ϑx = γ mc k 3k and then the field evaluated at that point using Eqn (13). The maximum angular excursion of the electron is given by 
K B1 B3 − ϑx,max = γ B0 3B0 For example, a 15% third harmonic component (B3 /B0 = 0.15) results in a 20% smaller maximum angle compared to the perfectly sinusoidal case. Figure 4.10 compares the variation of critical energy with horizontal angle in the two cases. 3.5

Wiggler parameter optimization

The main choice to be made in designing a wiggler for a particular application is that of the wiggler field and its period length. Figure 4.11 shows the relationship between peak field strength and ratio of period length to gap (g) for practical permanent magnet devices that have been constructed over the last few years at various laboratories. The maximum field obtained corresponds well with the following empirical formula [20] developed originally by Halbach. 
2 / g g B0 = 0.95 × 3.44 exp −5.08 + 1.54 (14) λ0 λ0 where an additional factor of 0.95 has been added to account for 3D effects and non-optimized dimensions etc. Using this relationship allows a graph to be drawn of the flux available as

124

R. P. Walker 2.5

B 0 (T)

2.0

1.5

1.0

0.5

0.0

0

2

4

6 0 /g

8

10

12

Figure 4.11 Field amplitude as a function of the period/gap ratio achieved in practical devices. Solid line from Eqn (14).

Flux (1015 ph/s/mrad/0.1%bw/m/GeV/A)

1.0 /E 2 = 0.2

0.8

0.6

0.5

0.4 1.0 0.2

x 10

2.0 0.0 0.0

5.0 0.5

1.0

1.5

2.0

B0 (T)

Figure 4.12 Photon flux as a function of wiggler field strength, for various normalized photon energies for a fixed gap of 20 mm.

a function of field strength for wigglers of a fixed gap (20 mm) and length for various photon energies, normalized to the storage ring energy (see Figure 4.12). The figure clearly shows that the optimum magnetic field strength varies with the energy of photons required. At relatively low energies a better optimum results from increasing the number of poles rather than the field strength. Higher energies, however, require the maximum available field strength and eventually favour the use of superconducting magnet technology.

Bending magnet and wiggler radiation

125

There are, however, several other factors that might limit the maximum field strength and/or number of poles of a practical wiggler design: •

•

•

4

Limitations on total power and power density that must be handled by the beamline components. Equations (70) of Chapter 3 give practical expressions for these quantities for a planar wiggler and it can be appreciated that very high power loads can be produced, especially in high energy rings. For example, in the ESRF the maximum permitted power that can be handled by the radiation absorbers is 15 kW and even a single 1.6 m device can exceed this limit with a field in excess of about 1.5 T; higher field devices must therefore be restricted to shorter lengths. Limitations on the opening angle of the radiation. For example, a small opening angle may be required in order to limit the power density on the vacuum chamber walls, or alternatively there may be a particular requirement for a sufficiently large angle in order that the radiation can be used by several experimental stations simultaneously. Effects on the storage ring operation, linear optics and emittance effects (see Section 4.3 of Chapter 1).

Wigglers for circularly polarized radiation

We have seen in the previous sections that circularly polarized radiation is produced vertically off-axis in a bending magnet; however, this is cancelled out in the case of planar undulators and wigglers. The desire for circularly polarized radiation with a higher flux and brilliance has, therefore, led to the development of a number of different types of insertion devices. Section 4 of Chapter 3 dealt with the radiation properties of elliptical undulators; here, we consider two special types of wigglers that are capable of producing circularly polarized radiation. 4.1

Asymmetric wigglers

An asymmetric wiggler [21] is a planar device with a single on-axis field component, but the cancellation of the circularly polarized component that occurs in a conventional device is avoided by employing positive and negative field poles of different strengths. As a result, circularly polarized radiation is emitted off-axis in the vertical direction as in a single bending magnet. The first asymmetric wiggler was constructed at HASYLAB in 1989 and employed a pure permanent magnet construction [22]. Figure 4.13 shows the result of a calculation of the field for such a structure. The radiation properties of an asymmetric wiggler can be understood with the aid of a graph of the variation of the field at the tangent point as a function of the horizontal angle. In a normal sinusoidal wiggler the result is a circle; the radiation emitted at a given horizontal angle has two source points per period with equal positive and negative field amplitudes at the respective tangent points. Figure 4.14 shows a typical graph in the case of an asymmetric wiggler, corresponding to the field distribution of Figure 4.13. It can be seen that the difference between the field values is greatest at zero horizontal angle and decreases with increasing angle. The circular polarization rate therefore follows the same trend, as can be seen in Figure 4.15. Another related feature of this device is that the total intensity peaks off-axis in the horizontal direction. It is clear that the highest circular polarization rate is obtained close

126

R. P. Walker 1.0 0.8 0.6

B (T)

0.4 0.2 0.0 –0.2 –0.4 –0.6 0

40

80

120 s (mm)

160

200

240

Figure 4.13 Typical variation of vertical field component with distance in an asymmetric wiggler.

1.0 0.8 0.6

B (T)

0.4 0.2 0.0 –0.2 –0.4 –0.6 –1.5

–1.0

–0.5

0.0 x (mrad)

0.5

1.0

1.5

Figure 4.14 Variation of magnetic field at the tangent point with the horizontal emission angle in an asymmetric wiggler, with the field distribution of Figure 4.13.

to zero horizontal angle and also off-axis vertically, and there is therefore a compromise between flux and degree of circular polarization. Figure 4.16 shows the integrated flux, circularly polarized flux and circular polarization rate for a particular case, with horizontal acceptance of ±0.25 mrad and vertical acceptance from 0.1 to 0.2 mrad. The characteristic feature can be seen, that is, the circularly polarized flux S3 changes sign at low photon energies when the lower field pole (with negative S3 ) contributes more than the higher field pole (with positive S3 ). On the other hand, at the highest photon energies the contribution of

Bending magnet and wiggler radiation

127

1.0

P3, S0 (rel. units)

0.8 0.15 0.6

0.1

0.4 z (mrad) = 0.05

0.2

0.0 –1.0

–0.5

0.0 x (mrad)

0.5

1.0

Figure 4.15 Variation of circular polarization rate as a function of horizontal and vertical angle (solid lines), for the asymmetric wiggler of Figures 4.13 and 4.14. Dashed line – total flux at vertical angle of 0.1 mrad; E = 5.3 GeV, photon energy = 10 keV. 1.0 P3

S0, S3, P3 (rel. units)

S0 0.5 S3 0.0

–0.5

–1.0 102

103

104

105

Photon energy (eV)

Figure 4.16 Integrated flux and circular polarization rate as a function of photon energy, for the same case as Figure 4.15; horizontal acceptance ±0.25 mrad, vertical acceptance 0.15 ± 0.05 mrad.

the lower field pole becomes negligible, resulting in increasing polarization rate, but rapidly diminishing flux. Further asymmetric wigglers have been built using the hybrid technology at LURE [23], HASYLAB [24] and ESRF [25]. A superconducting asymmetric wiggler has also been constructed for DELTA [26].

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4.2

Elliptical wigglers

Section 4 of Chapter 3 showed that circularly polarized radiation can be generated on-axis in an elliptical undulator, which consists of superimposed vertical and horizontal fields: Bx = Bx0 sin ks

Bz = Bz0 cos ks

ϑx = (Kx /γ ) sin ks

ϑz = −(Kz /γ ) cos ks

where Kx = eBz0 λ0 /2πmc and Kz = eBx0 λ0 /2πmc. In the pure elliptical case (Kx = Kz ) only the first harmonic is produced, with 100% circular polarization. However, good circular polarization rates (P3 ≥ 0.8) can also be produced on higher harmonics with slightly different K values (Kx /Kz ≥ 0.4). Taking this to an extreme, one arrives at the situation with a large K value in one plane, typical of a wiggler, with a much smaller value in the other plane, typical of an undulator. For practical reasons the vertical magnetic field is usually the strongest, since the vertical gap can usually be made much smaller than the horizontal one. In the case of such an ‘elliptical wiggler’ the radiation can, with certain restrictions, be approximated well by a series of bending magnets with vertical field Bz0 with a superimposed vertical deflection Kz /γ , of alternating polarity, at the position of each pole. It can be seen from the equations above that for observation angles off-axis in the horizontal plane the effective field at the tangent point, and the vertical deflection at that point, are given as follows:  2
Bz ϑz θx = 1− = Bz0 Kz /γ Kx /γ Bx θx = Kx /γ Bx0 Thus, provided the horizontal acceptance angle is not too large with respect to the maximum angular deflection, both the vertical field strength and vertical deflection angles can be approximated by their peak values, and the horizontal field at that position can be neglected, with respect to the much larger vertical field. The effect of the vertical deflection is such that (for example) radiation from positive poles are deflected vertically downwards while that from negative poles are deflected vertically upwards. On-axis, therefore, the circularly polarized component (S3 ), which changes sign both with the vertical observation angle and with dipole polarity, remains the same for each pole. Figure 4.17 shows the resulting variation of the circularly polarized component as well as the total flux as a function of vertical observation angle, for various vertical deflection angles, at the critical photon energy. As the deflection angle increases the total flux on-axis decreases while the circularly polarized flux increases and maximizes at a particular value. The polarization rate S3 /S0 however continues to increase. The range of vertical angles over which circularly polarized flux of a given sign can be obtained also increases with deflection angle. The optimum settings of deflection angle and acceptance angle depend on the photon energy and √ also on what is considered the parameter of merit: S3 , P3 or some combination such as S3 P3 (considered as corresponding to the signal-to-noise ratio in a dichroism experiment). Figure 4.18 shows the variation of on-axis circularly polarized component and total flux with vertical deflection angle at various photon energies. Lower photon energies, therefore, require larger deflection angles in order to maximize the circular polarization component. The first permanent magnet elliptical wigglers were installed in the Photon Factory and TRISTAN Accumulator Ring [27] and measurements carried out in the X-ray region of the

Bending magnet and wiggler radiation 1.00

129

a

0.75

b

S0, S3 (rel. units)

b c

0.50

a

0.25 c 0.00 –0.25 –0.50 –2.0

–1.5

–1.0

–0.5

0.0  z

0.5

1.0

1.5

2.0

Figure 4.17 Variation of circularly polarized component (solid lines) and total flux (dashed lines) as a function of vertical angle in an elliptical wiggler at the critical photon energy; vertical deflection angle Kz /γ = 0.25 (a), 0.5 (b), 1.0 (c). 1.5

S0, S3, P3 (rel. units)

b

c

1.0

b

b

a a

0.5

a

c

c 0.0 0.0

0.5

1.0 Deflection angle (Kz /)

1.5

2.0

Figure 4.18 Variation of circularly polarized component (solid lines), total flux (dashed lines) and circular polarization rate (dotted lines) as a function of vertical deflection angle; ε/εc = 0.1 (a), 1 (b), 3 (c).

spectrum have confirmed the main features of the radiation emission described above [28]. The idea of replacing the permanent magnet arrays in the above scheme with an electromagnet in order to allow a rapid switching of the polarization state was first mentioned in [29]. The first proposal for an electromagnetic wiggler combined a strong vertical permanent magnet with a weaker horizontal electromagnet and had a projected switching speed of up to 100 Hz [30]. A 7-pole device was later built by an APS/BINP/NSLS collaboration and

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installed in the NSLS X-ray ring [31], where it is now operating at 100 Hz switching rate [32]. A similar device with 32 full field poles designed to operate at 10 Hz has subsequently been installed in the APS. Another form of elliptical wiggler consisting of combined horizontal and vertical electromagnets and capable of a 100 Hz switching rate was installed recently in ELETTRA [33].

5 5.1

Geometrical properties of bending magnet and wiggler sources Introduction

The design of a high performance bending magnet or insertion device beamline generally involves a detailed ray-tracing analysis including all optical elements between the source and the sample. An accurate description of the geometrical properties of the source is clearly an essential starting point for this analysis. Various computer codes are available for carrying out such calculations, which also model various types of sources, for example SHADOW [34]. It is useful, however, to study the properties of the source as a separate topic, particularly in the case of wigglers since they can be strongly affected both by the extended source length and the relatively large acceptance angles, as well as by the electron trajectory. The main effects that can arise are given below: • • •

The ‘double source’ effect: if the oscillation amplitude in the wiggler is sufficiently large compared to the electron beam size, two separate sources can be seen at zero angle with transverse separation of ±Kλ0 /2πγ . The ‘depth of field’ effect gives rise to an increase in source size in both planes as a function of acceptance angle and wiggler length and causes the spatial distribution to become strongly non-Gaussian in shape. Under certain conditions the brilliance of the radiation may also increase less than linearly with wiggler length.

In the following we will examine the source properties in ‘phase space’, that is, using position–angle co-ordinates, as is common in the case of electron beam optics (see Chapter 1, Section 3). The large angular divergence of the radiation emitted by bending magnets and wigglers means that we can ignore diffraction and treat the problem using geometrical optics. Such a treatment is valid provided the resolution with which the phase space distribution is  ∼ calculated is larger than the limiting resolution of a diffraction-limited system, σx,z σx,z λ/4π . A phase space analysis of bending magnet radiation was presented in [35], while an analytical treatment of the optical properties of bending magnet and wiggler sources was given in [13]. Coïsson [36] presented results of numerical calculations for wigglers in the horizontal plane and further simulations of this kind were presented in [37,38]. 5.2

Phase space analysis

As the point of reference for the definition of the source properties, it is convenient to project the source to the centre of the arc of the trajectory seen by the observer in the case of a bending magnet, or to the centre of the wiggler in that case (s = 0). In [13] this was achieved by a series of separate transformations, whereas here we combined them into a single transformation according to the procedure presented in [37] and extended in [39]. The central orbit of the electron beam is defined by co-ordinates yc (s), yc (s), while the deviation of an individual

Bending magnet and wiggler radiation

131

electron with respect to this trajectory is given at position s in terms of the values at s = 0 as follows: y(s) = y0 + y0 s

y  (s) = y0

where y refers to either x or z. The transformation above is strictly valid for a field-free region; however, it is generally a sufficiently good approximation in a wiggler, or over the short distances that are involved in the case of a bending magnet. If at position s an electron emits a photon with angle yυ and this is projected backwards (or forwards) to the reference plane at s = 0, the final phase space co-ordinates are given by the following:  Y = y0 + yc (s) − yc (s)s − yυ s Y  = y0 + yc (s) + yυ

(15)

Thus, in general, by averaging over electron beam position and angle, photon emission angle and source length, −L/2 ≤ s ≤ L/2, the above equations allow the distribution of intensity in phase space to be calculated. In the following, the above equations will be used both in numerical simulations as well as to derive analytical expressions for source size and brilliance. Although the above equations seem quite straight-forward, some care is needed in their interpretation. For example, it might appear from the absence of the electron divergence y0 in the first equation that this quantity does not influence the source size. However, when one considers the effect on source position Y at a fixed angle Y  , it is clear that a variation in y0 does have an effect via the other parameters yc and yυ . In the horizontal plane, assumed to be the plane of bending, the electrons sweep out a large angle compared to either the electron divergence or range of photon emission angles. Evaluating the effect of a photon emission angle xυ on the source position X, at fixed angle X , at a position when the local trajectory has a radius of curvature ρ gives the result -X = ρxυ2 /2. Considering that the horizontal emission angle is of the order 1/γ , it is clear that in all cases of interest this effect is much smaller than the electron beam size and so can be neglected. The above is true irrespective of the source point s, and so is valid both for wigglers and bending magnets. The electron beam divergence, however, can have a larger effect, given by -X = x0 s and so in general Eqn (15) becomes X = x0 + χ(s) X  = x0 + χ  (s)

(16)

where for convenience, in what follows, the projection of the tangent to the electron trajectory to the position s = 0 has been defined as χ (s) = xc (s) − xc (s)s χ  (s) = xc (s) Because of the short effective length in a bending magnet, however, the x0 term can be neglected. In the vertical (non-bending) plane, the terms describing the central orbit of the electrons are zero. Since the photon emission angles generally exceed that of the electron beam divergence, the effect on the source size is similar to that in the horizontal plane, namely

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-Z = z0 s. As a result the zυ s term can be neglected in a bending magnet, but generally not in the case of a wiggler, and so Eqn (15) in this case becomes Bending magnet

Wiggler

Z = z0

Z = z0 − zυ s



Z =

z0

+ zυ



Z =

z0

(17)

+ zυ

The electron beam is well described by a two-dimensional Gaussian distribution in position and angle (in either plane):

γ y02 + 2αy0 y0 + βy02 1  Py (y0 , y0 ) = exp − (18) 2πε 2ε where ε is the emittance and α, β and γ are the ‘lattice functions’ which vary as a function of position around the ring circumference. Contours of constant probability thus form ellipses in phase space.√ Projections on the axes yield standard deviations of position and divergence √ given by σy = εβ and σy = εγ , respectively. In the special case of a symmetry point in the lattice the probability distribution can be written as follows:



2 2 y y 1 (19) exp − 02 exp − 02 Py (y0 , y0 ) = 2πσy σy 2σy 2σy and in this case the emittance is equal to the product of the rms size and divergence, ε = σy σy . In order to make analytical calculations of photon densities and source sizes etc., it is useful also to approximate the photon distribution as a Gaussian:

  √ θz2 d< d<  d<  = = exp − 2πσR Pυ (θ ) (20) d. dω/ω d. dω/ω θz =0 d. dω/ω θz =0 2σR2 In the above, an approximation has been used which preserves the correct peak value while using the best Gaussian approximation to the width, given by Eqn (6), in order to give correct values of brilliance and source size, rather than alternative definitions which give the correct integrated intensity. 5.3

Bending magnets

The electron trajectory in the case of a constant magnetic field is an arc of a circle with constant radius ρ, which for small s can be written as follows: xc (s) = −s 2 /2ρ

xc (s) = −s/ρ

Photons emitted from a variable position s are therefore projected onto a parabola in phase space defined by co-ordinates χ(s) = s 2 /2ρ, χ  (s) = −s/ρ. Since the intensity emitted is independent of s, Eqn (16) shows that the resulting phase space distribution is equivalent to a convolution of the electron beam ellipse with this parabola. The curvature of the parabola is, however, usually such that it dominates over the effect of the electron beam divergence. At a given angle of observation in the horizontal plane X the curvature of the trajectory corresponds to a variation in position of ρX2 /2, which is therefore greatest for a ring with

Bending magnet and wiggler radiation

133

large bending radius. Figure 4.19 shows the horizontal phase space distribution in the case of the ESRF for an acceptance of ±3 mrad. The maximum curvature in this case amounts to 106 µm, significant compared to the rms electron beam size of 126 µm; nevertheless, the effect on the integrated source distribution is almost negligible, as shown in Figure 4.20. In the vertical plane, Eqn (17) shows that the angular distributions of the electrons and photons must be convoluted, but that the position distribution remains the same. Using a Gaussian distribution for the angular distribution of the photons, the resulting photon

3 2

X ⬘ (mrad)

1 0 –1 –2 –3 –0.5

–0.4 –0.3 –0.2 –0.1

0.0

0.1

0.2

0.3

0.4

0.5

X (mm)

Figure 4.19 Horizontal phase space distribution of an ESRF bending magnet source; σx = 0.126 mm, σx = 0.108 mrad, ρ = 23.5 m. 1.0

Relative intensity

0.8

0.6

0.4

0.2

0.0 –0.5

–0.4 –0.3 –0.2 –0.1

0.0 0.1 X (mm)

0.2

0.3

0.4

0.5

Figure 4.20 Horizontal source size distribution for the case of Figure 4.19, at zero angle (dashed line) and integrated over ±3 mrad (solid line).

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R. P. Walker

distribution in phase space is therefore given as follows:   ∞ √ d<    I (Z, Z ) = 2πσ Pz (Z, Z  − θ )Pυ (θ ) dθ R d.dω/ω θ =0 −∞ where in general Eqn (18) must be used for the electron distribution since the source point is generally not a symmetry point. Integration over θ can be performed using the standard integral [40]  ∞  exp(−ax 2 − 2bx)dx = (π/a) exp(b2 /a) (21) −∞

to yield the following:  √ d<  I (Z, Z ) = 2πσR P (Z, Z  )  d. dω/ω θ =0 

where the probability function can be expressed in the same form as Eqn (18) using modified emittance and lattice functions (indicated by  ) as follows:   α  ε  = αε, β  ε  = βε, γ  ε  = γ ε + σR2 ε = ε(ε + βσR2 ), The effective source divergence, integrated over source position, is therefore given by  Qz = σz2 + σR2 . As an example Figure 4.21 shows the vertical phase space ellipses for an ELETTRA bending magnet source for various photon energies. Because of the relatively small electron beam emittance in the vertical plane the resulting ellipses are dominated by that of the photons, whose angular divergence changes significantly with photon energy, as already observed in Figure 4.2.

0.2 Ⲑ c = 0.1

Y1 (mrad)

0.1

1

5

0.0

–0.1

–0.2 –0.04

–0.02

0.00 Y (mm)

0.02

0.04

Figure 4.21 Vertical phase space ellipses for an ELETTRA bending magnet source at various photon energies; central ellipse – electron beam only; E = 2 GeV, B = 1.2 T, ε = 7 × 10−11 mrad, α = −2.92, β = −9.20, γ = 1.04.

Bending magnet and wiggler radiation

135

It also follows from the above analysis that the central brilliance of a bending magnet source is given as follows:   σR σR IZ (0, 0) d<  d<  B= √ = = d. dω/ω θ =0 2πε  σx d. dω/ω θ=0 2πσx σz (ε/β + σR2 )1/2 2π σx in standard units of photons/s/0.1% bandwidth/mm2 /mrad2 . Thus in the usual case that the photon divergence dominates over the electron distribution, the brilliance becomes simply  d<  1 B=  d. dω/ω θ =0 2πσx σz

5.4

Wiggler magnets

5.4.1

Horizontal plane

In the case of a wiggler the electron trajectory in the horizontal plane is given by xc (s) = a cos ks

xc (s) = −(K/γ ) sin ks

where a = Kλ0 /2πγ . In this case, the projection of the tangent to the electron trajectory onto the reference plane at s = 0 traces out a complex curve. For example, Figure 4.22 shows the result for a wiggler with 21 poles. At X  = 0 the two points correspond to the emission from positive and negative poles, separated by a distance 2a. Off-axis the emission from the different poles becomes more and more widely separated in position. The phase space distribution in this case is not simply a convolution of this pattern with the electron beam ellipse, because of the variation of intensity with position s. Since the vertical

6 4

X⬘ (mrad)

2 0 –2 –4 –6 –4

–3

–2

–1

0

1

2

3

4

X (mm)

Figure 4.22 Horizontal phase space projection of the electron trajectory in a wiggler; E = 2 GeV, B = 1.6 T, λ0 = 0.14 m, 21 poles.

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R. P. Walker

emission angle is also small, and it is common to accept most, if not all, of the flux in this plane, it is convenient to calculate the horizontal distribution integrated over vertical angle. The required phase space distribution can then be written as follows: 

I (X, X ) =



L/2

−L/2

 d< 1 Px X − χ(s), X  − χ  (s) ds dθx dω/ω ρ(s)

(22)

where ρ(s) is the local radius of curvature of the trajectory and Px (x, x  ) is the electron beam probability distribution given by Eqn (18) or (19). The expression for the flux is given by Eqn (9) in which y is also a function of s: y = ε/εc = ε/εco cos ks. The practical calculation of the phase space distribution can be made in either of two ways: (i) for each X, X an integral over s is performed, evaluating the electron probability for each value of s, or alternatively (ii) integrating over s, xo , xo , calculating X, X  for each combination of values according to Eqn (14), and histogramming the resulting intensity values. Figure 4.23 shows an example of a phase space distribution for the 9-pole SRS wiggler. Despite the large trajectory amplitude in the device (0.3 mm), no structure is seen in the phase space distribution because of the relatively large electron beam size (σx = 1 mm). The same device in ELETTRA, however, would have a much more complex distribution because of the smaller beam size, as shown in Figure 4.24. In this case, two sources are visible at zero angle, as well as each individual pole at large angle (the distribution is symmetric with respect to X = 0). It is clear from the shape of the phase space distribution that the effective source size increases with off-axis angle as well as with wiggler length. We will return to this point after developing analytical expressions for the central phase space density and for the source sizes. We first expand the electron probability distribution for given position s as follows, assuming

10

8

x1 (mrad)

6

4

2

0 –4

–2

0 x (mm)

2

4

Figure 4.23 Horizontal phase space for the SRS wiggler (E = 2 GeV, B = 2 T, λ0 = 0.2 m, 9 poles); σx = 1 mm, σx = 0.59 mrad.

Bending magnet and wiggler radiation

137

10

x1 (mrad)

8

6

4

2

0 –4

–2

0 x (mm)

2

4

Figure 4.24 As Figure 4.23 with the ELETTRA parameters, σx = 0.24 mm, σx = 0.029 mrad.

for simplicity that the wiggler is located at a symmetry point in the lattice: 

(X − xc (s))2 1 (X − xc (s) + xc (s)s)2  exp − Px (X, X , s) = exp − 2σx2 2σx2 (2π)σx σx Provided the horizontal acceptance is not too large, the emission is restricted to the regions close to each pole and so the flux and bending radius terms can be considered constant in Eqn (22). Using small-angle approximations xc (s) ≈ a and xc (s) ≈ s/ρ enables an integration over s to be performed for each pole, centred at position si , by means of the standard integral (Eqn (21)) giving the following:

  s )2 ρ (X ± a + X i P˜x (X, X  , si ) = Px (X, X  , s)ds = √ exp − 2(σx2 + si2 σx2 ) 2π(σx2 + si2 σx2 )1/2 where the ± sign refers to the emission from the positive or negative poles. With X = X  = 0 we obtain:

2 ρ a P˜x (0, 0, si ) = √ (23) exp − 2(σx2 + si2 σx2 ) 2π(σx2 + si2 σx2 )1/2 It does not appear possible to average this expression; however, by assuming a ≤ σx and L ≤ 2βx the following approximation can be obtained [39]: 0 1 ρ P˜x (0, 0, si ) = √ (24) 2 2 2π(σx + a + L2 σx2 /12)1/2 The final expression for the central phase space density using this approximation then becomes I (0, 0) =

Npole d< √ dθx dω/ω 2π(σx2 + a 2 + L2 σx2 /12)1/2

(25)

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R. P. Walker

To obtain an approximate expression for the source size, we first calculate the mean square value of X for a given X  and pole position si as follows: ∞ 2  0 1 ˜ −∞ X Px (X, X , si ) dX 2 X = ∞ P˜x (X, X  , si ) dX −∞

Using the standard integral (Eqn (21)) and the following [40] 
  ∞ π 1 b2 2 2 + 2 exp(b2 /a) x exp −(ax + 2bx) dx = a 2a a −∞ results in 0 1 X2 = (σx2 + si2 σx2 ) + (a + si X  )2 Averaging over the length ±L/2 and the range of angles ±Zx then results in the following:
Qx |±Zx =

σx2

L2 2 L2 2 σ + Z +a + 12 x 36 x

1/2

2

(26)

As an example for numerical calculations we take the ELETTRA wiggler, with period length 140 mm and field strength 1.6 T. Figure 4.25 shows the spatial distribution of flux in the horizontal plane when integrated over an acceptance of ±0.5 mrad for different wiggler lengths. The increasing source size and the non-Gaussian form for longer wiggler lengths are clearly visible. Figure 4.26 shows the integrated flux as well as the central value of flux density I (0, 0) and the peak spatial density (photons/s/0.1%bandwidth/mm) as a function of wiggler length. While the total flux and central value of flux density increase linearly with wiggler length, the peak spatial density increases less slowly, due to the increasing source

1.0

Intensity (rel. units)

0.8

0.6

0.4

0.2

0.0 –2.5 –2.0 –1.5 –1.0 –0.5

0.0 0.5 x (mm)

1.0

1.5

2.0

2.5

Figure 4.25 Spatial distribution of photon intensity in the horizontal plane, integrated over ±0.5 mrad, as a function of number of poles in the ELETTRA wiggler, 21, 41, 61. . .121 poles, B = 1.6 T, λ0 = 0.14 m.

Bending magnet and wiggler radiation

139

12

Intensity (rel. units)

10 8 6 4 2 0

0

25

50 75 No. of poles

100

125

Figure 4.26 Variation of integrated flux (squares), central phase space density (circles) and peak spatial density (diamonds) for the case of Figure 4.25, crosses – Eqn (25). 0.8

Source width (mm)

0.6

0.4

0.2

0.0

0

25

50 75 No. of poles

100

125

Figure 4.27 Source sizes for the case of Figure 4.25, circles – rms width, crosses – Eqn (26), squares – FWHM/2.355.

size. The values of I (0, 0) predicted by Eqn (25) are indistinguishable from the numerically calculated values. Figure 4.27 shows various estimates of width of the spatial distribution as a function of wiggler length. The numerically calculated rms values agree very well with the analytical estimate of Eqn (26) over the whole range. On the other hand, the equivalent standard deviation derived from the full width at half maximum agrees at smaller wiggler lengths, but deviates considerably for longer lengths, which again evidences the increased tails of the distribution compared to a Gaussian.

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R. P. Walker 1.0

Intensity (rel. units)

0.8

0.6

0.4

0.2

0.0 –6

–4

–2

0 x (mm)

2

4

6

Figure 4.28 As Figure 4.25, with fixed number of poles (57) and total horizontal acceptance angle (±0.5 mrad) with varying off-axis angles: 0 (centre), 0.5, 1.0, 1.5, 2.0 mrad (outer).

Increasing the angular acceptance has a similar effect on the spatial distribution as an increase in wiggler length. Moving a fixed angular acceptance off-axis, however, has a different effect as shown in Figure 4.28. Although in this case the total flux remains constant within 5% (the maximum angle of 2 mrad is small compared to the angular excursion of 5.3 mrad) the source size increases significantly and the peak spatial flux density decreases by a factor of 3.5 at 1 mrad and 8 at 2 mrad. In the case of the ELETTRA wiggler the central phase space density increases linearly with wiggler length, as shown in Figure 4.26, however, this is not always the case. Equation (25) √ shows that this occurs when the wiggler length exceeds roughly the value 12σx /σx . Figure 4.29 shows the same calculation as Figure 4.26 but with a value β = σx /σx = 1. In this case, the value of I (0, 0) increases less than linearly with increasing wiggler length. The approximate value given by Eqn (25) also becomes less accurate in this case. 5.4.2

Vertical plane

From Eqn (17) it follows that the vertical phase space distribution can be calculated as follows:  L/2  ∞  d< 1  Pz Z + θz s, Z  − θz ds dθz I (Z, Z ) = −L/2 −∞ d. dω/ω ρ(s) where the expression for the flux is given by Eqn (4). The calculation in this case follows the same lines as that for the horizontal plane, except that in general a second integration is involved. A further complication is the fact that the vertical distribution of emitted photons depends on s and hence on the horizontal angle X  which is mainly determined by the trajectory angle xc . Thus, since the horizontal acceptance ±Zx is not usually sufficient to collect all of the radiation in that plane, the integration over s should be limited to those values

Bending magnet and wiggler radiation

141

12

Intensity (rel. units)

10 8

6 4 2 0

0

25

50 75 No. of poles

100

125

Figure 4.29 As Figure 4.26 calculated with β = σx /σx = 1.

0.15

Y1 (mrad)

0.10

0.05

0.00 0.00

0.05

0.10 Y (mm)

0.15

0.20

Figure 4.30 Vertical phase space distribution for the ELETTRA wiggler, at the critical photon energy; σz = 0.014 mm, σz = 0.005 mrad.

for which −Zx ≤ xc (s) ≤ Zx . Figure 4.30 shows the vertical phase space distribution for the ELETTRA wiggler, calculated at the critical photon energy, for small horizontal acceptance. The strong increase in source size with off-axis angle is clearly seen. To proceed further analytically we use a Gaussian approximation for the vertical distribution of emitted photons, Eqn (20). The phase space distribution can therefore be

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R. P. Walker

written as 



I (Z, Z ) =

L/2



∞

−L/2 −∞

√   2πσR d<  Pz Z, Z  , θz , s ds dθz  d. dω/ω θz =0 ρ(s)

where the probability distribution for finding a photon at Z, Z  emitted at angle θz from a position s is given as follows: 
1 (Z + θz s)2  Pz (Z, Z , θz , s) = exp − (2π)3/2 σz σz σR 2 σz2


 θz2 (Z  − θz )2 × exp − exp − 2 2σz2 2σR For simplicity it has been assumed that the centre of the wiggler is at a symmetry point in the lattice. In general, all terms within the integral are functions of s but since the horizontal acceptance angle (±Zx ) is usually small with respect to the maximum deflection angle K/γ , the values of s that contribute to the integral are limited to the regions of the magnetic poles allowing the first two terms to be removed and approximated by values given at the pole centres. Integration over θz can be performed by means of the standard integral (Eqn (21)) to yield, after some simplification, the standard form for an ellipse:
 1 γ Z 2 + 2αZZ  + βZ 2  Pz (Z, Z , s) = exp − (27) 2πε 2ε with γ ε = (σz2 + σR2 ) βε = (σz2 + s 2 σR2 )

αε = sσR2   ε = s 2 σR2 σz2 + σz2 (σz2 + σR2 )

An equivalent result was obtained by Green [13]. Setting Z = Z  = 0 in the above gives Pz (0, 0, s) =

1 2π s 2 σz2 σR2 + σz2 (σz2 + σR2 ) 

(28)

 which can be averaged over s using the standard integral (x 2 + a 2 )−1/2 dx = sinh−1 (x/a) giving the result

σ  Lσ 1 z R sinh−1 Pz (0, 0) = Lπσz σR 2σz (σz2 + σR2 )1/2 Since in the usual case σR  σz and βz (= σz /σz ) ≥ L/4 the argument of the sinh−1 term is small permitting it to be expanded as follows: sinh−1 (x) ≈ x(1 + x 2 /3)−1/2 and therefore giving Pz (0, 0) =

1 2π(σz2 + σR2 )1/2 (σz2 + L2 σz2 /12)1/2

(29)

Bending magnet and wiggler radiation

143

Taking into account the fact that the region of integration is -s = 2ρZx for each pole, the approximate expression for the central phase space density is then  2Npole Zx σR d<  I (0, 0) = √ d. dω/ω θz =0 2π(σz2 + L2 σz2 /12)1/2 (σz2 + σR2 )1/2 Several expressions can be derived for the effective source size, depending on the angular acceptance. First, putting Z  = 0 in Eqn (27) gives directly the following result for the mean square value of Z: Z 2  =

s 2 σ 2 σ 2 ε = σz2 + 2 z R2 = Qz2 γ (σz + σR )

In the usual case with σR  σz and after averaging over s we obtain the following expression for the effective source size in the limit of zero acceptance angle:
Qz |Z  =0 =

L2 2 + σ 12 z

σz2

1/2

At the other extreme, integrating over all angles one obtains directly from Eqn (27) Z 2  = εβ = (σz2 + s 2 σR2 ) Averaging over s we obtain the following expression for the source size:
Qz |all angles =

σz2

L2 2 σ + 12 R

1/2 (30)

In intermediate cases we can follow the same method used above for the horizontal plane; we obtain, in the usual case σR  σz Z 2  = σz2 + s 2 σz2 + s 2 Z 2 Averaging over the acceptance angle −Zz ≤ Z  ≤ Zz would in general involve taking into account the intensity variation with angle; however, provided the acceptance is not too large (Zz ≤ σR ) this can be neglected to give the following over-estimate for the rms source size:
Qz |±Zz =

σz2

L2 2 L2 2 + σ + Z 12 y 36 z

1/2 (31)

Figure 4.31 shows the calculated source size distributions for a fixed number of poles (57) as a function of the angular acceptance for the ELETTRA wiggler. The effect of increasing angular acceptance is more marked than in the horizontal plane because of the much smaller electron beam size in this plane (13.5 µm compared to 240 µm). Beyond a certain acceptance the distribution becomes constant, since all of the flux has been collected. Figure 4.32 shows the total flux and peak spatial flux density for the same data as Figure 4.31. Figure 4.33 shows the calculated source width. The numerically calculated rms value increases rapidly with angular acceptance and then becomes constant. The initial increase is well approximated by Eqn (31), and the final value by Eqn (30). The value derived from the full width at half maximum is however much smaller, due to the non-Gaussian form of the distribution.

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Intensity (rel. units)

0.8

0.6

0.4

0.2

0.0 –0.5 –0.4 –0.3 –0.2 –0.1

0.0 0.1 y (mm)

0.2

0.3

0.4

0.5

Figure 4.31 Spatial distribution of photon intensity in the vertical plane as a function of vertical acceptance angle for the ELETTRA wiggler: 0.01 mrad (inner), 0.05, 0.1.. 0.5 mrad (outer). 1.2

Intensity (rel. units)

1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.1

0.2

0.3

0.4

0.5

Vertical acceptance (mrad)

Figure 4.32 Total flux (circles) and peak spatial density (squares) for the case of Figure 4.31, normalized to the maximum value.

5.4.3

Brilliance

Combining the horizontal and vertical planes one obtains the following general expression for the brilliance distribution:  L/2  ∞  d< 1   I (X, X , Z, Z ) = Px X − χ(s), X  − χ  (s) −L/2 −∞ d. dω/ω ρ(s)  × Pz Z + θz s, Z  − θz ds dθz

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145

Source width (mm)

0.20

0.15

0.10

0.05

0.00 0.0

0.1

0.2

0.3

0.4

0.5

Vertical acceptance (mrad)

Figure 4.33 Source sizes for the case of Figure 4.31; circles – rms width, squares – FWHM/2.355, crosses – Eqn (31), dashed line – Eqn (30).

The central brilliance (B) is then defined as the value of I (0, 0, 0, 0). It should be noted that it is not possible to perform the integrals separately for the two planes since, as already noted above, they are not independent: the values of s which contribute to the integral are strongly determined by the horizontal acceptance angle, whereas in the vertical plane this restriction is not present. To proceed analytically, we proceed as above, restricting the range of integration to the region of the poles and making a Gaussian approximation for the vertical angular flux density. After integration over the vertical angle we obtain √   L/2 2π σR d<  B = I (0, 0, 0, 0) = Px (0, 0, s)Pz (0, 0, s)ds d. dω/ω θz =0 ρ −L/2 Considering the integral as a sum of integrals over each separate pole, and assuming the vertical probability is constant for each, one obtains using Eqns (23) and (28)   Npoles  exp −a 2 /2(σx2 + si2 σx2 ) σR d<  B=   2 2 1/2 σ 2 (σ 2 + σ 2 ) + s 2 σ 2 σ 2 d. dω/ω θz =0 2 i=1 2π σx + si σx z z R i z R

1/2

(32)

An approximation, valid in the limit L ≤ 4βz , L ≤ 2βx , a ≤ σx , σz ≤ σR /2 is given by combining Eqns (24) an (29) as follows:  Npoles σR d<  B= d. dω/ω θ z=0 2π(σx2 + a 2 + L2 σx2 /12)1/2 (σz2 + L2 σz2 /12)1/2 (σz2 + σR2 )1/2 (33) Figure 4.34 shows the result of a numerical calculation of the brilliance as a function of length for the ELETTRA wiggler, at the critical photon energy. The brilliance in this case increases

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R. P. Walker 12

Brilliance (rel. units)

10 8

6 4 2 0

0

25

50 75 No. of poles

100

125

Figure 4.34 Brilliance as a function of number of poles in the ELETTRA wiggler; solid line – exact calculation, crosses – Eqn (32), plus signs – Eqn (33).

linearly with length up to about 40 poles and then begins to deviate. Although the absolute values of brilliance vary with photon energy, via the term d
References [1] Motz, H., J. Appl. Phys. 22, 527 (1951). [2] Purcell, E. M., Unpublished note July 1972, reproduced in ‘Wiggler Magnets’, Stanford SSRP Report No. 77/05, May 1997 pp. IV-18. [3] Trzeciak, W. S., IEEE Trans. Nucl. Sci. NS-18, 213 (1971). [4] Berndt, M. et al., IEEE Trans. Nucl. Sci. NS-26, 3812 (1979). [5] Barbini, R. et al., Riv. del Nuov. Cim. 4, 1 (1981). [6] Winick, H. and R. H. Helm, Nucl. Instr. Meth. 152, 9 (1978). [7] Hoyer, E. et al., IEEE Trans. Nucl. Sci. NS-30, 3118 (1983). [8] Artamonov, A. S. et al., Nucl. Instr. Meth. 177, 239 (1980). [9] Suller, V. P. et al., IEEE Trans. Nucl. Sci. NS-30, 3127 (1983). [10] Schwinger, J., Phys. Rev. 75, 1912 (1949). [11] Sokolov, A. A. and I. M. Ternov, Synchrotron Radiation (Pergamon Press, 1968). [12] Jackson, J. D., Classical Electrodynamics Ch. 14 (John Wiley & Sons Inc., New York, 1962). [13] Green, G. K., ‘Spectra and Optics of Synchrotron Radiation’, Brookhaven National Laboratory Report 50595 Vol. II, February 1977. [14] Kim, K. J., ‘Characteristics of Synchrotron Radiation’, Physics of Particle Accelerators Vol. 1, AIP Conf. Proc. 184, p. 567 (1989). [15] Hofmann, A., ‘Characteristics of Synchrotron Radiation’, Proc. CERN Accelerator School, Grenoble, April 1996, CERN 98–04, p. 1 (1998). [16] Hsu, K. T. et al., Nucl. Instr. Meth. A406, 323 (1998).

Bending magnet and wiggler radiation [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]

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Kostroun, V. O., Nucl. Instr. Meth. 172, 371 (1980). Walker, R. P. and B. Diviacco, Rev. Sci. Instr. 63, 392 (1992). Walker, R. P., Nucl. Instr. Meth. A335, 328 (1993). 6 GeV Synchrotron X-ray Source Conceptual Design Report, Supplement A, LS-52 (March 1986), Argonne National Laboratory. Goulon, J. et al., Nucl. Instr. Meth. A254, 192 (1987). Pflüger, J. and G. Heintze, Nucl. Instr. Meth. A289, 300 (1990). Barthès, M. et al., IEEE Trans. Magn. 24, 1233 (1987) and 28, 601 (1992). Pflüger, J., Rev. Sci. Instr. 63, 295 (1992). Chavanne, J. et al., Nucl. Instr. Meth. A421, 352 (1999). Schirmer, D., Proc. XV Int. Conf. High Energy Accelerators, Int. J. Mod. Phys. A (Proc. Suppl.) 2B, 2, 644 (1993). Yamamoto, S. et al., Rev. Sci. Instr. 60, 1834 (1989). Yamamoto, S. et al., Phys. Rev. Lett. 62, 2672 (1989). Onuki, H., Nucl. Instr. Meth. Phys. Res. A246, 94 (1986). Walker, R. P. and B. Diviacco, Rev. Sci. Instr. 63, 332 (1992). Gluskin, E. et al., Proc. 1995 Particle Accelerator Conference, IEEE Catalog No. 95CH35843, p. 1426. Singh, O. and S. Krinsky, Proc. 1997 Particle Accelerator Conference, IEEE Catalog No. 97CH36167, p. 2161. Walker, R. P. et al., Proc. 1998 European Particle Accelerator Conference, IOP Publishing, p. 2255. Lai, B., K. Chapman and F. Cerrina, Nucl. Instr. Meth. A246, 544 (1986). Sabersky, A. P., Particle Accelerators 5, 199 (1973). Coïsson, R., S. Guiducci and M. A. Preger, Nucl. Instr. Meth. 201, 3 (1982). Walker, R. P. and R. Coïsson, Proc. SPIE 582, 24 (1985). van Dorssen, G. E., H.A. Padmore and W. Joho, Proc. SPIE 2013, 104 (1993). Walker, R. P., Sincrotrone Trieste Internal Report ST/M-TN-89/24, July 1989. Gradshteyn, I. S. and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, New York, 1980).

5

Technology of insertion devices Joel Chavanne and Pascal Elleaume

1

Introduction

This chapter deals with the technological and engineering issues encountered during the manufacture of an insertion device. Section 2 deals with the properties of the various magnetic materials such as permanent magnets, iron and other soft materials. Section 3 presents the various methods used to predict and numerically compute the magnetic fields. Section 4 presents and compares the most important magnetic designs used to build permanent magnet undulators and wigglers such as the pure permanent magnet and the so-called hybrid structures. Section 5 presents the various methods of magnetic field measurement, including local and field integral measurements. Section 6 presents the various methods of shimming used to correct field errors, including both multipole and phase correction. Section 7 presents the issues encountered when designing the mechanical support structure which allows the tuning of the magnetic gap of a permanent magnet assembly under high magnetic force. Section 8 deals with the very special case of high field superconducting wigglers with a particular emphasis on cryogenic engineering. The presentation and many illustrations are strongly inspired by the ESRF experience. In the last twelve years of technical development, more than 80 insertion devices have been designed and built under the supervision of the authors.

2

Properties of magnetic materials

There are two main sources of magnetic fields which can be used to build undulators and wigglers. They are current coils and permanent magnet materials. In addition, iron and other soft materials can be used to drive and/or concentrate the fields. In this section, a short summary of the magnetic characteristics of the materials used to build the insertion devices is given. The magnetic properties of any material are usually described by means of the vectors  H and M  · B is the flux density or magnetic induction expressed in Tesla or kilo-Gauss B,  is the magnetization of (1 kG = 0.1 T), H is the magnetic field or magnetic excitation, M   the material. Both H and M are expressed in Ampere/metre. They are related to each other by the relation  B = µ0 (H + M)

(1)

 is zero, but in any other material the magnetization In the vacuum the magnetization M deviates from zero. It can be almost constant for any excitation H as in a permanent magnet material, or it can be proportional to the excitation as in paramagnetic, diamagnetic and

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 vs H relation (or equivalently ferromagnetic materials. Any material is characterized by a M   B vs H ). 2.1

Permanent magnet materials

The permanent magnet materials required to build insertion devices need to have both a large spontaneous magnetization and a large anisotropy responsible for the coercivity of the material. Both properties can be reached by a few alloys such as the binary compounds SmCO5 and Sm2 CO17 and, more recently, from the ternary compound Nd2 Fe14 B discovered in 1983. Other well known permanent magnet materials such as Alnico and Ferrites are almost never used due to either their low coercivity or low magnetization. Ferrite which is inexpensive is only competitive whenever a low field and a large volume of an almost uniform field is required [1] which is rarely the case for insertion devices. There are several methods to produce permanent magnets. The most popular is powder metallurgy which give magnets with high remanent field and/or high coercivity. The magnetic alloy is ground into powder and compacted in a controlled atmosphere inside a magnetic field in order to permanently align the magnetic grains. The magnet is then sintered in order to induce crystalline growth of the magnetic phase. The sintering is a slow temperature cycling up to 1000 ◦ C. A final annealing process improves the coercivity of the material. These permanent magnet materials are essentially anisotropic. The axis of the magnetic field applied during the manufacture is called the easy axis. The various steps of manufacturing leaves a permanent anisotropy in the material with different magnetic properties parallel and perpendicular to the easy axis. Let H and H⊥ be the components of the excitation H parallel and perpendicular to the easy  In some range of excitation H , axis. Similarly, one defines B and B⊥ for the induction B. the permanent magnet is characterized by the following relations: B = µ0 (χ H + H ) + Br

B⊥ = µ0 (χ⊥ H⊥ + H⊥ )

(2)

where µ0 is the vacuum permeability, Br is the remanent field and χ and χ⊥ are the susceptibility parallel and perpendicular to the easy axis. In the vacuum, both χ and χ⊥ are equal to zero. For most permanent magnet materials and for sufficiently small values of the excitation H , they are approximated in terms of the linear relative permeabilities µr, and µr,⊥ and Eqn (2) reduces to B = µ0 µr, H + Br

B⊥ = µ0 µr,⊥ H⊥

(3)

For large excitation, the susceptibilites are known to be non-linear. In particular, one defines the intrinsic coercivity HcJ and the coercive force HcB as the value of the excitation H for which the magnetization and the field are zero (respectively). This is illustrated in Figure 5.1. Table 5.1 presents the remanent field, relative permeabilities, coercive field and temperature coefficient of the most commonly used permanent magnet materials. The temperature coefficient is the rate of change of the remanent field with temperature. Its value given in Table 5.1 is the average in the temperature range of 20–100 ◦ C . For all types of materials, the coercivity can be changed by adding a few specific impurities during the manufacturing process. The higher the remanence the lower the coercivity. Note that if the permeabilities defined in Table 5.1 apply for any excitation H , the coercivity HcJ would be equal to Br /(µ0 (µr, − 1)) which is a very high value. Unfortunately, if one drives a magnetic material towards large negative fields, the susceptibility ceases to behave linearly. This occurs because the various magnetic domains constituting the material flip their magnetization in

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0M, B

µ0M||

B|| = 0 (H|| +M||)

HcJ

O

HcB

H

Figure 5.1 Relation between the field, excitation and magnetization in a permanent magnet material. Table 5.1 Remanent field, relative permeabilities, coercive field and temperature coefficient of the most commonly used permanent magnet materials Material

Br [T]

µr,

SmCO5 Sm2 CO17 NdFeB

0.9–1.01 1.04–1.12 1.0–1.4

1.05 1.05–1.08 1.04–1.06

µr,⊥

HcJ [kA/m]

10−2 /◦ C

1.15–1.17

1500–2400 800–2000 1000–3000

−0.04 −0.03 −0.10

an irreversible manner. As a result, if one increases the excitation again, the material follows a different characteristic of B vs H with lower induction B. It is therefore important to avoid such a permanent demagnetization. In this respect, the coercivity gives an indication of the level of excitation where it occurs. The reader interested in the numerical implementation of B (H, T ) can look at [2]. Permanent magnets lose remanence under exposure to electron beams. In this respect Sm2 CO17 is much more resistant than NdFeB under exposition to several GeV electrons [3]. 2.2

Soft magnetic material

Soft magnetic materials present a large susceptibility and a remanent field as close as possible to zero. The large susceptibility enables them to guide the flux line from the source (permanent magnet or coil) to the region of interest with low losses. They are essentially isotropic but the M vs H characteristic is non-linear. The magnetization tends to a limit Ms as H goes to infinity. The saturated magnetization (or polarization) µ0 Ms reached for large H is of the order of 2 T for most useful materials. At low excitation H , the magnetization is a linear function of the excitation and one defines the relative permeability µr and the differential permeability µdiff according to µr =

B µ0 H

µdiff =

1 dB µ0 dH

(4)

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(b)

Flux density (T)

Fe–Co 0.1 Pure Fe

Low carbon steel

0.01

100

101

102

103

Magnetic field H (A/m)

104

Relative permeability r

Pure Fe 1

104 Fe–Co 103 Low carbon steel 102 100

101

102

103

104

Magnetic field H (A/m)

Figure 5.2 Dependence of the flux density B and relative permeability as a function of the field H for three representative soft magnetic materials.

Figure 5.2 presents the B vs H characteristics and the permeability of iron, cobalt–iron and a low carbon steel. Clearly the differential permeability reaches a maximum value at some non-zero excitation, it then drops to 1 for large excitation. The saturated magnetization reaches 2.35 T for cobalt– iron whereas it is 2.15 T and 2.08 T for the iron and low carbon steel (respectively). Due to its high saturated magnetization, cobalt–iron is the material of choice for the manufacture of high field wigglers. Because its manufacturing process is rather complex, it is more expensive than iron or low carbon steel which are, therefore, preferred in applications which do not require a high field.

3

Magnetic field computation

In this section, we review the various methods of magnetic field computation used when designing insertion devices. They include analytical integration made directly from Maxwell’s equations and numerical methods such as the finite element method and the integral method. In the rest of this chapter, we shall use the following set of orthogonal axes. Os is the longitudinal axis parallel to the average electron beam velocity. Ox and Oz are orthogonal to each other and both orthogonal to Os. Ox (Oz) are also referred to as the transverse horizontal (vertical) axis. In a conventional planar undulator or wiggler the main component of the magnetic field is vertical. Any vector V is written as V = (Vx , Vz , Vs ) where Vx , Vz and Vs are, respectively, the horizontal, vertical and longitudinal components.

3.1 3.1.1

Analytical expressions Two-dimensional fields

In two dimensions there are a number of (almost) exact results concerning periodic magnetic fields. These results are of great importance and are summarized below. Many expressions derived in this section are taken from the PhD thesis by X. Marechal [4].

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3.1.1.1 GENERALITIES

In this section we restrict ourselves to two dimensional (2D) magnetic fields which can be  z, s) = (0, Bz (z, s), Bs (z, s)) where both Bz and Bs are independent of expressed as B(x, the horizontal coordinate x. A vertical field undulator with magnets and poles built with large horizontal dimensions of the magnets and poles is a good candidate for such an approximation. Let us consider the magnetic field produced by the static current density J = (J, 0, 0). From  B = 0 and ∇  × B = µ0 J, one derives the following Maxwell’s equations (in SI units), ∇ differential equations for Bz and Bs : ∂Bz ∂Bs ∂Bz ∂Bs (5) + =0 − = µ0 J ∂z ∂s ∂s ∂z In the region of space where the current density is equal to zero, the conditions set by Eqn (5) are the necessary and sufficient conditions for the complex function B = Bz + iBs to be ! √ an analytical function of the complex variable y = s + iz in which i = −1 is the usual ˜ complex number. Conversely, any particular complex analytical function F of the complex coordinate y = s + iz is a possible magnetic field in a current-free region,!Bz (Bs ) being the ˜ real (imaginary) part of F . To ease the distinction between real and complex quantities, in ! quantity y is written as y. the following, any complex ˜ 3.1.1.2 CURRENTS

A filament infinite wire of current I intersecting the (s, z) plane at the origin is a particularly simple example of a 2D magnetic field for which the field solution B (y) is derived from ! ˜ Eqn (5) as µ0 I B (y) = (6) 2π y ! ˜ ˜ Let us now consider an infinite array of current . . . , +I, −I, +I, −I, +I, . . . parallel to the Ox axis and intersecting the Osz plane at the positions z = 0 and s = n(λ0 /2) with n = . . . , −2, −1, 0, 1, 2, . . . , λ0 being the spatial period. In such geometry, the magnetic field is obtained by summing over the field contribution from all currents [5]:   µ0  1 I µ0 I I −I −I + +  + · · ·=   B= ··· + (y − λ0 ) λ0 sin 2π(y/λ ) y ! 2π y − (λ0 /2) y + (λ0 /2) 0 ˜ ˜ ˜ ˜ ˜ (7) Equation (7) defines a magnetic field which is a periodic function of the longitudinal coordinate s. One can therefore expand B as a Fourier series in s and obtain ! 
ε = 1 if z > 0 2µ0 I  s + iz B = −iε (8) exp 2iπn λ0 λ0 ! ε = −1 if z < 0 n=1,3,5,... Assuming that each current has a finite rectangular cross section with dimension tz and ts along the Oz and Os axis, one derives the Fourier series of the magnetic field by integrating Eqn (8) over the cross section of the conductor: 

 ε = 1 if z > Hs /2 2µ0 I  2iπn ts tz B = −iε exp ε F nπ , nπ (s + iz) λ0 λ0 λ0 λ0 ! ε = −1 if z < −Hs /2 n=1,3,5,... (9)

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We have assumed that the centre of each current conductor intersects the Ozs plane at z = 0. The function F (s, z) is given by sin(s) sinh(z) (10) s z To build an undulator, one places two parallel arrays of such conductors vertically separated by a gap g as shown in Figure 5.3. By summing the contribution from both arrays, and redefining the origin of the Oz and Os axes in the middle of the gap, as shown in Figure 5.3, one derives the following expression for the magnetic field which is only valid in the region of space located between the two arrays: 


tz ts s + iz 4µ0 I  g + tz B = Bz +iBs = F nπ , nπ sin 2πn exp −nπ λ0 λ0 λ0 λ0 λ0 ! F (s, z) =

n=1,3,5,...

(11) One easily extracts the vertical and longitudinal field components from Eqn (11) by making use of the following identity:




 s + iz s z s z sin 2π n = i cos 2πn sinh 2πn + sin 2πn cosh 2π n λ0 λ0 λ0 λ0 λ0 (12) The magnetic field contains all odd harmonics of the period λ0 but it is dominated by the first harmonic (n = 1) whose expression in the median plane (defined by z = 0) is 


ts s 4µ0 I g + tz tz Bz (s, 0) = F π ,π sin 2π Bs (s, 0) = 0 exp −π λ0 λ0 λ0 λ0 λ0 (13) Electromagnet undulators are almost never built as shown in Figure 5.3. One usually inserts an iron yoke between and below the conductor which significantly increases the peak field and modifies the harmonic content. Unfortunately, one must use a numerical method to compute the field in such geometry. Equations (11) and (13) are nevertheless useful to estimate

z

ts

tz g

O

s

0

Figure 5.3 Undulator made of an array of infinite parallel thick conductors.

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the field from a superconducting undulator for which the yoke becomes fully saturated and makes a negligible contribution. It is also useful to derive the field from a pure permanent magnet undulator as we shall see in the next section. 3.1.1.3 PERMANENT MAGNETS

Let us consider the (very important) magnet assembly shown in Figure 5.4. It is made of two parallel arrays of permanent magnets (upper and lower arrays). The magnetization M in each magnet block is uniform and rotated by 90◦ from one magnet to the next. The magnetization in the upper and lower magnet are symmetric in such a way that, in the median plane, the vertical field components from each array have the same sign while the longitudinal field cancels. Let t be the thickness of each array, λ0 the period and δ the air gap between two adjacent blocks. All magnet blocks have the same dimensions and are infinite in the direction Ox orthogonal to Figure 5.4. The magnetic field can be computed by replacing the magnetization  × nˆ flowing at the surface of the magnet blocks. nˆ is of the block with a current density Js = M a unit vector orthogonal to the surface of the magnet. These surface currents are orthogonal to the magnetization plane and are periodic functions of the longitudinal coordinate s. The magnetic field can be computed by integrating Eqn (9) over the surfaces of the magnet blocks which are parallel to the magnetization. Placing the origin of the axis in the middle of the gap (see Figure 5.4), the magnetic field in the space located between the upper and lower magnet arrays can be written as B = Bz + iBs !

 g sin((π n/4) − (nπ δ/λ0 )) = 2µ0 M exp −nπ λ0 (nπ/4) n=1,5,9,... 


s + iz t sin 2πn × 1 − exp −2πn λ0 λ0 




(14)

The field only contains harmonics n = 1, 5, 9, . . .. As for the undulator discussed in the previous section, the field is dominated by the fundamental harmonic (n = 1). Neglecting the air space δ between blocks and assuming a height of blocks equal to half the period z

 t

s

g

O

t

0

Figure 5.4 Pure permanent magnet undulator made up of four magnets per period.

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(t = λ0 /2), one obtains the following expression for the first harmonic of the field in the medium plane: 

s g sin 2π Bz (s, 0) = 1.72 µ0 M exp −π λ0 λ0

Bs (s, 0) = 0

(15)

Increasing the block height t to infinity would only increase the peak field by 4%. The period of a single array (upper or lower) shown in Figure 5.4 is made up of four blocks, with a rotation of 90◦ of the magnetization from one block to the next. One can generalize Eqn (14) to magnet stuctures made of p blocks per period in which the magnetization is rotated by 360/p◦ from one magnet block to the next. The result is B = Bz + iBs !

 g sin((π n/p) − (nπ δ/λ0 )) = 2µ0 M exp −nπ λ0 (nπ/p) n=1,p+1,2p+1,...


 t s + iz × 1 − exp −2πn sin 2πn λ0 λ0 




(16)

This expression was first derived by K. Halbach [6]. The larger the number of magnets per period, the less harmonics in the field. In the case δ = 0 and in the limit of an infinite number of magnets per period, which corresponds to a magnetization continuously rotating as a function of s, the field only contains the harmonic n = 1 and it is 10% higher than in the case of four blocks per period which field is given by Eqn (14). In a real device, the splitting of the magnet blocks into smaller pieces to accomodate a larger number of blocks per period, and the multiplication of small air spaces between blocks result in a loss of peak field. In addition, the smaller size of the magnet blocks together with a greater number of pieces results in a more expensive magnet array. For these reasons, the four blocks per period of Figure 5.4 is almost universally used. It is quite economical to build and gives a peak field on harmonic 1 close to maximum without significant perturbation by the higher harmonics. In a real situation the blocks have a finite transverse width Lx which slightly reduces the magnetic field as compared to infinite blocks. Numerical simulations show that if δ = 0 and t = λ0 /2, the deviation of the peak field is smaller than 1% whenever λ0 /g ≤ 5 and Lx /g ≥ 5. In Eqns (14)–(16), the magnetization M is assumed to be uniform inside each block and independent of the magnetic gap. This is a reasonable assumption if the blocks are made of SmCO5 , Sm2 Co17 or NdFeB; however, the magnetization at any point in a magnet presents a small variation with the local magnetic field. Taking into account the non-unit permeabilities µ and µ⊥ of the material, as defined in Section 2, a numerical simulation made on the magnet array of Figure 5.4 with t = λ0 /2 shows that if λ0 /g ≥ 3 and if µ < 2, one may approximate the magnetization by µ0 M ≈ Br exp(−(µ − 1)/2) independently of µ⊥ . The term exp(−(µ − 1)/2) accounts for a few percent correction (3% for NdFeB magnets) and one usually neglects this, assimilating the magnetization µ0 M to the remanent field Br . 3.1.1.4 COMPARISON BETWEEN CURRENT AND PERMANENT MAGNET UNDULATORS

Let us compare the peak field Bˆ I created by the current undulator shown in Figure 5.3 with the peak field Bˆ M created by the permanent undulator of Figure 5.4. Making use of Eqns (13)

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and (15) and assuming the same magnetic gap, period and height t = tz = λ0 /2 and no air space between conductors or magnets, their ratio is Bˆ I λ0 J I /λ0 = 0.45 ≈ 0.11µ0 Br M Bˆ M

(17)

where J is the current density in the conductors. For a 30 mm period and assuming NdFeB magnets with Br = 1.2 T one requires a current density of 284 A/mm2 to reach the same peak field in both devices. Such a current density cannot be reached with room temperature conductors due to the Joule effect and the necessary cooling of the conductors which carry the current. It is, nevertheless accessible to superconducting coils. Note that the shorter the period, the higher the current density required to compete with the permanent magnets. It is only for sufficiently large periods that both technologies may compete for the highest peak field. Nevertheless, for ultra-short period undulators with period length smaller than 10 mm, one may face some difficulties in assembling the small pieces of permanent magnets while electromagnet undulators do not cause any major difficulty [7]. 3.1.2

Three-dimensional fields

Contrary to the 2D case, analytical expressions of three-dimensional (3D) magnetic fields are not available even in the simplest geometry. The field computation can only be done by one of the numerical methods described in Sections 3.2 and 3.3. Nevertheless, one may derive a useful approximation of the field in some limited range of space by making use of Maxwell’s equations. For example, let us consider the case of a conventional vertical field undulator producing a field which is symmetric and perpendicular to the median plane Oxs. Assuming that the field is independent of the horizontal coordinate, we have seen in the previous section that the vertical field Bz around the median plane can be expressed as Bz = Bˆ 0 cosh (kz) sin (ks)

(18)

where Bˆ 0 is the peak field in the median plane and k = 2π/λ0 . Due to the finite horizontal width of the magnet blocks (or conductors), one expects the field to be maximum at some horizontal position which is taken as the origin of the Ox axis. It is of interest to look for a solution of the form Bz = Bˆ 0 cos (kx x) cosh (kz z) sin (ks s)

(19)

with ks = 2π/λ0 leaving kz and kx free for the moment. The cos (kx x) has been selected to  × B = 0, one derives the describe the lateral decay of the field away from x = 0. From ∇ other components of the field: Bx = − Bs =

kx ˆ B0 sin(kx x) sinh(kz z) sin(ks s) kz

ks ˆ B0 cos(kx x) sinh(kz z) cos(ks s) kz

(20) (21)

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 · B = 0 imposes the following condition on kx , kz and ks : The other Maxwell equation ∇ kz2 = kx2 + ks2

(22)

Note that Eqns (19)–(22) satisfy Maxwell’s equations in the whole space. However, comparing with the field of a real undulator, it appears that they are an approximation which is only valid in the vicinity of (x, z) = (0, 0). The dependence of the peak field B 0 on the gap between the magnets and/or current in the conductors cannot be determined without any description of the magnet and coil geometry. This approximation can nevertheless be useful to predict the focussing properties of the undulator field on the electron beam since these properties only depend on the field expansion close to the main electron beam axis which coincides with Os. At the ESRF, we have also found them useful for the interpretation and correction of the signals given by Hall probes during the field measurement. Similar analytical results could be derived for a helical or ellipsoidal field undulator by summing over two orthogonal components with the same period but different phase. 3.2

Finite element method

3.2.1

Introduction

The finite element method has been widely used in many different areas of physics and engineering including mechanical structure, magnetism, electrostatics, fluid dynamics, heat propagation and so on. Its application to electric and magnetic fields together with a comparison of the various techniques of field computation is given in [8] and [9]. There exist a large number of finite element codes available to perform 2D or 3D field computations of magnetic fields [10,11]. In the field of accelerator magnets, the most popular is POISSON [12] which is one of the earliest 2D codes and belongs to the public domain. Nowadays, it is still in use but it tends to be replaced by one of the commercially available 2D or 3D packages such as ANSYS [13], MAGNET [14], FLUX3D [15], MAXWELL [16] or TOSCA [17] which provide more pre- and post-processing features and a more friendly user interface. Most finite element codes are organized around four main modules: pre-processor, mesh generator, solver and post-processor. Each module is usually run successively. •

•

The pre-processor is the first module to be executed. In this module the shape and magnetic properties of each object (magnet block, iron piece, coil etc.) are defined. Making use of the symmetries, the user restricts the volume of computation and defines the conditions to be satisfied by the field on each boundary (field either normal or orthogonal to the boundary). The smaller the volume the shorter the time for solving and the less memory is required. The next operation consists in partitioning the volume of interest in a number of smaller sub-volumes. The individual sub-volumes (also called elements) are usually triangular or quadrilateral in 2D and tetrahedrons or parallelepipeds in 3D. When solving, the magetic field is computed at each vertex point sustaining each sub-volume. These vertex points are called nodes. The operation of partitioning is also called mesh generation. It is one of the most delicate processes which largely determines the precision and accuracy of the field computation. Having too many elements results in an excessive memory requirement and long CPU time for solving. However, having too few elements gives

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J. Chavanne and P. Elleaume poor precision. It is common practice to vary the mesh size from one region to another. One would use the smallest mesh size at places where the field gradient is strong or in the vicinity of the region where one aims for a high precision in the field. The mesh generation used to be a very tedious operation requiring the user to define manually each element and its list of nodes. Nowadays, most packages provide an automatic or semi-automatic mesh generation which greatly simplifies this task. It is nevertheless important to make a visual inspection of the resulting mesh using some graphical utility. It is common to carry out the operation of solving a few times with different meshes until a satisfactory precision is reached. The next operation concerns the solving. This is an automatic process which requires very little input from the user. It may take from a few minutes to a few hours of CPU time depending on the number of nodes. When the solver terminates execution, the field potential (scalar or vector, depending on the code being used) is known at each node. In the post-processor, the user makes linear plots, contour plots etc. of the field inside the computation volume. The field at any point is determined by a suitable interpolation of the potential computed at each node by the solver. Other important quantities can be computed by the post-processor such as the field integrals or magnetic forces.

Some codes use the finite difference method, which, in many respects, can be considered as a particular case of finite elements for which all elements have a rectangular shape (in 2D) or parallellepipedic shape (in 3D). Even though, a finite difference code is much easier to write than a finite element code, it does not offer as much freedom in the description of geometries with complex shapes. This limits the field of application of finite difference codes. One should mention that, even though a lot of progress has been made in the past ten years, the use of a 3D finite element code is still not trivial. The effort required for a novice user is significant. Besides the careful study of substantial documentation, one usually needs several days of training with experts and a lot of practice before mastering or even making a correct use of the code. This explains why many people also use the analytical approach (see Section 3.1) or integral approach (see Section 3.3), whenever it is possible. 3.2.2

Application to a hybrid undulator

In this section, we shall illustrate the various steps of the computation of a 2D magnetic field from a hybrid undulator by finite elements. The geometry is shown in Figure 5.5. The structure is made of two arrays symmetrically positioned with respect to the median plane. Each array is periodic and alternates between a narrow pole made of iron (or some other high permeability material) and a thick permanent magnet. For a given period, the optimum thickness ratio between the magnet and the pole and the difference in height is the result of the maximization of the peak magnetic field in the medium plane obtained from the finite element computation. Making use of the periodicity and numerous symmetries, one can restrict the field computation domain to a rectangle in the Osz plane (see Figure 5.5). The longitudinal size Ls along Os is equal to a quarter of the period, while the vertical size Lz extends from the middle of the gap accross the magnet block and iron pole to a region in air where the magnetic field is sufficiently weak (see Figure 5.5). The next step is to partition (or mesh) the surface of the rectangle. Figure 5.5 presents a partitioning made with triangles using an automatic mesh generator. Note that each material (air, magnet and iron) has been meshed separately in such a way that all triangles are assigned to a single material. As discussed earlier, the nodes of the triangles are the points at which the vector potential (which has a single component

z

O

s

B II = 0

Magnet

B⊥ = 0

B II = 0

Iron

B II = 0

Figure 5.5 Computation of the 2D magnetic field produced by a hybrid undulator using finite elements. Making use of the symmetries one reduces the field computation region to the rectangle shown below. Boundary conditions are applied to each side of the rectangle. Each sub-region corresponding to one material (air, magnet or iron) is partitioned into a mesh of triangles. The vector potential is solved on the node.

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along Ox) is solved. The next step is to specify the boundary conditions applied to each face of the rectangle. To establish it, one must already have some understanding of the path followed by the flux line. Since the rectangle has been cut along the symmetry lines, the field is either parallel or perpendicular to the boundary. There is the difficulty of the lower border of the rectangle located in the air outside the undulator. In fact, the field in this zone is small and the field computed in the magnetic gap is rather insensitive to the type of condition applied to this boundary once it is located sufficiently far away from the magnet block. The boundary conditions are either a constant potential (equivalent to B perpendicular to the boundary) or potential with zero normal derivative (equivalent to B parallel to the boundary). The next operation consists of assigning the magnetic characteristics B (H ) for the magnet block and iron pole. The description is now complete and solving can take place. The magnetic field at any point located inside a triangle is derived by interpolating the vector potential computed at the nodes of the triangles. Other shapes of elements (rectangle, polygons etc.) could be used instead of a triangle; however the triangle has the advantage of simplicity. One also very often uses second order triangles for which six nodes are defined: the three summits and the middle of each of the three sides of the triangle. The interpolation is then more accurate and, as a result, a fewer number of triangles are usually required to reach a given accuracy. So far we have looked at the periodic part of the field. If one is interested in the field at the end of the undulator, one must define a computation region with a much larger dimension because fewer symmetries are present. It must extend vertically and longitudinally sufficiently far away from the last magnet and poles to reach a region where the field vanishes. The computed field loses some of its accuracy due to these additional arbitrary parameters selected by the user. One of the most important issues in designing the end of an insertion device concerns the field integrals I(x, z) which are defined as  ∞  I(x, z) = B(x, z, s) ds (23) −∞

Because the central field is periodic it does not contribute to I(x, z) and only the extremities contribute. These integrals produce an undesirable angular deviation of the electron trajectory as it crosses the magnetic field. Unfortunately, field integrals cannot be computed correctly from a 2D finite element code. One must use a 3D code. In a 3D code the field computation region becomes a volume rather than a surface. The volume is typically meshed into a set of tetrahedrons. Since the field integrals are computed by numerically integrating the local field and because the field is only computed in a bounded region of space, it is difficult and requires many iterations to compute the field integrals with a 3D finite element code. 3.2.3

Principle

It is out of the scope of this chapter to give a full description of the finite element method applied to magnetostatics. There are a number of books dedicated to this topic, see for example [18]. We shall nevertheless summarize the basic principles. The magnetic energy in a volume . originating from an ensemble of static currents and materials is given by

  B E= H dB − JA d. (24) .

0

where J is the current density, A the vector potential and H and B are, respectively, the magnetic field strength and magnetic induction. The material property is described by the

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function  H )) B = µ0 (H + M(

(25)

 is the magnetization vector which depends on H . From a variational principle, one where M determines the equilibrium field distribution by expressing that the partial derivative of the energy E with respect to each component of the vector potential A at each node is equal to zero. This determines a set of linear equations in the vector potential which can be inverted to give the solution. The associated matrix has the property of being symmetric and sparse. It means that most of its elements are zero with the exception of a few of them located close to the diagonal. Special algorithms are used to efficiently store sparse matrices in computer memory and invert them. The solution in terms of the vector potential is particularly well adapted to the 2D case where A has only one component different from zero. However, in 3D, one must deal with each of the three components of A separately. To decrease the dimensionality of the set of linear equations, one usually reformulates the energy in terms of the scalar potential as follows:

  H  E= B dH d. + B n[ ˆ dV (26) .

0

V

where V is the outer surface surrounding the volume . and nˆ is a unit vector orthogonal to the surface V. [ is the scalar potential from which the field is derived:  H = H cur − ∇[

(27)

where H cur is the field produced by current sources alone. H cur is computed by direct integration of the Biot and Savart law (see Section 3.3.1). One then derives a set of linear equations by expressing ∂E/∂[ = 0 at each node. Once the sparse matrix is inverted, the vector or scalar potential is known at each node and one determines the magnetic field at any point using some interpolation. In problems involving non-linear materials such as soft magnetic material close to saturation, one operates by iteration using, for example, a Newton– Raphson method. At each iteration a part of the matrix is reconstructed assuming a material which behaves linearly around the field strength H obtained from the previous iteration. 3.3

Integral method

Another popular numerical method to compute the magnetic field created by an ensemble of permanent magnets, iron pieces and coils is based on an integral approach. In the simplest form of the integral approach, all magnets and iron pieces are subdivided into a sufficient number of small volumes over which the magnetization is assumed to be uniform. The field created by each uniformly magnetized volume at any arbitrary point is computed analytically. The magnetization in the iron is determined as a result of an iterative process. 3.3.1

Currents

The magnetic field created by a current circulating in a conductor can be computed using the Biot–Savart law:  dl × (r − r )   B(r ) = µ0 J dS (28) |r − r |3

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 where r is a point running along the condutor, J is the current  per unit surface, dI is a longitudinal integration along the direction of the current and dS is a 2D transverse integration made over the cross section of the conductor. Equation (28) can be computed by means of a direct numerical integration. However, for a simple geometry, the integration can be performed analytically. This is the case of the parallellepipedic conductor with uniform current distribution [19]. In the important case of a circular arc of conductor with rectangular cross section, the integration over the cross section can be performed analytically and the longitudinal integration along the arc is made numerically. Alternatively, one can treat the arc as a succession of linear segments for which a 3D analytical integration can be performed [20]. Any coil with rectangular cross section can usually be considered as an assembly of straight and curved segments with identical cross sections. A coil with a non rectangular cross section can be approximated by an assembly of coils with smaller rectangular cross section. 3.3.2

Field from a uniformly magnetized volume

3.3.2.1 INTRODUCTION

Very generally, for any shape, the 3D magnetic field H = (Hx , Hz , Hs ) produced by a  = (Mx , Mz , Ms ) at a point P can be uniformly magnetized volume with magnetization M written in the matrix form:   Qxx Qxz Qxs  with Q =  Qzx Qzz Qzs  H = QM (29) Qsx Qsz Qss The matrix depends on the shape of the volume and its position with respect to the point P where the field H is computed. For a uniformly magnetized volume, Q is always symmetric and satisfies  0 if P is outside the volume Tr(Q) = Qxx + Qzz + Qss = (30) −1 if P is inside the volume 3.3.2.2 PARALLELLEPIPEDIC VOLUME

In the simple case of a magnet with parallelepipedic shape, analytic expessions of Q are derived by means of a 3D integration of a dipole field over the volume of the block. The result is 
2 zj sk 2 1  i+j +k+1 2 2 1/2 Qxx = (31) (−1) atan (xi + zj + sk ) 4π xi i,j,k=1

Qxz = Qzx

  2 2 1 i+j +k  log  = (sk + (xi2 + zj2 + sk2 )1/2 )(−1) 4π

(32)

i,j,k=1

with x1,2 = −xP ∓ wx /2 z1,2 = −zP ∓ wz /2 s1,2 = −sP ∓ ws /2

(33)

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z wx ws wz O

x

s

Figure 5.6 Parallelepipedic magnet block.

where (xP , zP , sP ) is the position of the observation point P assuming that the centre of the parallelepiped is placed at the origin of the Oxzs frame. (wx , wz , ws ) are the dimensions of the parallelepiped with respect to the Ox, Oz and Os axis (see Figure 5.6). Qzz is deduced from Qxx by the following circular permutation of the coordinates: xi → zj , zj → sk and sk → xi . Similarly, Qss , Qzs = Qsz and Qsz = Qzs are deduced from Qzz , Qxz and Qzx using the same permutation. The field integral I(x, z) defined in Eqn (23) can also be written in the matrix form:   Gxx Gzx 0  with G =  Gxz Gzz 0 I = GM (34) 0 0 0 For a uniformly magnetized volume, G is symmetrical. In the particular case of a uniformly magnetized parallelepipedic source with Os parallel to one face of the parallellepiped, G can be derived analytically:

 2 2 zj ws  xi ws  i+j i+j Gxx = Gzz = (−1) atan (−1) atan 2π zj 2π xi i,j =1 j =1   (x12 + z22 )(x22 + z12 ) ws Gxz = Gzx = log 2π (x12 + z12 )(x22 + z22 )

(35)

where x1,2 and z1,2 are given by Eqn (33), the coordinates xP and zP being the coordinates of any point of the line over which the field integration is carried out. 3.3.3

Generalization

So far we have restricted the integration direction to be parallel to one face of the parallellepiped but one can also obtain analytical expressions for G in the more general case of arbitrary orientation of the parallellepiped with respect to the integration axis [21]. Expressions for the matrix Q and G can be derived analytically for an arbitrary polyhedron shape [22]. A polyhedron is a volume bounded on all sides by planar polygons. Such expressions together with all those mentioned in Section 3.3 can be derived more easily using the symbolic computation facilities built in the Mathematica software [23]. Since most

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permanent magnet materials of interest have a relative permeability close to 1, the magnetization M can, in a first approximation, be assimilated into the remanent field of the magnet. As a result, the 3D field created by an ensemble of permanent magnets can be computed very simply and quickly by summing over the contribution from each magnet volume. With modern personal computers, one typically computes the three components of the field produced by a single parallellepipedic magnet within 10–25 µs. 3.3.4

Soft materials

In any material, the magnetization at any point is a function of the local field through the  H ). The local field can be computed as the sum of contributions from material property M( all other currents and magnets. Let n be the total number of small volumes and let H i and  i be the average field and magnetization in the volume i. Assuming the volume to be small M enough for the magnetization to be almost uniform, and approximating H i by the field in the geometrical centre of volume i. H i is a solution of H i =

n 

 j (Hj ) + H i,cur Qij M

(36)

j =1

where H i,cur is the field created by all currents in the centre of the volume i and Qij is a 3 × 3 matrix which determines the field contribution to the centre of volume i originating from the magnetization in volume j . We shall first assume a linearity of the material property  j (H j ) in all volumes j , namely, M  j (H j ) = M

Rj + χj Hj µ0

(37)

Such a material definition can be used to describe either a linear soft material or a linear permanent magnet. For a soft material R j = 0 and χj is a diagonal 3 × 3 matrix with all coefficients in the diagonal equal to the susceptibility. For a permanent magnet material, R j is the remanent field of the material constituting the volume j and χj is a 3×3 matrix describing the linear susceptibility. Permanent magnet materials are anisotropic and one diagonalizes χj according to   0 0 χj, χj,⊥ 0  χj =  0 (38) 0 0 χj,⊥ Where χj, (χj,⊥ ) is the linear susceptibility parallel (perpendicular) to the direction of the remanent field R j . Substituting Eqn (37) in Eqn (36), one obtains H i =

n  j =1

Qij χj H j +

n  j =1

Qij

R j + H i,cur µ0

(39)

 i in the volume i can be From Eqn (39), the field H i and therefore the magnetization M obtained:     n n   R R R j  i = j + χi H i = j + χi   + H k,cur  M (40) -ik  Qkj µ0 µ0 µ0 k=1

j =1

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where (-ik ) with i, k = 1, . . . , n is a 3n × 3n matrix which is the inverse of the matrix  i is known (δik − Qik χk ). (δik ) represents the identity matrix. Once the magnetization M in each volume i, the magnetic field at any point in space is obtained from Eqn (29). It is clear from Eqn (40) that the sources of fields are either the currents or the remanent field of the permanent magnets. Equation (40) is easy to implement numerically but, unfortunately, the matrix (Qik ) and therefore (-ik ) are full and for or a few hundreds volumes, the matrix inversion becomes impractical. Another method of solving is to iterate Eqn (39) many times starting from H i = 0. One stops the iteration when H i becomes stationary. The process only converges if the susceptibility is small compared to 1 which is only suitable for permanent magnet materials but useless for any type of soft material with a large susceptibility. One can also use a relaxation method which consists in iterating a modified version of Eqn (39) which has the same stationary solution but for which convergence takes place for any value of the susceptibility. Such a method has been derived in [21] for both linear and non-linear materials. 3.3.5

Discussion

To our knowledge, the first computer code making use of the integral approach to solve problems which includes iron is GFUN-3D [24]. It was written in the 1970s and, for sometime, it was the only 3D magnetostatic code available and was reported to be successful. Nevertheless, the development of GFUN-3D suffered from a lack of computer resources (memory and CPU) available at the time it was written and from the implementation of a matrix inversion rather than relaxation which severely limits the number of volumes. Since then, finite element codes have been massively and successfully developed. Some mixed finite element/integral methods have been developed [8]. The method described above is nowadays called a volume integral method. Most finite element codes can handle a few hundred thousand elements while codes based on the integral approach hardly use more than one or a few thousand elements. The integral approach has been somewhat eclipsed and is essentially used for iron-free structures made of parallellepipedic volumes of permanent magnets. Recently, the computer code Radia [21,22,25] was written at the ESRF. It implements the integral approach with a relaxation method. It accepts volumes with polyhedron shapes and non-linear materials. It makes use of the mirroring technique to simulate planar boundary conditions in order to save memory. Taking the example of a simple 1 T hybrid wiggler with seven poles with its extremities, Radia solves the peak field with a 1% absolute precision within five seconds of CPU time on a high end personal computer. For field geometry largely opened towards infinity, the integral method is significantly more efficient that the finite element method. This comes from the fact that in the integral approach one only needs to subdivide the iron volumes and to a lower extent the permanent magnets’ volumes while with finite elements, one must segment the whole space including the air which may represent an enormous number of elements if the region is not bounded. Moreover, for a given precision, one requires much less sub-volumes of iron in the integral approach as one needs with finite elements. Ratios higher than 100 are frequent. The integral methods do not suffer from the truncation at infinity; in particular, the field integrals are accurately computed by summing over the contributions from each small volume as described in Eqns (34) and (35). The main limitations of the integral approach are the memory and CPU time which grow proportionally to the square of the number of elements. The matrix Qij occupies more than 72 MB of memory for 1000 volumes. Another limitation is the discontinuity of the field when crossing the boundary between two volumes of soft material. This limitation can, in principle, be removed by applying some kind of interpolation.

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Due to the large number of parameters influencing the precision of the computation and the large difference in the definition and solving of any problem, it is difficult to make a fair comparison between the two approaches. In addition, the criteria of comparison should take into account the pre- and post-processing capabilities and the quality of the user interface of the code which largely determine the time required to build, debug and solve a problem. This time can be much longer than the solving time. Both approaches are probably suitable in many situations. However, in the particular case of the design of the extremities of a permanent magnet insertion device, we have found that the integral approach is dramatically more precise and, therefore, quicker for estimating the field integrals. Indeed, finite element and integral methods are complementary. Finite elements are particularly well suited to bounded geometries while integral methods are more efficient for open geometries. The usefulness of a computer code does not only have to deal with the efficiency of the numerical method but also with the friendliness of the user interface. In this respect a user friendly code based on the integral method is much easier to write than a finite element code. A detailed comparison of the advantages and drawbacks of each method is out of the scope of this chapter. For further reading on this topic one can look at [26] and the references therein.

4

Magnetic design of permanent magnet insertion devices

There exist, essentially, two kinds of magnetic structures used to build a permanent magnet insertion device with a periodic vertical field. They are the pure permanent magnets (PPM) and the hybrid (HYB) structures. During the design phase one faces two important issues, namely the central field which is periodic and the termination where the periodicity is broken. We shall discuss both structures. 4.1 4.1.1

Periodic part Description

Figure 5.7 presents a 3D view of the commonly used PPM and HYB periodic structures. The PPM structure is made of four magnets per period with magnetization parallel to one face of the magnet and rotated by 90◦ from one magnet to the next. The HYB structure can be viewed as a PPM structure in which the vertically magnetized magnet has been replaced by a narrow piece of high permeability steel that we shall call the pole. In the HYB structure the field seen by the electron beam comes from both the magnet and the magnetization of the pole. The pole is magnetized under the action of the field of the permanent magnet blocks. For a given size of the magnet blocks, one usually determines the optimum dimensions of the poles by maximizing the peak field in the middle of the magnetic gap. This optimization results in a narrow pole with horizontal and vertical dimensions slightly smaller than those of the magnet blocks as shown in Figure 5.7. 4.1.2

Magnetic field

The magnetic field produced by a hybrid structure is slightly larger than that produced by a pure permanent magnet structure. Figure 5.8 presents a comparison of the peak magnetic field and its first harmonic of the Fourier decomposition for a pure permanent magnet (four magnets/period), a hybrid structure with poles of ARMCO iron (inexpensive) and a hybrid structure with poles made of Vanadium Permendur (highest performance and cost). In all

HYB

PPM

Magnet (NdFeB, Sm2Co17,...)

Pole (Steel)

Figure 5.7 Schematic of the periodic part of a Hybrid (HYB) and Pure Permanent Magnet (PPM) insertion device creating a vertical magnetic field. The electron beam propagates in the middle of the magnetic gap between the upper and the lower magnet arrays.

0.85

Magnetic field (T)

0.80

Hybrid (Vanadium Permendur)

0.75

Hybrid (ARMCO Iron)

0.70 Gap/0 = 0.314 Lx magnet = 2 0 First harmonic Peak field

0.65

2

3

Pure permanent magnets

4 Magnet volume (N03 )

5

6

Figure 5.8 Peak and first harmonic of the field as a function of the magnet volume for a PPM structure, a HYB structure with pole made of ARMCO iron and a HYB structure with pole made of Vanadium Permendur. In all cases the width of the magnet is equal to twice the period and the ratio of the gap to the period is 0.314. For each volume of magnet, all free parameters are optimized to maximize the peak field.

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three cases, the horizontal size of the magnet blocks was kept equal to twice the period λ0 and the ratio of gap over period is equal to 0.314. The field is expressed as a function of the total volume of magnets normalized to the product of the number of periods N times the cube of the period. For a given volume of the magnets, the remaining free parameters (height, width and thickness of the pole, height of the magnet) are re-optimized each time to produce the highest peak field. The magnet material is NdFeB with 1.1 T remanent field with the relative permeability given in Table 5.1. The field produced by the pure permanent magnet device reaches a limit which is close to the prediction made using Eqn (14). Besides harmonic 1, the next contribution comes from harmonic 5, which is, however, low. As a consequence the peak field can be assimilated to the field of the first harmonic. In the hybrid structure, the field grows continuously as one increases the volume of the magnet. For a large magnet volume, the peak field differs from the first harmonic by a few per cent due to the presence of harmonic 3. Figure 5.9 presents the variation of the field as a function of the ratio of the gap over period assuming a volume of magnet per period equal to 2λ30 . The permanent magnet material is the same as in Figure 5.8. For a given gap, the longer the period, the larger the field difference between the pure permanent magnet structure and the hybrid structure. The difference is more pronounced on the peak field than on the first harmonic. One should mention that the field of the hybrid structure can be further increased by a few per cent by using wedge poles [27] or by adding small magnet blocks on the side of the poles. The choice of a horizontal width of the magnet equal to twice the period was made to ensure that the field is close to the maximum available for infinitely wide magnet blocks. However, this usually results in a very massive and expensive structure requiring a lot of magnetic material. Many designers optimize their hybrid structures with a magnet width somewhere between λ0 and 2λ0 , resulting in a negligible reduction of the field.

2

Magnet volume = 2 N03 Lx Magnet = 2 0 First harmonic Peak field

Hybrid (Vanadium Permendur) Magnetic field (T)

1

9 8 7 6 5 4

Pure permanent magnet

3

2

0.1 0.2

0.4

0.6 Gap/0

0.8

1.0

Figure 5.9 Peak and first harmonic of the field as a function of the ratio of the magnetic gap to period for a pure permanent magnet structure and a hybrid structure with pole made of Vanadium Permendur. In both cases the horizontal width of the magnets is equal to twice the period, and the volume of magnet per period is equal to 2λ30 .

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Field termination

4.2.1

Generalities

A very large majority of insertion devices are such that the field within a single period only contains odd harmonics, the fundamental being the dominant one. In such devices the shape of the field profile where the field is negative is identical to that where the field is positive. As a result, the horizontal and vertical field integrated over a single period is zero. At the termination, the periodicity is broken and some field integral may be generated which depends on the magnetic gap. The main difficulty in the design of a field termination is the control of these field integrals. In almost all situations, the design of the field terminations is a full 3D magnetostatic problem. A number of designs of undulator or wiggler terminations have been published and implemented, see [28–32]. We shall develop this topic in further detail and try to present a methodology applicable in a large range of situations for implementing field integral free terminations. It is important to mention that if one builds an insertion device in such a way that the field of the upstream and downstream terminations are of opposite signs, their field integral contributions cancel. Nevertheless, a non-zero double field integral is usually observed which also induces some distortion of the closed orbit in the whole of the storage ring’s circumference. In addition, such a variation of the second field integral induces a modification of the direction of emission of the radiation which can be a problem for an undulator beamline where the light is collected over a very small aperture. For both these reasons, it is preferable to correct the field integral variations with gap separately for both ends. The correction can be made with coils localized in the terminations and powered by currents. This correction scheme, known as the active correction method, is fairly straightforward; however, it complicates the control system as it is prone to failures and may generate some heat. For these reasons, the passive correction scheme, where the field integral variations with gap are corrected or minimized at the design level, is preferred by most designers. This has been made possible with the availability of 3D magnetostatic software. If one uses segmented undulators (see Section 7.3), one needs a special magnetic design of the ends to ensure a proper phasing of the radiation produced in the adjacent undulator segments. This brings additional complexity to the design. A large number of different terminations have been designed for the ESRF insertion devices. In the course of this work, we have found it very useful to split the field integral of a complete insertion device as follows: I = Iup + Idown + Iint

(41)

in which the vector I = (Ix , Iz ) holds the horizontal (Ix ) and the vertical (Iz ) field inte Iup , Idown and Iint are computed or measured gral components defined by Eqn (23). I, along an axis parallel to the magnet array in the middle of the magnetic gap g. They essentially depend on g but their dependence on the horizontal position of the beam may also be of interest. Iup (Idown ) is obtained by opening the gap to a large value and measuring separately the field integrals at a distance g/2 of the upper (lower) magnet array. In other words, Iup (Idown ) is the field integral contribution of the upper (lower) magnet array assuming the lower (upper) magnet array is removed. Iint is called the interaction field integral and Eqn (41) can be considered as its definition. It is generated through the following process. The upper magnet array produces a magnetic field on the lower magnet array. The magnetization of the magnet blocks of the lower magnet array is modified

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due to this additional field. This additional magnetization produces an additional field integral to the electron beam. In principle, one should repeat this process many times between the upper and lower magnet array. For pure permanent magnet undulators, Iint is usually rather low to the point that little attention has been paid to it by most designers, even though, for some NdFeB undulators, it may result in field integral variations out of tolerance. Iint can be very large for hybrid devices and presents a significant variation with the magnetic gap. Whereas both Iup and Idown are rather sensitive to magnetization, machining and positioning errors of the magnet blocks and poles, Iint is largely insensitive to these errors and is, therefore, very difficult to correct by a shimming process. The only efficient way to compensate Iint is to use an active correction. To conclude, let us say that the goal of the designer is to optimize the dimensions and positioning of the magnet and pole of the end structure to minimize the variations of Iint vs gap as much as possible.

4.2.2

Pure permanent magnet structures

Figure 5.10 presents three different designs built at the ESRF to terminate a pure permanent magnet undulator segment. Only the lower magnet array is presented. Structure A is the classical and simplest termination in which the last magnet block is vertically magnetized and its longitudinal dimension is equal to one half of the other blocks

A

B

Additional end H block

C

Blocks with tilted magnetization

Figure 5.10 Three magnet designs of the termination of a PPM undulator. Design A is the simplest. Design B corrects the field integral variations vs gap. Design C corrects both the field integral and phase variations with gap.

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(λ0 /8). This structure would give no contribution to Iup , Idown and Iint if the relative permeabilities µr, and µr,⊥ of the magnets were equal to zero. In the case of NdFeB magnets, the effect of the permeabilities results in field integrals of typically 10–100 G cm per extremity, depending on the gap and period. Structure B is derived from A by adding a horizontally magnetized block of a longitudinal dimension that is approximately equal to 3λ0 /20. In this structure the interaction is reduced resulting in a field integral variation with a gap four times smaller than that of A. The optimum dimension of the last magnet block depends on the choice of material (through the permeabilities) and on the period of the structure. This structure has another advantage: the trajectory in the undulator presents a negligible horizontal offset upon crossing the end of the structure. The only problem with this structure is that it does not allow a proper phasing of two adjacent undulator segments. Structure B has been used at the ESRF for undulator periods between 20 and 70 mm. Structure C is the most sophisticated. Some magnets are magnetized at an angle in the 30–45◦ range with respect to the horizontal plane. This structure simultaneously corrects the field integral variations vs the gap and the phasing [33]. The phase variations with gap are minimized for a distance between the undulator segments equal to a few millimetres thereby reducing any possible mechanical interaction between the segments. Structure C has been used at the ESRF for undulator periods between 20 and 42 mm. To illustrate this discussion, Figure 5.11 presents some computations (and field measurements) performed with the RADIA code on the three types of end structures A, B and C for a 40 mm period undulator. The nominal height of the blocks is 20 mm and the horizontal width is 55 mm. The magnets are made of NdFeB with a remanent field of 1.15 T. The permeabilities are those given in Table 5.1. The various plots present the vertical field integral and double field integral predicted for a single end section as a function of the magnetic gap. For a full undulator, one must take into account both extremities. For a symmetric (antisymmetric) configuration of the end structures, where the sign of the field is the same (inverted) at each end, the field integrals are twice those of Figure 5.11 (zero) and the double field integrals are zero (twice those of Figure 5.11). Clearly structures B and C give a much lower field integral variation with gap than structure A. This is even more striking on the plot of the interaction vs the gap. Note the strong dependence of the interaction on the gap. Figure 5.12 presents the phase shift on the fundamental vs the gap induced at the junction between two undulators equipped with end structures of types A and C. The expression of the phase shift is easily derived from Eqn (99) of Chapter 3. δs is the longitudinal air gap between the last magnets of each segment. Note that, to minimize the phase shift, structure A requires a distance δs as small as possible – the value of 0.5 mm has been selected as a reasonably small value which still allows an independent tuning of the gap in each segment. Structure C was optimized in order to minimize the phase shift around an air gap δs = 5.6 mm. As one increases the air gap δs, the phase shift increases. Structure A (C) gives an incremental phase shift variation with gap of 0.7◦ /0.1 mm(2◦ /0.1 mm) within the most useful gap range of 14–25 mm which makes structure C more tolerant to positioning errors. The phase shift is responsible for a reduced angular spectral flux and brilliance. The higher the harmonic, the more important the reduction. Compared to an ideal undulator of the same total length, the spectrum from a two segment undulator equipped with end structure A (C) presents some deviation at harmonics 7 (15) and higher. In addition, structure C presents the advantage of generating a zero phase shift at the minimum gap where the spectrum is rich, with many harmonics.

J. Chavanne and P. Elleaume

5 0 –5 –10 –15 –20 –25 –30 –35 –40

B

10 Field integral (G cm)

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C

A Lower magnet jaw

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10 × 103 Double field integral (G cm2)

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B 40

80 120 Gap (mm)

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8

A Trajectory offset

6 4

C

2

B

0 40

80 120 Gap (mm)

160

200

Figure 5.11 Vertical field integrals and double field integrals vs gap induced by one end structure of type A, B or C as defined in Figure 5.10. The undulator period is 40 mm. The upper left graph presents the field integrals produced by a magnet’s single jaw. The upper right graph presents the computation for a fully assembled structure, together with the result of a measurement. The crosses are the measurement on a real device. The lower left curve presents the interaction’s vertical field integral (as defined in Section 4.2.1) and the lower right curve presents the field integral which is proportional to the displacement of the average trajectory inside the undulator.

4.2.3

Hybrid structures

The proper design of the end section is more delicate for a hybrid device than for a pure permanent magnet. The main reason is the high permeability of the steel constituting the pole which results in a large cpu time during field computation. A few years ago, hardly anybody would try to make a 3D simulation of the end section of a hybrid device. One systematically implemented an active correction. Passive correction of hybrid devices is now possible if one uses a 3D code and a few hybrid undulators and wigglers have been implemented with passive correction. One must, nevertheless, stress the fact that the field integrals produced by a hybrid device are sensitive to the ambient field. More precisely, the ambient field which is generated by the earth, ion pumps, nearby iron structures etc. induces some magnetization in the poles of the undulator. This extra magnetization produces some extra field integral on the electron beam path which depends on the gap. For this reason, one may want to keep an active correction scheme running even if the extremities have been correctly designed and tested in the magnetic measurement laboratory. The problem is that the ambient field in the measuring laboratory may be different from the one present in the storage ring tunnel.

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2 C : s = 5.6 mm

Phase shift (deg)

0 –2 –4 –6 –8

A : s = 0.5 mm

–10 –12 20

30

40

50

60 70 Gap (mm)

80

90

100

Figure 5.12 Phase shift on the fundamental induced at the junction of two undulator segments equipped with extremities of types A and C. δs is the longitudinal air gap between the last magnets of each segment.

End pole

End magnet A

End magnet B Air gaps

Figure 5.13 Field termination of a hybrid undulator. The last magnet and poles have different thicknesses and are spaced with air gaps. The optimum thickness and air gap are obtained by minimizing the vertical field integral interaction with gap.

Figure 5.13 presents a termination of a simple hybrid undulator implemented at the ESRF. It is optimized for minimizing the field integral vs gap. The longitudinal thickness and the air gap between the last magnet and poles are the free parameters of the optimization. The optimum depends essentially on the undulator period and horizontal width of the magnets and poles. A tentative step has been taken towards implementing simultaneously the phasing and the field integral correction on a hybrid undulator [34]. The final performances are not as satisfying as those of structure C in Figure 5.10 The terminations shown in Figure 5.13 have been used at the ESRF for hybrid devices with periods ranging from 23 to 70 mm. The results obtained on a 34 mm period undulator are shown in Figure 5.14. In this undulator the horizontal × vertical × longitudinal dimensions

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Double field integral (G cm2)

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Full structure

25 20

Lower jaw

15 10

Measurement

5 0 –5

Interaction 40

80

120 Gap (mm)

160

200

900 800 700 600 500 400 300 200 100 0 40

80

120 Gap (mm)

160

200

Figure 5.14 Simple and double field integral computed for a 34 mm period hybrid undulator terminated as shown in Figure 5.13. The dotted curve is the result of a measurement of the interaction on a real device following a multipole shimming.

of the magnets (poles) used in the periodic part are 65 × 35 × 13 (45 × 25 × 4) mm. The end magnet B (end pole) shown in Figure 5.13 has a reduced thickness of 7.7 mm (2.4 mm). The air gaps on both sides of the last pole are set to 3.5 mm. The pole is made of low carbon steel. The magnets are made of NdFeB with a remanent field of 1.15 T. The permeabilities are those given in Table 5.1. As seen in Figure 5.14, the full structure is expected to have a maximum field integral variation per extremity of 30 G cm with a very low interaction. During the manufacture of the real device, some field errors were encountered and multipole shimming was applied (see Section 6.2). It was possible, by shimming, to reduce the field integral to a much lower value than that predicted for the full structure. In other words, because of the low interaction, the final field integrals could be corrected by shimming to a lower value than those predicted by the 3D magnetostatic computation (made without shims). 4.3

Comparison between the PPM and HYB structures

The advantages of the PPM type are the simple field computation for both the central part and the extremities and the easy implementation of end sections which are both phase and field integral compensated for any gap value. This gives the possibility of segmenting very long undulators. The cost of a PPM type undulator is also slightly less and finally the field integrals are insensitive to any variation in the ambient field. The main advantage of the HYB type undulator or wiggler is the higher peak field that can be reached for a given gap and period (see Section 4.1.2). This is of particular importance for high field wigglers with long periods where a 30% extra peak field can be reached. The peak field advantage is reduced (5–10%) for a low or medium field undulator. In addition, the higher peak field usually comes from harmonics 3 and 5 which present a reduced interest for the radiation. In the past, some people have reported larger field errors from PPM type undulators compared to HYB types. A detailed comparison is difficult as different types of errors appear in each device and the importance of the errors depends on the manufacturing process of the magnet blocks. The experience at the ESRF is that if one uses highly uniform magnet blocks, no significant difference in field errors is seen between the HYB and the PPM type undulators before shimming. In all cases, a multipole shimming would further remove any difference. In view of this, the strategy followed at the ESRF is to manufacture high field wigglers of type HYB and undulators of type PPM. The lower field of the PPM undulators compared to

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that of the HYB type undulator can easily be compensated by slightly increasing the period by 2–5% which has negligible consequences on the brilliance and flux.

5

Magnetic field measurement

The techniques used for magnetic field measurement of conventional accelerator magnets such as dipole, quadrupole and sextupole magnets have been addressed at a number of workshops and in papers [35,36]. Because of the peculiar nature of the undulator field, the technique must be slightly modified to apply to insertion devices. The main differences between insertion devices and conventional magnets are the small gap, great length and the field shape with large longitudinal gradient. The most important field measurement is probably the field mapping along a straight line as close as possible to the electron trajectory. The field integrals can be deduced from a numerical integration of the field measured along this line. For a number of reasons, this method is slow and inaccurate. In many laboratories, people use special measuring benches dedicated to the precise measurement of the field integrals. Section 5.1 describes the local field measurement technique and Section 5.2 describes the field integral measurement technique. 5.1 5.1.1

Local field measurement Mechanical set-up

Figure 5.15 presents a view of a measuring bench used at the ESRF to sample the magnetic field of an insertion device. It is a three-axis bench equipped with a Hall probe sensor. The longitudinal range of measurement must be at least equal to the length of the insertion device plus 0.5–1 m to account for the fringe field. The measurement range is typically between 2.5 and 6 m. Such a bench is suitable for measuring most insertion devices excluding certain exotic devices which do not provide any lateral access (see Section 5.1.3). For insertion devices made of NdFeB magnet, the longitudinal gradient dBz /ds may reach 75 (150) T/m at a magnetic gap of 20 (10) mm. Consequently, a longitudinal mispositioning of 1 µm results in a field error of 0.75 (1.5) G. A high level of accuracy is, therefore, required in the positioning of the longitudinal axis. Depending on the field geometry, one or all three components of the field must be sampled every 0.3 mm (for small gap and small period devices) to a few millimetres for high field wigglers. Assuming a typical 1 mm sampling distance and a 6 m measurement length results in 6000 data points. The stop and go type of measurement, where the sensor is moved by 1 mm, stops, waits for the field to be measured and then starts a new step, is inadequate. The reason for this is the significant time required for the correct damping of the vibrations induced on the sensor during each acceleration and deceleration. One prefers the on-the-fly method for which the sensor is moved at an almost constant velocity while the field and the longitudinal position of the sensor are continuously recorded. The longitudinal position can either be read from an optical encoder or a laser interferometer. With a typical speed of 10 mm/s, a 6 m long undulator is measured in 10 min. A higher speed is possible but ultimately the measured field loses precision due to both the vibrations and the low signal-to-noise ratio generated by the short measuring time of the Hall voltage. By measuring on-the-fly, one shortens the measuring time to the point where the Hall probes and the associated current power supplies do not need any stringent temperature and current stability over a long timescale. Any temperature variation between two scans results in a small relative difference of both the peak field and the numerically computed field

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z-axis

Hall sensors Spindle

x-axis

s-axis

Guide rail

Tunable wedges

Linear encoder (optical ruler)

Figure 5.15 View of a Hall probe bench dedicated to the field measurement of undulators and wigglers.

integral (0.1%/degree for NdFeB). For long undulators, one nevertheless needs to minimize any temperature gradient along the structure. For this, a simple air conditioning system is adequate. The longitudinal motion is usually made with a DC or stepper motor coupled to a long ball bearing screw. For a very long measuring range, a belt or a linear motor are preferable. Around the central axis of a conventional insertion device, the transverse gradients dBz /dx and dBz /dz are small, as a result a 0.01 to 100 mm accuracy is usually sufficient for both the horizontal and vertical translation stages. Helical undulators may require more precision. 5.1.2

Magnetic field sensors

The most frequently used sensor for magnetic field measurement is the Hall probe. It is essentially a small piece of semiconductor material such as InAs or GaAs in which one makes use of the property that the transport of the charges is modified by an external magnetic field. More precisely, the electric field E in the material is related to the current density j and the magnetic field B by the following relation [37]:      E = ρ j + C1 j × B + C2 B B · j + C3 jB 2 + C4 T j + o B 2

(42)

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where ρ is the resistivity of the semiconductor. ρ j determines the dissipation by the Joule  is the main Hall contribution which is responsible for an electric effect and ρ(C1 j × B) field in the plane orthogonal to both the current and magnetic field. The term proportional to C2 is the planar Hall effect. The term proportional to C3 is known as the magnetic resistance. 2 δ along the crystallographic axis. o(B 2 ) represents some T is a tensor defined by Tk1 = Bk1 k1 terms containing the magnetic field with a power higher than 2. The Hall voltage V measured   by a voltmeter is the electric field integrated over the thickness of the crystal: V = E dl.  All Following the Joule term ρ j, the next most important term is the Hall term C1 j × B. other terms in Eqn (42) are small perturbations. They depend on the temperature and, for a very precise measurement of the absolute field, one must regulate and/or compensate for the temperature variations. Let us assume that the current I is directed along x and the collecting electrodes which measure the voltage V are sensitive to the electric field along s, Eqn (42) reduces to V = kρI (C1 Bz + C3 Bs Bx + · · · )

(43)

where k is a constant related to the dimensions of the crystal. We assume here that the axes of the crystal are directed along x, z and s. The voltage is proportional to the current in the crystal. In practice kρC1 and the other coefficient which relates the measured voltage to the various powers of the Bz component are measured by calibrating the Hall probe with respect to an NMR probe in a uniform field. The planar Hall contribution kρI C3 Bs Bx is more difficult to measure. This may induce serious perturbations in the interpretation of the field measurement especially in the determination of the horizontal field component of a vertical field undulator. In a vertical field undulator, the horizontal field integral deduced from a numerical integration of the local field values may be largely offset due to the planar Hall perturbation. This problem is particularly severe for a helical undulator. At the ESRF, the problem has been addressed by both measuring (and compensating for) the planar Hall coefficient by means of the undulator field itself which is known from design and by selecting probes which present a small planar Hall effect. Large variations of the planar Hall coefficient inside the same batch of probes from a given manufacturer have been observed. This may come from a small difference in the shape or connection of the collecting electrodes around the crystal. Besides the linearity and the planar Hall coefficient, the other figure of merit of a Hall probe is its voltage/field characteristics. For a given field, the higher the voltage the less sensitive the voltage measurement is to the various sources of noise. Finally, probes with small dimensions are of interest for very short period undulators. To summarize, the Hall probe is the easiest to use; however, it suffers from non-linearities, planar Hall effect and temperature variations. Another alternative is to use a very small coil made of many turns of a wire of small diameter. Such a coil is connected to a voltmeter and the time integrated voltage and the displacement of the coil are recorded simultaneously. The field component B along the axis of the coil is deduced from the effective surface S of the coil and the time integrated voltage V (t) using
 t2 3 B= V (t) dt S (44) t1

The advantage of this method over the Hall probe type is its linear response. It suffers from the perturbation induced by any small voltage (thermal, inductive etc.) induced in the connections of the coil to the voltmeter. It can be minimized by operating a large surface coil which requires a very small diameter of conductor to minimize the volume. The effective surface S can be

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calibrated against NMR by rotating the coil in a uniform magnetic field. Another difficulty is in the timing and triggering of the voltmeter’s electronics. The voltmeter internal noise is ultimately the limiting parameter. Recently, such a coil has been used to measure the field from a 7 T superconducting wiggler [38] and for a 3.6 T permanent magnet wiggler with a 6 mm magnetic gap [39]. We believe that such methods have a high potential and are worth developing further. A variant of the measurement has been used by a few groups which consists in integrating the voltage over a full period of the undulator. This reduces the peak voltage and, depending on the type of voltmeter being used, may improve the precision. The drawback is that it smooths out the field. 5.1.3

Special techniques

As discussed in the previous section, the most widely used technique for field measurement is scanning a Hall probe inside the field. For some insertion devices, where access is limited, such as some helical undulators with magnets occupying all sides of the transverse plane or for very long undulators, this technique is impractal or too expensive. Alternatives are the pulsed stretched wire [40] and vibrating wire methods [41]. For the pulsed wire method, one stretches a single conductor made of CuBe inside the gap of the device under measurement. A short pulse or step of current is sent to the wire. Under the combined action of the current and magnetic field, the wire is distorted transversally and some mechanical vibrations are generated which propagate along the wire. The measurements of these oscillations of the order of one micron are made outside the undulator gap using highly sensitive sensors. The main advantage of this method is its simplicity, low cost and minimum geometrical constraints. In addition, both the local field and field integrals can be measured directly by sending either a pulse or a step of current. The most serious drawback is the lack of linearity in the dynamic range of measurement of the detector. A 1 G precision over a 1 T field requires a precision and linearity of 0.01% which is difficult to achieve in view of the small wire displacement. Other difficulties in the method include the natural dispersion of the wave propagation and sagging of the wire under the gravitation field. It is interesting to note that the sagging of the wire can be indirectly measured by recording the fundamental frequency, F1 , of vibration of the wire which is related to the vertical profile z(s) of the wire through the relations [41,42]  s g 1 z(s) = cos π 32 F12 L

F12 =

T 4µL2

(45)

where g = 9.81 m/s2 is the gravitational acceleration, L is the length, T the tension and µ the density of the wire. The origin of the s coordinate is taken from the centre of the wire. The higher the tension, the high the frequency F1 , the lower the sag. When the ratio of the tension over the section of the wire exceeds the yield stress the wire lengthens irreversibly or breaks. One can monitor the frequency F1 to quantitatively follow the mechanical status of the wire. In this respect one of the best materials for minimizing the sag is the BeCu alloy which gives, at the breaking threshold, a sag of 0.5 mm for a 6.0 m long wire. Even though this method has been found suitable for measuring field integrals with reasonable accuracy, the precision is not sufficient to correctly predict the undulator spectrum from the local field measurement [43]. The vibrating wire method is a variant in which one measures the frequency response around each resonance mode of oscillation of the wire. This can be done with a shorter wire; however, it requires a longer acquisition time.

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Field integral measurement

5.2.1

Field integrals

A typical set-up used at the ESRF to measure the transverse field integral produced by a wiggler or an undulator is shown in Figure 5.16. It is made up of a pair of manipulators between which some coil or wire is stretched. The insertion device under measurement is placed between the manipulators, the coil or wire is then aligned on the electron beam path in the magnetic gap. The manipulators are equipped with three linear stages allowing a translation of the wire along the x-, z- and s-axis. If a coil is used (as shown in Figure 5.16), one adds some rotation stage with the axis of rotation parallel to the electron beam direction. In the flipping coil method a long coil is used which has a typical transverse size of 5–10 mm (depending on the magnetic gap of the device). To increase the sensitivity, it is made up of several turns. The coil is connected to a voltmeter which integrates the voltage during the rotation or displacement. By rotating the coil along its symmetry axis and recording the voltage, one determines the integrated multipole content of the magnetic field. The same technique using a solid rigid coil has been used for many years to measure conventional dipole, quadrupole and sextupole magnets. A rigid coil gives less noise (no vibrations) and is more easily calibrated but, due to the small magnetic gap, it is much more expensive and difficult to have it straight enough to eliminate the field integral offsets. A measurement of

z-axis

Slave

Multiturn coil

x-axis

Master s-axis

Figure 5.16 View of a field integral measuring bench.

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the vertical field integral is made by placing the coil in the horizontal plane and by integrating the voltage V (t) induced during a flipping of the coil by 180◦ through the rotation stages. The vertical field integral Bz ds between both manipulators is derived from 

 Bz ds =

t2

* V (t) dt 2N a

(46)

t1

where N is the number of turns and a is the width of the coil. To eliminate the perturbation induced by constant voltages (thermal, noise etc.), one uses a slightly more elaborate protocol. The voltage is measured when flipping back and forth both ends of the coil by 360◦ by steps of 90◦ . The sum of all eight voltages recorded gives an estimate of the perturbing sources. Subtracting the corresponding voltages recorded during the positive and negative rotation of the coil, one eliminates the contribution from the perturbing sources. From the resulting four quantities, one derives the vertical and horizontal field integral as well as the integrated skew quadrupole component. The small width of the coil which is imposed by the small gap results in a low sensitivity of the coil to multipoles higher than the dipole. As a result, the most precise measurement of the integrated quadrupole, sextupole, and so on is made through a numerical differentiation of the vertical and horizontal field measurements (integrated dipole) made at various horizontal positions in the middle of the gap of the insertion device. Using fast rotation and linear stages, one requires about 10 min to measure the field integrals over 30 equally spaced horizontal positions of the coil which are needed for an efficient multipole shimming (see Section 6.2). This time period can be reduced to 1 min by performing an onthe-fly measurement [44]. The field integrals are sampled on some axis, usually the symmetry axis of the magnetic structure. Then the coil is displaced back and forth in the horizontal plane with a constant velocity. From the integrated voltages measured on-the-fly during the motion of the coil, one deduces the field integral variation with respect to that recorded on the reference axis. Because of the expected low integrated multipole content of the field, the precision desirable for the linear (rotation) stages is only of the order of 0.1 mm (0.1◦ ). Such a low precision figure allows the use of a high speed of 100 mm/s (45◦ /s) for the translation (rotation) which minimizes the contribution from the perturbing voltage sources and reduces the measuring time. The main source of errors in the field integral measurement comes from small defects of the wire of the coil which induce a small fluctuation of the width a(s) of the coil at the position s. These errors result in an absolute offset in the measured field integrals which can easily be too great. To reduce them, one can stretch the wire as much as possible (below the yield stress or breaking point) and one may average two or more field integrals measured for several displacements of the longitudinal position of the coil with respect to the field. A longitudinal displacement by one half period is particularly efficient because it reverses the offsets induced by the periodic part of the field. Another source of errors comes from the imperfect knowledge of the width of the coil and from vibrations. They result in a scaling error and a noise. Experience shows that for a length of coil shorter than 6 m, if the period of the eigen-mode of vibration is sufficiently longer than the period of the magnetic field, the vibrations and the gravitational sag are not a serious source of errors. The sag and the spatial period of vibrations can be monitored by measuring the vibration frequency of the wire as discussed in Section 5.1.3. For a 3 m long undulator with 0.6 T peak field, the field integrals measured with a flipping coil bench typically reach a 1–2 G cm repeatability and a 5–10 G cm absolute precision. The stretched scanning wire is sometime preferred because it is free from the offset and scaling errors. In this technique the coil is replaced by a single or multiple stretched wire.

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If a multiple wire is used, they are connected in a loop with a return line fixed in space in a region where there is preferably a low magnetic field. Following a small horizontal displacement δ of the wire, one deduces the field integral from the time integrated voltage V (t):   t2 * Bz ds = V (t) dt N δ (47) t1

where N is the number of wires. The scaling precision only depends on the precision of the linear stage. Ultimately, it can reach 1 × 10−4 which makes it suitable for the measurement of conventional dipole or quadrupole magnets. Due to the effective large surface of the single or multifilar loop, it is more sensitive to electromagnetic interference than the flipping coil measurement. By continuously scanning the horizontal position of the wire, one may operate on-the-fly to accelerate the measurement of the vertical field integrals. However, if one is interested in both a vertical and horizontal field integral measurement with on-thefly acceleration, one comes to a protocol made of a sequence of both horizontal and vertical displacements which is nearly as time-consuming as the on-the-fly flipping coil measurement described earlier. 5.2.2

Double field integrals

The double field integrals which are defined by Eqn (49) can be measured using either the flipping coil or the scanning wire set-up. In the flipping coil set-up, one rotates one single end of the coil in such a way that the two wires constituting the coil cross each other in the centre between the two manipulators. In this position, one then performs a field integral type of protocol measuring the integrated voltage as one flips simultaneously both ends of the coil by 180◦ . The second  t vertical field integral Jz is deduced from the first integral Iz and the integrated voltage t12 V (t) dt by 
 t2 * L V (t) dt 2N a (48) Iz − Jz = 2 t1 where N is the number of turns, L is the longitudinal length of the coil and a is its width measured at the position of the rotation stage. With the scanning wire technique, one proceeds with a sequence of displacements of both ends which consists in a pure translation or a pure rotation of the wire around the vertical axis intercepting the wire in the centre between the two manipulators. A proper linear combination of the recorded voltages allows the determination of both Iz and Jz . A similar sequence of displacements in the vertical plane gives the simple and double horizontal field integrals Ix and Jx . 5.3

Other measuring equipment

During the assembly of a permanent magnet insertion device, one often needs to characterize the magnetization and its orientation for each block. The method consists in placing the magnet block in the middle of a so-called Helmholtz coil. The magnet is flipped inside the coil while the voltage induced in the coil is integrated in a voltmeter. As the voltage measurement is largely insensitive to the precise positioning of the magnet block in the centre of the coil, the technique is precise and absolute if one uses a sufficiently large diameter for the coil compared to the magnet block dimension. To obtain all three components of the magnetization, one

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must repeat the measurement with different orientations of the magnet block. Characterizing the few hundred magnet blocks required to build an insertion device can be time-consuming. It can be accelerated using a remote-controlled rotation. A faster measurement can be made by means of a three-axis fluxgate integrator located at 20–100 cm from the magnet block. Fluxgate integrators are commercially available field sensors (see for example [45]) which operate with a low magnetic field, typically 10 G full scale. As a result one can derive the average magnetization by measuring the total dipole of the multipole expansion made with respect to the centre of the block. At a sufficient distance, all multipoles essentially produced by the shape of block vanish, leaving a dominant dipole term which is proportional to the volume of the block times the magnetization. To eliminate the perturbation by the earth’s magnetic field one subtracts two measurements made before and after flipping the block. The series of measurements on a large number of blocks can be made in a simpler and faster way but the absolute scaling of the magnetization is not as accurate. This can be remedied by measuring the absolute magnetization on a few blocks using a Helmholtz coil and then rescaling the data. The results of the measurement of the three components of magnetization of all blocks is used to pair the magnet blocks in order to locally cancel the field errors. There is no worldwide consensus on the best pairing technique – some people use a stimulating annealing [46] others use some deterministic procedure based on a series of sorting. Our experience shows that the characterization of the magnet block errors in terms of the three components of average magnetization only, is insufficient to produce a complete removal of the errors. More effort is needed to reach the field specification. This can be made by spending more time at the initial characterization of the individual blocks [47] and/or by shimming. The strategy used at the ESRF consists in pairing the blocks as simply and economically as possible to minimize the initial field errors following the first assembly and then proceed to a mechanical and magnetic shimming of the assembled structure.

6

Magnetic field shimming

6.1

Magnetic field specifications

Two important categories of field errors exist in an insertion device: the field integral errors and the phase (or spectrum) errors. 6.1.1

Field integral errors

An ideal insertion device must not change the closed orbit and the dynamics of the electron beam in the storage ring for any value of the magnetic gap (for a permanent magnet device) or any value of the current (electromagnet device). To achieve this goal a number of conditions must be met. The field integral I(x, z) = (Ix , Iz ) defined by Eqn (23) and their derivatives with respect to the transverse coordinates x and z must be equal to 0. In addition, the second field integrals J(x, z) = (Jx , Jz ) defined by J(x, z) =



∞



∞

−∞ −∞

 B(x, z, s  ) ds  ds

(49)

must also be equal to zero. A practical limit on the values tolerated for these integrals is usually set by limiting the rms closed orbit distortion around the ring circumference below

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1/10th of the rms beam size. This condition results in the following constraints for the simple and double field integrals:  εx [nm] Iz [G m] ≤ 0.3E [GeV] sin(π νx ) βx [m]  εz [nm] Ix [G m] ≤ 0.3E [GeV] sin(π νz ) (50) βz [m]  Jz [G m2 ] ≤ 0.3E [GeV] εx [nm]βx [m] sin(π νx )  Jx [G m2 ] ≤ 0.3E [GeV] εz [nm]βz [m] sin(π νz ) where E is the electron energy, εx (εz ) is the horizontal (vertical) emittance, βx (βz ) is the horizontal (vertical) beta function and in the middle of the insertion device and νx (νz ) are the horizontal (vertical) tunes of the lattice (see Chapter 1). To establish Eqn (50), we have assumed that the centre of the insertion device is a symmetry point of the lattice. For an insertion device installed on a so-called high beta straight section of the ESRF (E = 6 GeV, εx = 4 nm, εz = 0.04 nm, βx = 36 m, βz = 2.5 m, νx = 36.44 and νz = 14.39), the limits set by Eqn (49) are Iz ≤ 0.6 G m

Ix ≤ 0.21 G m

Jz ≤ 21 G m2

Jx ≤ 0.54 G m2

(51)

The variations of Ix , Iz , Jx and Jz with the gap (current) for a permanent magnet (electromagnet) insertion device can be corrected with some correction coils (active correction) and/or by applying a multipole shimming to the magnetic structure (passive correction). The advantage of the multipole shimming approach is that it is also correct for the higher order multipoles such as the quadrupole, sextupole and so on, and it simplifies the control system.

6.1.2

Phase errors

The next important class of error is the phase or spectrum error. Any deviation of the magnetic field from perfect periodicity results in an incomplete constructive interference of the undulator radiation. The angular spectral flux on the peak of the harmonics of the spectrum is reduced by these errors. The higher the harmonic number the stronger the reduction. Contrary to the field integral errors which are detrimental to all other users of the ring through the closed orbit distortion, the phase errors are only detrimental to the user of the undulator. The phase shimming (or equivalently spectrum shimming) corrects these errors. Performing a phase shimming is only important for the insertion devices being used as an undulator but it is useless for those used as wigglers.

6.1.3

Mechanical and magnetic shims

The field integral and phase errors observed in permanent magnet insertion devices are due to a number of reasons. They include positioning errors of the magnet and poles, small differences of magnetization value and direction from one magnet block to the next. These

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errors are easily characterized and compensated for. Another very important source of field errors are the inhomogeneities of the magnetization inside a volume of a single block. The magnetization vector may change its direction from one part to the other of the magnetic blocks. These errors are created during the manufacturing process. They are both difficult to characterize and to correct. They can be reduced considerably by selecting an appropriate manufacturing technique such as the transverse or longitudinal die pressing whereby the magnet blocks are pressed one by one inside a very homogeneous field. For small quantities of magnets, the most economical method of production consists in cutting a large number of small pieces out of a large volume of magnet. However, this is known to result in large inhomogneities of the magnetization within the blocks and should, therefore, be avoided whenever possible. Following the manufacture of the blocks, one usually performs a measurement of the magnetization vector along all three directions with a Helmholtz coil or a fluxgate magnetometer as discussed in Section 5.3. From such a measurement, one can perform some pairing and determine a magnet assembly sequence that minimizes the peak field fluctuations from one period to the next. As a result, the field integral and phase errors are usually lower by a factor of 2–3 compared to a random assembly sequence, but in many situations, they are still a factor of 10–100 higher than the tolerances. One way to measure the errors is to perform a shimming of the assembled structure. The shimming is based on the fact that a small local modification of the magnetic field can be done by either moving a magnet or a pole (mechanical shim) or by placing some thin piece of iron at the surface of the magnet (magnetic shim). Magnetic shims become magnetized under the surrounding nominal field and their magnetization operates like an additional small source of field. For hybrid devices, the magnetic shims also modify the magnetization in the pole. Magnetic shims always short-cut some part of the flux that would normally go through the gap and as a result they induce a reduction of the peak field measured in the middle of the gap. In addition, the contribution to the field and field integrals from two magnetic shims located close to each other do not add linearly. For these reasons, one prefers the use of mechanical shims to magnetic shims whenever possible. Figure 5.17 presents the typical shape and dimensions of the magnetic shims used in HYB and PPM assemblies. It is recommended that the shimming process be performed separately on the upper and lower magnet arrays. Later when both magnet arrays are put together in the final assembly slight modifications may be necessary. Of course, one may be tempted to save time by applying the shimming procedure at the final assembly stage. This is, however, a delicate operation as a single shim placed on the upper or lower magnet array may correct one kind of field error. However, some other components of the field induced by the shim are opposite whether the shim is placed on the upper or lower magnet array. Another way to look at the question is to recognize that a shim should be placed as close as possible to the source of errors. Therefore, to correct a faulty magnet located on the lower magnet array, a shim must be placed on the same magnet array and in the vicinity of the faulty magnet. This applies to both mechanical and magnetic shims. Therefore, in the following we shall only discuss the multipole and spectrum shimming of a single jaw of magnet. Shimming is important for permanent magnet insertion devices because of the significant fluctuation of the magnetization within a block and from one block to the next. It is usually not necessary for electromagnet devices for which the field errors only come from machining errors which are easier to control. The shimming process presented in the following sections is only relevant to conventional permanent magnet planar undulators or wigglers producing an almost sinusoidal vertical field on the electron beam axis.

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HYB Spectrum shim Multipole shims

PPM

Spectrum shim Multipole shims

Pole Magnet

Magnet H Magnet V

Figure 5.17 Multipole and spectrum shims in a PPM and HYB undulator.

6.2

Multipole shimming

The primary purpose of the multipole shimming of an insertion device is to ensure a negligible field integral (horizontal and vertical) and negligible integrated multipole variations for any gap setting of the insertion device. To achieve this, one necessary condition (but not sufficient) is that the field integrals produced by any single jaw of magnets do not change over a range of vertical coordinates extending from half the minimum gap to half the maximum gap above the magnet array and over some range of horizontal coordinates. The process is strongly influenced by the fact that, due to Maxwell’s equations, the complex quantity I = Iz + iIx is ! which does an analytic function of the complex variable w = x + iz over any region of space ˜ not contain any magnetic source (current, magnet, iron). Therefore, over any closed contour delimiting a region with no magnetic source, I satisfies the Cauchy integral !  1 I (v) I (w) = (52) ˜ ˜ dr dv 2iπ v−w ! ˜ ˜ ˜ ˜ An important consequence is that if one component of the vertical field integral, Ix or Iz , is known on a horizontal line defined by z = z0 and if the magnet blocks are only present on a single side of the plane defined by z = z0 , then both components can be computed from Eqn (52) for any point in the half space free of magnets. In particular, assuming a magnet array located below z = z0 , if Iz = 0 over the line z = z0 , both Ix or Iz are equal to zero in the half plane z > z0 . In real life, the field is always the sum of a contribution from the magnet jaw and the ambient field (field produced by the earth and modified by the environment). The ambient field is usually weak and fairly uniform over the horizontal and vertical range of interest. As a result, if one manages to make the integral Iz equal to a constant over the line

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z = z0 , both components will be constant over the half plane z > z0 , the constant being due to the ambient field. Because the field integrals produced by a magnet array vanish at infinity, one can limit the range of horizontal coordinates over which the field needs to be measured and corrected. At the ESRF, we generally use a horizontal range equal to twice the horizontal size of the magnets and the field integrals are sampled with a horizontal spacing equal to the distance z0 between the plane of measurement and the surface of the magnet array. To summarize, the multipole shimming consists in flattening the field integral variations with the horizontal coordinate at the minimum vertical distance z0 above the magnet array. If this is performed over a sufficiently large range of horizontal coordinates, both Ix or Iz present negligible variations with x and z in the half plane defined by z > z0 . The process is carried out in two steps. First, one should determine the signature of the shims available. Second, one should fit Iz (x, z0 ) vs x in terms of the shim signatures. Because of the errors in the shim positioning and of errors in the measured signature of the shims, one may need a few iterations. Figure 5.18 presents the field integral signature of a mechanical shim (left plot) and magnetic shim (right plot) placed at a distance of 5.5 mm above the surface for a single jaw of a hybrid structure with a 34 mm period. The computation is made with the RADIA code. The (horizontal, vertical and longitudinal) dimensions of the magnet (pole) are 65, 35 and 13 mm (45, 25.5 and 4 mm). The dimensions of the shim are 10, 0.1 and 6.5 mm. The magnetic shim is placed next to a pole as shown in Figure 5.17 and at horizontal distance x from the centre of the structure. In this computation, both the pole and shim are made of ARMCO steel. The magnet blocks are made of NdFeB. These signatures are also easily derived experimentally by subtracting the field integrals measured with and without shim. The effect of the magnetic (mechanical) shims is proportional to the thickness (displacement). It is clear from Figure 5.18 that the field integrals generated by the magnetic shims are much more localized than those induced by the mechanical shims. During the assembly process it is difficult to avoid any accidental displacement of a pole or magnet and the field integrals measured immediately after the first assembly almost always look like the one shown in the left plot of Figure 5.18. Their correction is easily derived. Following this initial correction with mechanical shim, the residual field integrals are corrected by a series of magnetic shims. Note that the field integrals are unchanged if one displaces a shim along the longitudinal axis by a distance exactly equal to an integer number of periods. The fitting process just predicts the total thickness of magnetic shims that must be placed at a certain horizontal coordinate x1 . To avoid any important reduction of the magnetic gap locally, it is Pole displaced Horizontally horizontally

15

20

Field integral (G cm)

Field integral (G cm)

30

10 0 Pole displaced Vertically vertically

–10 –20

Vertical Horizontal

–30 –40

–20

Shim X = 10 mm

Vertical Horizontal

10

Shim X = 20 mm

Shim X = 0 mm

5 0 –5

–10 0

20

Horizontal position (mm)

40

–40

–20

0

20

40

Horizontal position (mm)

Figure 5.18 Field integral signature of a mechanical shim (left plot) and magnetic shim (right plot) for a single jaw of a hybrid structure with 34 mm period. The field integrals are computed at a vertical distance of 5.5 mm above the surface of the poles. The magnetic shims are placed at various horizontal position x.

Technology of insertion devices

250

400

Horizontal Vertical x=0

Before shimming

200

Field integral (G cm)

Field integral (G cm)

300

150 100 50 After shimming

0 10

15

20

25

z (mm)

200

187

Horizontal Vertical z = 5.5 mm

0 –200

After shimming

Before shimming –400

30

35

40

–40

–20

0

20

40

x (mm)

Figure 5.19 Field integral of a single jaw of HYB undulator before and after shimming as a function of the vertical coordinate z for x = 0 (left plot) and as a function of the horizontal coordinate x for z = 5.5 mm (right plot).

good practice to spread the total thickness of shims over the adjacent periods. The choice of the period to receive the shim is usually not very important, however, it can be derived from a measurement of Bz (x1 , z0 , s) as a function of s with a Hall probe. The field integrals obtained on this hybrid undulator before and after the multipole shimming are shown in Figure 5.19. Clearly, before shimming the field integrals on-axis (x = 0) present a significant variation with the magnetic gap and a strong integrated multipole is seen. After shimming both the field integral variation with the gap and the multipole have been dramatically reduced and the field integrals are nearly constant and equal to the contribution of the ambient field. After shimming each jaw of magnets separately, the jaws are brought together on both sides of the electron beam axis to build the final undulator. As already discussed in Section 4.2.1, the field integral measured on the final assembly differs from the sum of the field integrals measured on each magnet jaw separately. The difference is called the interaction integral. It is usually small for a PPM undulator (10–100 G cm depending on the gap and period) but it can be quite significant for a HYB undulator or wiggler (30–1000 G cm). It strongly varies with the gap. This is due to the change of magnetization induced by the lower (upper) magnets on the upper (lower) poles and magnets. It has a weak dependence on the horizontal coordinate and, for conventional devices with sinusoidal field, it is generated in the ends of the magnetic structure. As discussed in Section 4.2, it can be corrected either with coils or by a proper magnetic design of the ends. Such field integrals can be very large for asymmetric wigglers if no precaution is taken during the design phase to minimize them. Using the multipole shimming, the undulators produced at the ESRF have an integrated dipole <20 G cm, integrated quadrupole <10 G and integrated sextupole <10 G/cm2 for both the normal and skew components and any useful value of the magnetic gap without correction coils. 6.3

Spectrum shimming

Figure 5.20 presents a series of pulses observed in the horizontal component of the electric field produced by a single electron as it propagates through a planar undulator. For an ideal undulator, the pulses are equidistant, namely Tn are the same for every nth half period. Because of the periodicity of the electric field and of the narrowness of the pulses, the Fourier spectrum is made up of a series of peaks with a large number of harmonics. A phase

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J. Chavanne and P. Elleaume 1.5 Tn + 1

Tn

1.0

Tn + 2

MV

0.5 0.0 –0.5 –1.0 –1.5

1.0

1.2

1.4

1.6 nm

1.8

2.0

2.2

Figure 5.20 Horizontal electric field produced on-axis by a single 6 GeV electron propagating through an undulator with a 34 mm period and a 0.7 T peak field.

400 × 1015

Ideal undulator 3.5° rms phase error

Ph/s/.1%/mrad2

300

200

100

0 0

10

20

30

40

50 keV

Photon energy

Figure 5.21 On-axis spectrum produced by a 6 GeV filament electron beam in a 30 period undulator (K = 2.2, period = 34 mm) for an ideal field and for a 3.5◦ rms phase error. Only the odd harmonics are visible.

error corresponds to a small variation of the time Tn from one period to the next. The Fourier spectrum of such a non-periodic field is similar to that of the ideal undulator except that the intensity on the high harmonic numbers is reduced as shown in Figure 5.21. One usually characterizes these errors in terms of the dimensionless phase shift ϕn rather than the time delay Tn . The normalization of the phase ϕn is such that for an ideal undulator, it increases by π over a half period of the undulator. ϕn and Tn are related to each other by ϕn =

4πcγ 2 Tn λ0 (1 + K 2 /2)

(53)

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For a direction of observation located on-axis of the undulator, the phase advance ϕn between two successive pulses can be expressed as

 2γ 2 (n+1)λ0 /2 2 π ϕn ≈ 1+ ϑx (s)ds (54) (1 + K 2 /2) λ0 nλ0 /2 where K is the deflection parameter of the undulator. ϑx (s) is the horizontal angle of the electron trajectory at the longitudinal coordinate s which can be expressed as a function of the vertical field Bz (s) according to (see Chapter 2):  s e ϑx (s) = (55) Bz (s) ds γ mc −∞ in which e, m and γ mc2 are the electron charge, mass and total energy and c is the speed of light. To derive Eqn (54), one assumes that the field errors are small and that the field without errors is Bz (s) = Bˆ cos (2πs/λ0 ). Any field error δBz (s) localized around the coordinate s0 results in a change of the angle ϑx (s) and of the phases ϕn for all values s > s0 . If the error ∞ is such that it produces no field integral ( −∞ δBz (s)ds = 0) then the angle ϑx (s) and the phases ϕn are only modified in the vicinity of the field error. Conversely, by means of some shims one can inject some field errors which correct locally the phases ϕn and equalize them. As a result, the intensity on the high harmonics is restored. This process is called phase or spectrum shimming. As for the multipole shimming, one distinguishes the mechanical shims (pole and magnet vertical movement) from the magnetic shims (extra pieces of iron placed as shown in Figure 5.17). One first measures (or computes) the signature of a shim in terms of its modification on the local magnetic field and its effect on the time delays ϕn . The signature of a shim is proportional to the displacement (mechanical shim) or to the thickness (magnetic shim). Figure 5.22 presents the signature computed with RADIA for several shims placed on the 34 mm hybrid undulator defined in Section 6.2. Note that the vertical displacement of a pole is the most efficient to modify the field and therefore the phase advance ϕn ; however, it does not produce a zero field integral. To remove

Vertical field (G)

10

0

–10

–20

Pole lowered by 0.05 mm Magnet lowered by 0.1 mm Shim 30 ×13 × 0.1 mm –40

–20

0 20 Longitudinal position (mm)

40

Figure 5.22 Field modification induced by a vertical displacement of a pole, by a vertical displacement of a magnet and by an iron shim on the field of a 34 mm period hybrid undulator. The arrows indicate the position and polarity of the poles.

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the field integral, one must simultaneously displace two successive poles with alternate polarity by the same amount. The field modification produced by any shim placed on the nth half period affects primarily ϕn but also ϕn−1 and ϕn+1 . As a result, the correction of each period is slightly coupled with the adjacent ones. One may overcome this by inverting a band matrix or one may ignore it and perform a few iterations. As discussed previously, the phase shimming should be done separately on each jaw at a height above the magnets equal to one half of the minimum gap. The variation of the phase fluctuations with gap is usually weak and the spectrum shimming at a larger gap is unnecessary because the higher harmonics decay rapidly and are, therefore, of reduced interest as one increases the gap. Note that if the mechanical structure has been designed to accommodate a small vertical displacement of the magnet and pole, the phase shimming can be performed by means of mechanical shims only excluding any magnetic shims. After phase shimming, an undulator routinely produced at the ESRF has an rms error between 1◦ and 2◦ for any gap values. Lower values could be achieved by applying a few more iterations of shimming, but, if one takes into account the electron beam emittance and energy spread, the associated improvement of the spectral properties is marginal. Assuming a filament electron beam and a Gaussian phase error independent from one half period to the next, it has been shown in Chapter 3 that the reduction of the angular spectral flux on the harmonic n of the spectrum is proportional to exp [−n2 σ 2 ] where σ is the rms phase error.

7 7.1

Support structures for permanent magnet IDs Introduction

We have seen that the permanent magnet technology is the most efficient and therefore most economical to produce a high magnetic field over a short period. It is for this reason that more than 90% of the IDs installed on the various synchrotron sources are made of permanent magnets. The magnetic field on the electron beam path produced by an assembly of permanent magnets is changed by modifying the distance between the magnet arrays and the electron beam. To do this, one usually fixes the magnet assemblies onto two rigid girders located symmetrically above and below the electron beam. The space between the girders is varied symmetrically with respect to the horizontal plane containing the electron beam. The purpose of this section is to derive the specifications for the manufacture of these girders and to review the technical solutions that can be applied to build a complete support structure. 7.2

Magnetic force

The magnetic force F between the upper and lower girders is the most important mechanical constraint in the design of the support structure. It is numerically obtained by integrating the Maxwell tensor over an infinite plane surface P located somewhere between the two girders: 1 F = µ0


P



B2 (B n) ˆ B − nˆ 2

dS

(56)

where nˆ is a unit vector normal to the surface P, B is the magnetic field on the surface and µ0 is the vacuum permeability. In the usual case, where the upper and lower magnet arrays are symmetrical about the horizontal medium plane of the electron beam, it is useful to integrate the Maxwell tensor over the median plane P . The field is then perpendicular to P

Technology of insertion devices and Eqn (56) reduces to  1 B2 F = dS 2µ0 P 2

191

(57)

The field B depends on both the horizontal and longitudinal coordinates measured in the medium plane. In the case of a sinusoidal field with peak field B0 , length L and width W , Eqn (57) becomes F =

1 2 B LW 4µ0 0

(58)

Equations (57) and (58) are only valid for conventional undulators and wigglers with a vertical field symmetric with respect to the median plane. In the following, we shall limit the presentation to this case; however, one should keep in mind that for a helical undulator or any other exotic insertion device, the force must be computed from Eqn (56). For a L = 5 m long conventional wiggler with W = 10 cm and a 2 T peak field, Eqn (58) predicts an enormous force of 4 × 105 N. For the opposite case of a 1.7 m long undulator with a 0.7 T peak field and a width W = 5 cm, the force is reduced to 8.3 × 103 N. As one increases the magnetic gap of such a device, the field and therefore the magnetic force between the girders decreases. The strong force between the upper and lower magnet arrays results in a vertical deflection of the girder as shown in Figure 5.23. The magnetic gap and therefore, the peak field, vary along the length of the girder. The relative change of the peak field δB0 /B0 can be related to the gap change δg and the undulator period λ0 by the relation δB0 δg ≈ −π κ B0 λ0

(59)

where κ is a dimensionless factor greater than 1. For most undulators, it is close to 1 but can be as high as 2 for very high field hybrid wigglers. δg is equal to twice the deformation of a single girder. To minimize the fluctuations of peak field, one must keep δg as small as possible. This is achieved by optimizing the longitudinal position of the supporting point of the girder (symbolized by a dark triangle on Figure 5.23) and by selecting a large cross section. Let L be the length of the girder, the minimum deflection δg is reached when the

g /2

Electron beam

Figure 5.23 Deformed girder under the influence of the magnetic force between the upper and lower magnet arrays. The dotted shape corresponds to the non-deflected girder. The distortion is enlarged for better clarity. The dark triangles symbolize the supporting points of the girder.

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supports are placed at a distance of 0.22 L from the end of the girder. The minimum value of δg is [48]

δg ≈ 6 × 10−4

F L3 YI

(60)

F is the total force between the girders, Y is the Young’s modulus of the material consituting the girder and I is the moment of inertia. Note that if the girder is supported from its ends the deflection δg would be 43 times greater. In the following, we shall assume the optimum placement of the girder support. For a girder having a rectangular cross section with height h and width b, the moment of inertia can be expressed as [48]: I=

1 3 bh 12

(61)

We shall not deal with other girder profiles commonly used such as the I cross section because, for a given moment of inertia, they require a larger cross section. The field fluctuations δB0 /B0 along the girder induced by the magnetic force can be derived from Eqns (59)–(61). To minimize the field fluctuations, one needs short girders with a large cross section. At this stage, it is important to differentiate the cases of the wiggler from the undulator. A common (but not necessary) criterion for a wiggler is to maintain δB0 /B0 below 1%. Assuming B0 = 2 T, L = 5 m, W = 0.1 m, κ = 2 and a square section, it results in a section of 270 mm (355 mm) for a stainless steel (aluminium) girder. For a constant deflection and a square cross section, the section b = h scales proportionally to the wiggler length L. For an undulator, the maximum field fluctuation allowed δB0 /B0 is related to the phase errors induced on the radiation. It can be numerically simulated by computing the spectrum from an electron travelling through a sinusoidal field which slowly varies in amplitude according to the deflection profile of the girder and Eqn (59). We refer the reader to Chapter 2 to derive such a computation. Taking the case of a 35 mm period undulator with deflection parameter K = 2.2, the calculation shows that a maximum fluctuation δB0 /B0 of 0.3% (0.9%) is required for an undulator length of 5 m (1.6 m) if one does not want to reduce the angular spectral flux of the fifth harmonic by more than 20%. Making use of Eqn (59), it corresponds to a gap fluctuation of 33 µm(100 µm). This requires a minimum section of 175 mm (230 mm) for a stainless steel (aluminium) 5 m long girder with rectangular cross section. The section is reduced by a factor close to 4 for a 1.7 m long undulator. So far, we do not take into account the deflection of the girder under its own weight which may not be negligible for a 5 m long wiggler case. One should also consider that the methodology that we have just described to derive the minimum cross section of the girder is somewhat simplistic and, if possible, most designers take a 1.5–2 times larger cross section to eliminate this problem. An important consequence is that the longer the girder, the larger the section and the tighter the machining tolerances. To minimize the section, 5 m long girders are usually made of stainless steel while 1.7 m long girders can be built cheaply of aluminium with a smaller cross section. 7.3

Segmentation of the support structure

Most storage rings of the third generation built in the 1990s can accommodate a large number of 4–6 m long insertion devices. Several approaches have been pursued concerning the engineering of the support structures. The classical approach consists in using a single support

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structure with the necessary long girders. Another approach has consisted in segmenting the support structure into two or three shorter ones placed in series along the straight section. The advantages of this segmentation are the following: each small carriage is more simple and economical to manufacture than a large one; the full installation of a straight section can be spread over time by installing only a single segment for the beamline commissioning and finishing the equipping of the other segments later. It is also one way to manage the high heatload of the synchrotron radiation. A user may gradually increase the heatload in its beamline by closing one, two or three segments of undulator at a time. For example, if limited by heatload, a user would operate the three segments simultaneously when the current is low (e.g., single bunch operation) and only operate a single segment when the ring is injected with a large current. It also provides the flexibility of installing several insertion devices of different periods and different types (wigglers, planar undulators, helical undulators . . .) on a single beamline. Flatness errors of the vacuum chamber may prevent the closing of the magnetic gap to the minimum value; by segmenting the support into two or three units, one just needs to take care of the flatness errors over a shorter length of chamber and as a result a smaller magnetic gap can be reached. This point is particularly relevant to welded stainless steel chambers. The manufacturing tolerances of a support structure with 1.7 m long girders are easily met. As a result, a standard type of carriage can be manufactured in series to accommodate any undulator or wiggler of a conventional type. Significant reduction in cost can be reached from such a series production when a large number of insertion devices are required. Nevertheless, there are some drawbacks in the segmentation of the support structure. Some of the tasks to be performed during the manufacturing and testing of the insertion devices are multiplied by the number of segments. This concerns the control system (motors, encoders, etc.) and the magnetic field measurements. Probably the most important drawback is the necessary phasing of the individual undulator segments. For a filament and mono-energetic electron beam travelling through an ideal undulator, the spectral brilliance is multiplied by four when the undulator length is multiplied by two. Without any phasing precautions (random phasing) between the undulator segments, the gain of four becomes two on average. However, due to the finite electron beam emittance and energy spread, the interest in phasing disappears especially at high photon energies. For example, at the ESRF, the factor 4 shrinks to 3.1 at 3.5 keV and 2.3 at 20 keV for two 1.6 m long segments. Several approaches have been successfully developed to phase the PPM undulators (see Section 4). The case of hybrid undulators is much more severe since large field integrals are always created at the junctions between the segments. As a result, the segmented undulator approach applied with success at the ESRF and ELETTRA has never been applied successfully to segmented HYB undulators. An intermediate approach has been followed at SPring8 and BESSY II. It consists in joining two or three similar short segments with a single motor driving the gap of all segments simultaneously. The advantage is the relaxation of mechanical tolerances and automatic phasing. However, the user only has a single degree of freedom (the gap) as opposed to the two or three degrees of freedom offered by the segmented approach (the gap in each segment). 7.4

Description of a support structure

Only a few mechanical descriptions of support structure have been published (see [50] and [51]). The purpose of the support structure is to position and maintain the girders (and therefore the magnets) parallel to each other within a gap range of ten to a few hundred millimetres and under a magnetic force ranging from a few Newtons to a few 105 . We assume,

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C-shape Girders and magnets

Vacuum chamber

Figure 5.24 Schematic of a H-shape and C-shape support structure. In the H-shape support structure, the support is on both sides of the vacuum chamber resulting in a perfectly horizontal magnet surface but at the cost of needing to break the vacuum in the chamber during installation. With the C-shape, installation can be done without breaking the vacuum but one cannot avoid a small rotation of the girders under the influence of the magnetic force which results in a small gap change between the internal and external sides of the vacuum chamber.

for the moment, that the magnet blocks are in the open air outside the vacuum chamber where the electron beam circulates. Two types of structures are commonly used: the H-shape and the C-shape (see Figure 5.24). In the H-shape structure the load of the upper girder is transmitted to the ground using support columns placed laterally on both sides of the vacuum chamber. It is the simplest solution, it has nevertheless the drawback of needing to remove the vacuum chamber whenever one installs or removes a support structure. This drawback is corrected in the C-shape structure in which the support of the girder to the ground is provided on a single side of the vacuum chamber, the other side being free to support the vacuum chamber. In a C-shape structure, due to the strong magnetic force, and the asymmetry in the support, the magnetic gap measured on the corner of the girder opposite the supporting frame is slightly smaller than the gap measured in the middle of the girder. Nevertheless, if one optimizes the mechanical design to reduce this gap difference as much as possible one retains tolerable values hardly exceeding a few tens of microns. In this respect, it is common and useful practice to perform a computer simulation of the distortion of the whole structure using finite elements. Nowadays, most permanent magnet insertion devices are supported by a C-shape structure. In the following we shall continue the description of a C-shape structure by taking the example of the ESRF standard structure which is presented in Figure 5.25. In line with the choice of segmented support (described earlier), it is short and holds a 1.6 m long girder. Up to three identical structures can be placed together along the straight section to support a 5 m long insertion device. An air space of about 10 cm is left between the girders of two adjacent structures. The structure is a tall one in order to accommodate a magnetic gap as large as 300–500 mm depending on the height of the permanent blocks which are fixed on the girder. Such a large gap allows the baking of the stainless steel or aluminium vacuum chamber without having to remove the support structure. The short girders are made of aluminium with a square cross section of 137 mm only. The dovetails are used to fix the permanent magnet assemblies to

Technology of insertion devices

90° gearbox

Springs

195

Stepper motors

Guiding rails

Upper girder

Ball bearing leadscrew

Lower girder

Mechanical stop

Dovetail to fix magnetic assemblies

Welded framework

Height and tilt adjustment

Figure 5.25 The ESRF standard C-shape support structure. Dovetails are used to fix the permanent magnet assemblies to the girders.

the girders. The drive system is composed of two motors connected to two high precision ball bearing leadscrews through a 90◦ gearbox. An equal rotation angle of the two motors produces a parallel vertical displacement of the upper and lower girders while a different rotation angle generates a different gap at both ends of the girder which is called a taper. The choice of using two motors instead of a single motor is dictated by the need to change, by remote control, both the magnetic gap and the taper between the girders. The resolution in the positioning of the gap between the girders is 1 µm. This is required for a proper tuning of the peak of any undulator harmonic to the photon energy selected by the monochromator. Another important figure of merit is the backlash observed in the measured gap upon reversal of the motor rotation. It is typically of the order of 20 µm. Its correction can be done electronically or by cycling the gap changes. The magnetic gap and taper are read by means

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of two multiturn absolute encoders fixed on the bottom extremities of the high precision ball bearing lead screws. A more precise method would be to place a linear encoder directly attached to the girders. However, at such locations, the encoders are more difficult to shield against hard radiation present all around the vacuum chamber when the electron beam is circulating in the ESRF ring. Stepper motors have been selected due to the simplicity of the control. Brushless and cw motors have been used successfully by other designers. They tend to give a higher torque which translates into a higher speed during gap motion. On the ESRF support structure, the magnetic force between the girders can be partially compensated at small gaps by means of a set of Belleville’s springs. Such springs allow a reduction of the torque requirement on the motors and a higher speed during gap changes. 7.5

In-vacuum undulators

With a conventional insertion device, the difference between the minimum magnetic gap and the internal aperture of the chamber is typically 3–5 mm depending on the length of the insertion device and manufacturing technology. This corresponds to the thickness of the wall of the vacuum chamber combined with the flatness error. In some situations it is essential to operate with the smallest possible magnetic gap. This is particularly crucial for high energy undulators made with a small period – two solutions are known. One consists in implementing a so-called variable gap chamber. Special flexible joints enable the internal aperture of such a chamber to be changed while the magnet assemblies are in the air and remain in contact with the chamber wall. The first chamber of that kind was tested at SSRL [49] and later on in the MAX I and the ESRF rings. The reduction of gap occurs through the use of a thinner chamber wall and through the reduction of the internal aperture which is set to the minimum value compatible with a reasonable lifetime of the stored beam. Due to the complex shape of the joint and the delicate soldering process, only a few attempts have been made and one prefers the other alternative which consists in placing the magnetic assemblies inside the vacuum. The very first developments of in-vacuum undulators took place in the early 1980s [52,53]. Following the installation of a 3.6 m long device at the photon factory [54], SPring8 has massively invested in this technology [55,56] with many devices 4.5 m long and a record 27 m long in-vacuum undulator. A much shorter undulator (0.32 m) has been operated with a minimum gap of 3.3 mm (minimum value set by a maximum lifetime reduction of 10%) [57]. A 1.6 m long in-vacuum undulator is in operation at the ESRF with a minimum gap of 5 mm (lifetime reduction of 5%) [58]. An ultra short 11 mm period 3.3 mm gap device has been installed on the NSLS X-ray ring [59]. As the technology becomes more common, many synchrotron radiation facilities are now planning the installation of in-vacuum undulators. Figure 5.26 presents a 3D view of the in-vaccum undulator built at the ESRF. To minimize the diameter of the vacuum tank which holds both the magnet assembly and the girders, a segmentation of the support structure must be applied. With in-vacuum undulators, one maximizes the brilliance and the range of tunability for a given beam stay clear of the electron beam. The drawback is a higher cost and lack of flexibility. To avoid overheating of the magnet blocks, the magnet surface of an in-vacuum undulator must be covered by a thin conducting sheet which carries the image current of the electron beam circulating in the vacuum. Because of the power deposited by the wake field of the electron beam in the undulator extremities (also called RF fingers), one must provide some water cooling of the magnet assemblies to avoid any risk of irreversible demagnetization of the material. A number of precautions must be taken to ensure the compatibility of all materials with the ultra high vacuum environment. In particular, the magnet block must be coated. TiN and

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Magnet gap adjustment

Magnet cooling Ion pump 4001/s

Ion pump 2301/s

Ele c

tro

nb

ea

ium an tor Tit lima p b su pum

ium an tor Tit lima b mp u s pu

m

Ion pump 2301/s

Verticle movement adjustment

Figure 5.26 3D view of an ESRF in-vacuum undulator.

Nickel have shown good results at SPring8 and ESRF with vacuum below 10−10 following a maximum 120◦ C baking of the magnet array. The replacing of the NdFeB magnetic material by Sm2 CO17 , which is less temperature and radiation sensitive, is an issue. For a typical 20 mm period the associated reduction in peak field corresponds to a gap change of 1 mm.

7.6

Exotic type of support structures

In the previous section, the various components of a conventional support structure were described. We shall now briefly review the more exotic type of support structure. There is another method of varying the field of an undulator which consists in longitudinally displacing one magnet girder with respect to the other one. This concept is called the adjustable phase undulator [60]. One magnet array is fixed against the vacuum chamber while the other girder is just moved longitudinally over a range equal to half a period. If the two girders are in phase, their field contributions to the electron beam path add and reach a maximum value. On the other hand, if the two girders are out of phase, their field contributions are equal but with opposite signs resulting in a cancellation of the magnetic field. By continuously changing the phase one can tune the peak field to any value between zero and the maximum.

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The magnetic force required to displace the girder is similar to that required for changing the gap in a conventional structure; however, the whole support structure is much more compact and can be significantly less expensive to build. There are a few drawbacks to this type of structure. The magnetic field at the ends does not vanish completely for any phase configuration. Some field integrals are generated at the ends for both the HYB and PPM technologies which require the use of correction coils. A number of different experiments can be performed on a single beamline. The optimal insertion device may change from one experiment to the next between an undulator or a wiggler or simply between two undulators of different periods. There are two ways to solve this. The simplest method consists in using a segmented approach and gathering several segments of different characteristics on a single straight section. The drawback is that only a single short segment is used at a time. An alternative is to build a support structure that can accommodate several magnetic assemblies. In this respect, one must distinguish the multi-undulator [61] concept and the revolver [62] concept (see Figure 5.27). In the multi-undulator concept, a few undulator assemblies equipped with their associated girders are placed side by side symmetrically with respect to the horizontal plane of the electron beam. A lateral translation of the whole structure results in one particular magnet assembly being at the exact vertical position above and below the vacuum chamber ready for a magnetic gap tuning. The drawback of this structure is the large space needed on both sides of the vacuum chamber for all the magnetic assemblies. In the revolver concept, up to four magnetic assemblies are fixed on a single girder with a square cross section. By a symmetric rotation of the upper and lower girders, one brings the desired magnetic assembly in position in front of the vacuum chamber; the magnetic field is then tuned by varying the magnetic gap between the two girders. The main drawback of the revolver concept is the impossibility of supporting the girder from any point other than the ends. For the same

Revolver support Muti-undulator support

Vacuum chamber

Figure 5.27 Two schemes allowing the use of four different magnetic assemblies on a single beamline. The magnetic assemblies are schematically represented in grey. In the multi-undulator support the magnet arrays are placed side by side and the whole support structure is displaced to bring the required magnetic assemblies above and below the vacuum chamber. In the revolver-support structure, a large cross section girder is equipped with a magnetic assembly on each of its four faces. The girder is rotated to bring the required magnetic assembly in front of the vacuum chamber. In both structures, the magnetic gap is varied to tune the magnetic field.

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cross section, this results in a deflection under the magnetic load up to 43 times greater than that of a conventional girder with optimized supporting. For this reason, revolver structures tend to be bulky and limited to the use of low field insertion devices of moderate length.

8

Electromagnet insertion devices

As discussed in Section 3, for a given period and gap, the field produced by a room temperature electromagnet undulator is smaller than that produced by a permanent magnet array. As a result most designers have concentrated on permanent magnet technology and only a few electromagnet undulators or wigglers have been built. In recent times electromagnet technology has regained some interest in the context of fast field of polarization switching which is possible with low inductance coils. Field ramping times between a few milliseconds and a few tens of milliseconds have been reached. 8.1

Magnetic design

The issues in magnet design are quite different from those of permanent magnet insertion devices. In many respects, their optimization is similar to the optimization of a conventional quadrupole or sextupole magnet. It 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 alternate polarization. In most cases, as for any iron dominated magnetic structure, the peak field Bˆ in the middle of the gap g between the two coil arrays is deduced from Ampere’s law as µ0 I Bˆ ≈ g

(62)

where I is the number of ampere turns between two adjacent poles in one of the yokes. The gap between the upper and lower yokes is fixed and the field is tuned by changing the current I . Equation (62) is valid on condition that the number of ampere turns and the pole shapes are designed in such a way that no saturation takes place. The Bˆ vs I characteristics always present a linear part for sufficiently low currents and saturates at high current. The issue of saturation is the most delicate and usually requires a numerical field computation as discussed in Section 3. The maximization of the peak field Bˆ usually results in a pole shape with a larger cross section at the bottom (close to the steel base plate) than at the top (close to the gap) as shown in Figure 5.28. In a periodic structure, the most important harmonics for the field are the fundamental and the third harmonics. Some harmonic suppression can be obtained by properly shaping the pole tip in the longitudinal direction. 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 as shown in Figure 5.28. Passive correction is difficult to achieve because of the great sensitivity of the field to the shape of the yoke at the ends. One usually operates electromagnet insertion devices with at least a vertical and sometimes also a horizontal field integral correction. The correction is applied by a small power supply connected to a set of coils placed at the ends. It is not uncommon that such a magnetic structure presents a hysteresis behaviour of the field integral induced at the ends. Hysteresis is difficult to correct and most designers apply a

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Pole

Conductor

Cooling pipe

Insulator

Base plate

Figure 5.28 Longitudinal cut of the end of an electromagnet undulator. A single yoke is presented.

symmetric field configuration to cancel the field integral induced by the upstream termination with that induced by the downstream termination, leaving a residual double field integral. A unique opportunity of the electromagnet undulator is the potential rapid change of the magnetic field. To achieve this, one must use a laminated yoke which reduces the eddy current losses and the coil must have a low inductance which implies a large current and large cross section of conductor and a few turns. The cooling is then best achieved by circulating some water in a pipe in the middle of the copper or aluminium conductor as shown in Figure 5.28. Such a coil structure is built by bending a single long straight conductor. With a well designed water cooling pipe, the current density in the copper can reach, in extreme cases, 10–20 A/mm2 but most designers select a more conservative 5–10 A/mm2 current density. An alternative choice is to build individual racetrack shaped coils placed over each pole. The coils are then connected in series. This solution is of more interest for large period devices.

8.2

Performances

For a given gap and short period, the peak field reached by an electromagnet undulator is lower than that of a permanent magnet device by a factor of 2–5 depending on the period. This can be partly explained by the discussion in Section 3.1.1.4. The field limit is set by the cooling of the copper conductor. One way to eliminate this difficulty is to operate the coil in a superconducting mode as described in Section 9. Because of their low field, electomagnet undulators are rarely built as a synchrotron source. Exceptions include applications which require a fast switching of the magnetic field and applications which require a long period (low photon energy). Elliptical wigglers and helical undulators with fast switching polarization have been built with both horizontal and vertical electromagnet fields or by combining permanent magnets and electromagnets [63–68].

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9 9.1

201

Superconducting insertion devices Superconducting high field wigglers

Some applications which require high energy photons can only be performed if the source has a sufficiently high critical energy. The critical energy Ec is expressed as a function of the electron energy E and the magnetic field B as Ec [keV] = 0.665E 2 [GeV]B [T]

(63)

Clearly high electron energy is an efficient way to reach high critical energy. However, for a given infrastructure, the electron energy is already fixed by the design of the storage ring and it is usually much cheaper and more flexible to increase the critical energy by building a special insertion device with a high magnetic field rather than converting the whole ring to operate at a higher energy. A hybrid permanent magnet structure has been built at the ESRF that produces a peak magnetic field higher than 3 T within a 11 mm magnetic gap. Even though, the peak field can be made even higher by operating with a smaller magnetic gap, we believe that 4 T is not far from the maximum field achievable from a permanent magnet hybrid wiggler operated at a reasonably small gap without reducing the lifetime of the stored electron beam. To reach higher magnetic fields one must use the superconducting coil technology. The engineering issues are quite different from those of a permanent magnet insertion device. Due to their operation at 4 K (the liquid temperature of Helium), superconducting wigglers are significantly more expensive to build and operate than permanent magnet devices. As a consequence they represent a very marginal part of insertion devices in operation throughtout the world. As discussed in Section 3.1.1.4, the longer the period, the greater the field of a superconducting wiggler compared to a permanent magnet wiggler. In terms of peak field, the turning point between both technologies takes place at very short periods somewhere between 10 and 20 mm. So far, only large period devices have been operated that can produce a much higher field than any other technology even though medium period undulators (30– 50 mm) could, in principle, justify the use of the superconducting technology if cost is not an important issue. One should mention that due to their high peak field Bˆ and, usually long spatial period λ0 , the radiation is sent over a large total horizontal angle θx : θx [mrad] ≈ 0.095

Bˆ [T]λ0 [mm] E [GeV]

(64)

which ranges from 6 mrad (for a 4 T field, 100 mm period device operated with 6 GeV electrons) to 240 mrad (for a 10 T, 250 mm period device operated with 1 GeV electrons). This large angle has a number of consequences. First, it enables the splitting up of the beamline to many different experimental stations thereby reducing the incremental cost per experimental station. This is the basis on which a number of superconducting wigglers have been installed on low or medium energy rings. A number of third generation sources face a new issue. The achievement of small emittance and the tuning of the betatron functions in the ID straight sections has resulted in a rather long straight section leaving some space for the necessary quadrupoles and sextupoles on both sides of the insertion device. As a consequence, the distance between the middle of the straight section and the exit of the downstream bending magnet can be rather long (9 m at the ESRF) and the large angle of emission of the radiation results in a large size of the radiation beam in the vacuum chamber inside the bending magnet requiring special absorbers. The simplest superconducting wiggler is made up of three poles

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3 2 1 0 –1 –300

–200

–100

0 100 S (mm)

200

300

Figure 5.29 Magnetic field of the ESRF 3-pole superconducting wiggler.

(see Figure 5.29). The central pole presents a high field while the two side poles have a lower field in such a way that the total field integral is equal to zero. More flux is obtained with a longer wiggler made up of five or seven poles. The number of poles is in general odd and the field is symmetrical with respect to the middle longitudinal coordinate in order to produce a zero second field integral. The simultaneous vanishing of the field integral and double field integrals ensures that the electron trajectory is not deviated and not displaced upon traversing the wiggler. The manufacture of superconducting magnets is nowadays a sophisticated but well understood engineering task. A good introduction can be found in [69]. Superconducting magnets have been produced in large series for NMR applications and for high energy particle accelerators. One difficulty in the manufacture of a superconducting wiggler is the rather complicated field geometry in which the magnet is traversed by a small aperture chamber to let the electron beam pass through the field. Due to the complex engineering and the manufacture on a one-by-one basis, it is not rare to see superconducting wigglers failing to reach some of the main specifications. Superconducting wigglers with a peak field as high as 7 T have been routinely operated [70] and wigglers with a 10–12 T peak field are under manufacture [71]. A cold bore medium period (70 mm) medium field (3.5 T) multipole wiggler is installed on MAX II [72]. Such small periods medium field wiggler are very attractive for medium energy synchrotron sources. 9.2

Superconducting short period undulators

Another area of interest for the superconducting technology concerns short period undulators. The first one was built and operated in the first Free Electron Laser experiment in Stanford [73]. Further attempts were made at NSLS [74], in connection with the LCLS Project [75], and a prototype superconducting in-vacuum undulator of period 3.8 mm has been tested [76]; a larger period with full field tunability suitable for a storage ring is under construction [77]. Figure 5.30 presents a comparison of the deflection parameter K reachable as a function of the period for the permanent magnet technology and the superconducting technology. Each curve is drawn for a particular magnetic gap. The continuous (dotted) lines correspond to the superconducting (permanent magnet) technology. The computation for the superconducting technology assumed no iron piece and a LHC type NbTi cable with a critical current density of 2.7 kA/mm2 at 5 T and 4.2 K. The geometry is that of Figure 5.3 with a conductor cross

Technology of insertion devices 3.0 Superconducting NbTi @ 4.2° K 2.7 kA/mm2 Wy = 0.35 × Per. Wz = 0.50 × Per. No Iron

Deflection parameter K

2.5

4

25 12

Permanent magnet Br = 1.05 T Wz = 0.50 × Per. 4 Magnets/Per.

2.0

30

15

6 8

203

20

10

35

1.5 4 6

1.0

8 10

12

15 20

0.5

25

30 35

0.0 0

10

20

30

40

50

Period (mm)

Figure 5.30 Deflection parameter K vs period for different gap and technology. The continuous (dotted) lines correspond to a superconducting (permanent magnet) undulator. Each curve is labelled with the magnetic gap.

section of 0.35 × period (longitudinally) and 0.5 × period (vertically). The computation is based on a PPM geometry with Sm2 CO17 material (Br = 1.05 T). The choice of Sm2 CO17 rather than NdFeB is somewhat conservative and justified by the higher resistance to baking and to radiation damage which is desirable for the targeted small gap devices. It is clear from Figure 5.30 that the larger the period, the larger the field ratio between both technologies. This was anticipated from the field computation derived in Section 3. Taking the example of a 6 mm gap device which is a realistic value for a storage ring and assuming a target K of 1.5, one deduces that the period required for a superconducting (permanent magnet) undulator is 14 (21.5) mm. As a result the spectrum produced by both devices will have the same harmonic content but will be energy shifted by a ratio 21.5/14 = 1.54. Shifting the spectrum to higher energies is precisely the interest of superconducting undulators and can justify the larger complexity and cost when compared with permanent magnet technology. Note that a slightly higher field (or equivalently lower period for a given K) can be reached if one uses an iron yoke made of high permeability steel. The ideal material is Holmium which has a saturation at low temperatures close to 4 T (compared to 2.2 T for cobalt steel). One can also increase the field by operating at a temperature lower than 4.2 K. A possible simplification of the engineering of superconducting undulators has been suggested in [78]. It consists in embedding an array of steel blocks inside the field of a solenoid (staggered undulator). Such an undulator based on a superconducting solenoid has been built and successfully operated in a Free Electron Laser experiment. Staggered superconducting

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undulators present a number of advantages over in-vacuum superconducting devices. The solenoid can be made with a large diameter and does not need to be in-vacuum, the cryogenic insulation becomes much simpler, opening the way to a reduction of the running cost through the use of small integrated refrigerators. The tolerance of machining and positioning of the steel blocks are relaxed [79]. In recent years staggered undulators have been envisaged for storage ring operation [80]. From Section 3, assuming that the iron pieces are magnetized uniformly according to the solenoidal field, the first harmonic field B1 produced by a staggered undulator can be approximated as


g B1 ≈ Bsol coth −π λ0





sin(πf ) h 1 − exp −2π πf λ0

(65)

where Bsol is the solenoidal field, h is the height of the iron piece assumed to be parallellepipedic, g is the magnetic gap, λ0 is the undulator period and f is the ratio of the non-magnetic material to the period. The solenoidal field Bsol required to saturate the iron piece is in most useful geometries Bsol ≈ Bsat /2. Using cobalt steel (Vanadium Permemdur), the field from a staggered undulator is similar to the one of a NdFeB PPM undulator. By adding permanent magnets in the space between the steel pieces and/or by using some steel with higher saturation (e.g. Holmium), one may significantly increase the field by a factor up to 2 [81]. For storage ring operation, one should note that the strong solenoidal longitudinal field is superimposed on the undulator field which generates a coupling of the horizontal and vertical betatron oscillations. Such a coupling can, in principle, be compensated by gathering two solenoids with opposite polarity in tandem in the same straight section. 9.3 9.3.1

Engineering issues Magnetic design

The magnetic design of a superconducting wiggler is more involved than that of a room temperature electromagnet or permanent magnet wiggler. One of the difficulties comes from the superconducting material which loses its superconductivity above a critical magnetic field. This critical field is a function of the material being used, the current density in the wire of the coils and the temperature. The loss of the superconductivity in any part of the coil results in a sudden dissipation of heat which by conduction forces the surrounding region also to transit from superconductivity to normal resistivity. After starting, the process grows extremely quickly until the whole magnet becomes resistive. This phenomenon is called a quench, it is rather sudden and extends over a fraction of a second during which the whole energy stored in the static magnetic field (typically of the order of 50–1000 kJ) is converted into heat with the corresponding vaporization of a part or of the whole of the liquid helium. This brutal process introduces significant thermal stress to the magnet structure which, in extreme cases, may then be permanently damaged. At the design stage, the quench event must be simulated and proper precautions taken to make sure that the magnet can survive this. The designer must make sure that, at every point in the coils, the magnetic field does not exceed the critical field. In reality, some safety margin must be allowed with respect to this criterion as the current densities and magnetic field are incomparably higher than those in room temperature magnets and they introduce large forces and stresses inside the coil. If, for some reason, the stress induces a small displacement some dissipation takes place due to the friction which heats up the coil and may trigger a quench. This is made worse by

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the low specific heat of all materials at low temperature. In fact, during the commissioning of the magnet structure, one records a large number of quenches (magnet training). The successive quenches usually take place at increasingly higher field until the nominal field is reached. It is believed that a displacement of 1 µm may induce enough heat to quench the whole magnet. It is important, for economic reasons, to limit the number of quenches required to reach the nominal field. To do so the coil is usually pre-stressed at a value greater than the stress induced by the field and current densities, thereby eliminating as much as possible any risk of motion of the coil when the field is turned on. The pre-stressing is done at room temperature and must take into account the differential thermal contraction between the copper material of the coil and the supporting structure usually made of aluminium or steel. During a quench, eddy currents are generated in all conducting pieces due to the rapid decay of the coil current. To avoid any risk of mechanical damage, for warm bore magnets, the copper shield in place between the electron beam tube and the helium vessel (see next section) is usually perpendicularly slotted in the natural direction of flow of the eddy currents. Most superconducting magnets use NbTi wires stabilized with copper to wind the coil. The critical field of NbTi at low current is around 10.6 T at 4.2 K and atmospheric pressure which is adequate to build a superconducting wiggler with a field on the electron beam lower than 7–8 T. For higher magnetic fields one must use Nb3 Sn, which is a more delicate technology, or reduce the helium temperature by lowering the pressure of the helium bath. The coils are normally made in a racetrack shape with the long side perpendicular to the electron beam direction. The coils are placed on both sides of the electron beam. It is easy to demonstrate from the Biot–Savart Law that the magnetic field is always higher on the inner side of the coil rather than on the outer side. The choice of current density in the coil must be compatible with the highest field existing at any place in this coil. In this respect, it is common to split the coil into two parts, one being placed in the inner side with a low current density, while the outer coil is operated with a higher current density which is acceptable as the magnetic field is lower. In this respect Figure 5.31 presents the coil set-up of the ESRF superconducting wiggler [82], for a 5 T peak field on the electron beam, the inner side racetrack coil is operated with a current density of 170 A/mm2 whereas all the other coils are operated with 340 A/mm2 . For very high field wigglers, current densities as high as 500 A/mm2 are not uncommon. This must be compared with the typical 3–10 A/mm2 used in room temperature electromagnets. The use of an iron yoke is favourable since it increases the magnetic field on the electron beam and it provides an attractive vertical magnetic force on the coil which helps in supporting the coil and prevents any risk of collapse of the coil onto the nearby electron beam tube. The iron yoke also strongly reduces the leakage of the magnetic field outside the magnet. However, it is not as essential to use an iron yoke as it is in room temperature electromagnets since the yoke becomes saturated and its contribution to the nominal field seen by the electron beam does not grow linearly with the coil current. In fact, in superconducting wigglers, the field contribution from the coils without any iron is always greater than the field contribution from the magnetized iron excluding the coil. This is in contrast to the case of a room temperature electromagnet wiggler for which the direct contribution of the coil current to the nominal field excluding the yoke is usually negligible. As a result, except for very high magnetic fields, there is little penalty to remove the iron yoke. This reduces the total mass of the magnet assembly and gives more space to organize the coil supporting and prestressing. Without any iron yoke, the confinement of the magnetic field can be performed by wrapping the cryostat tank with iron plates a few centimetres thick which short-cut the flux and avoid any important field leakage except around the electron beam inlet and exit

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–100 0

100

50

Z

0

–50

–100 0 Y

100

Figure 5.31 Coil set-up used in the ESRF superconducting wiggler. The dark inner racetrack coils are powered with a current density of 170 A/mm2 whereas the other coils are powered with 340 A/mm2 .

ports. If a superconducting wiggler is intended to be operated at different nominal fields, one must make sure that the field integral is corrected for any value of the nominal field. To do so, one usually installs some dedicated (superconducting or resistive) coils connected to a small power supply. Another important advantage in removing the iron yoke from the cryostat is the reduction of the variations of the field integral vs nominal field. This allows a reduction of the compensation current. One should nevertheless mention that all superconducting magnets present abnormal field integrals which are generated by the so-called persistent currents. The persistent currents are current loops which close their path inside a single superconducting wire. The diameter of the superconducting wires embedded in the copper matrix is usually of the order of 10–20 µm; as a result, these loops are very narrow but they can be as long as a few tens of centimetres. They produce a negligible contribution to the magnetic field but their contribution to the field integrals is far from negligible. Figure 5.32 presents a plot of the field integral observed on the ESRF wiggler as one cycles the nominal field from 0–4 T and back to 0. The field integral, as a function of the nominal field, draws some sort of hysteresis loop. At 0.5 T nominal field, one observes a 1 T mm difference in the field integral whether it is recorded during a ramping up or down of the nominal field. This is 10 times larger than the maximum field integral variation allowed to maintain the closed orbit distortion along the ring circumference within a tenth of the rms beam size. Therefore, this must be corrected, and in addition, if one wants some repeatability, one must continuously cycle the nominal field between the same values. Note that the largest irreversibility in the loop takes place for a nominal field below 2 T. Even though the mechanism is rather well understood, the computation of these field integrals is difficult and is never done for superconducting wigglers. One efficient way to reduce it is to use superconducting cables made of filaments of NbTi with the smallest diameter possible.

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Field integral induced by persistent currents

0.5 Field integral (T mm)

207

0.0

–0.5

–1.0 0

1

2

3

4

Nominal field (T)

Figure 5.32 Field integral observed on the ESRF superconducting wiggler as a function of the nominal field. The measurement is made by continuously cycling the nominal field between 0 and 4 T.

9.3.2

Cryogenics

Figure 5.33 presents a schematic view of the main components present in the cryostat of a superconducting wiggler. The superconducting coils are immersed inside a liquid helium bath at atmospheric pressure which maintains the coil around 4.2 K. The main cryogenic issue is to minimize the heat leak towards the surroundings which are at a room temperature of 300 K. As expected from the second principle of thermodynamics, the efficiency of any liquefier is severely limited and as a rule of thumb one may consider that the compensation of 1 W of heat leak at 4 K requires 1 kW of electrical power at room temperature to perform the necessary liquefaction of the helium gas. This ratio largely depends on the liquefier and should be regarded only as an order of magnitude calculation. The design of the cryostat holding the super-conducting coil is therefore strongly driven by the minimization of the heat inleak to the helium vessel containing the coil. In what follows, a short description is given of the standard techniques used for all types of superconducting magnets to minimize the heat leaks. The helium vessel is leak tight and usually made of stainless steel, it is isolated from the outer tank of the cryostat by vacuum. The pressure of the cryogenic vacuum is around 1 × 10−6 to 1 × 10−7 mbars which is sufficient to eliminate the heat leak through thermal conduction in the residual gas. The next most important heat leaks occur through the black body radiation of the outer tank towards the helium vessel. To reduce it, a copper shield is placed in the cryogenic vacuum between the wall of the outer tank and the helium vessel (see Figure 5.33). The copper shields are usually thermally connected to a liquid nitrogen vessel which maintains the temperature of the shield close to 77 K. As a result, the power radiated by the wall of the outer tank does not fall on the helium vessel but is evacuated by the shield and results in a boil-off of liquid nitrogen. The helium vessel is nevertheless exposed to the black body radiation coming from the shield at 77 K. The power per unit surface radiated by a surface at a temperature T is proportional to T4 (neglecting the effect of emissivity). Therefore, at first order, the effect of the 77 K shield is to reduce the power incident on the helium vessel by the

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J. Chavanne and P. Elleaume Nitrogen filling line Helium filling line

Current leads Liquid nitrogen vessel

Cryogenic vacuum Liquid helium vessel

Copper shield

Beam stay clear

Electron beam

Storage ring vacuum

Magnetic gap

Copper shield

Superconducting coils and yoke

Figure 5.33 Schematic of the main components present in the cryostat of a superconducting wiggler.

factor (300/77)4 = 230. It is made at the cost of the necessary re-liquefaction of the boiled nitrogen. Nevertheless, nitrogen liquefaction is more economical than helium liquefaction. An important issue is the temperature of the wall of the vacuum chamber. A cold bore chamber allows the reduction of the magnetic gap to the smallest value compatible with the electron beam stay clear and the thickness of the wall of the helium vessel. A cold bore chamber has the disadvantage of exposing the helium vessel to any additional heatload originating from the electron beam such as beam dump or wake field that may either quench the magnet or significantly increase the helium losses. In this respect, a more conservative design (warm bore) consists in leaving a narrow gap with cryogenic vacuum and a shield between the wall

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of the vacuum chamber and the wall of the helium vessel. Then, any power deposited on the wall of the chamber is not transferred to the helium vessel but carried out to the adjacent flange by conduction. Such a design typically increases the magnetic gap by 10–20 mm and can be undesirable from a magnetic point of view especially when aiming at very large magnetic fields. To further reduce the heat leak by radiation, the helium vessel and the shield are usually wrapped with a multilayer insulation made by alternating thin aluminium sheets and a thin thermal insulator (Mylar or Kapton). The multilayer insulation has an overall low emissivity and re-absorbs the radiation; as a result it reduces the heat leak by radiation. To maintain the desired vacuum after sealing off the cryogenic vacuum, a getter material such as activated charcoal is normally employed to adsorb gases that desorbs with time from the surfaces. The use of a simple liquid nitrogen cooled shield is a standard technique and usually results, for superconducting wigglers, in a heat leak in the 1.5–3 W range corresponding to a liquid helium consumption of 2–4 l/h. When state-of-the-art technology is used to reduce all other sources of heat leak, radiation still remains the primary source of heat loss. A lower heat leak can be achieved by introducing a second shield at some intermediate temperature between 77 and 4.2 K. Using a second shield around 20 K, reduced helium consumption of 0.35 l/h has been achieved on the NSLS 11-pole wiggler [84] and on the three pole wiggler of the ESRF. The easiest way to cool the second shield is by using the helium vapour freshly evaporated from the helium vessel (NSLS, MAX). Another method consists in introducing a two stage refrigerator in the cryostat and thermally connecting the shields to each stages [82,83]. This solution completely avoids the use of liquid nitrogen during operation. A further sophistication consists in implanting an in situ re-liquefaction of the boiled helium. This can either be done by means of a Joule–Thomson closed loop cycle equipped with counter flow heat exchanger or by using the newly developed cryo-coolers from Sumitomo, Leybold or Cryomech which nowadays produces from 0.5 to 1.5 W at 4.2 K either using a Gifford MacMahon type engine or using pulsed tubes. The old style use of a central liquefier outside the cryostat is being reconsidered in many laboratories. Experience at BESSY and ESRF shows that the use of an in situ refrigerator reduces significantly the operation costs. Nevertheless, due to their limited cooling capacity, the use of such refrigerators is only possible if a low heatload on the shields and on the cold mass is achieved. In particular, it implies the use of a warm or medium temperature electron beam chamber which is detrimental to the magnetic performances. Another important source of heat leaks are the current leads. The current leads span the temperature interval between room temperature and helium temperature. The heat leaks occur through thermal conductivity in the material and through heat deposited through the Joule effect by the current flowing in the lead. The minimization of the leaks through conduction is done by reducing the section of the lead. Conversely, the minimization of the leaks by Joule effect is done by increasing the section of the lead. An optimum diameter exists which is proportional to the current and results in about 1 mW of losses at 4 K per Ampere of current. On both the BESSY and ESRF wigglers, a high temperature superconducting material (HTS) such as BSCCO and YBCO have been used in the 4–20 K part of the lead. Due to their low thermal conductivity, they allow a significant reduction of the losses at 4 K. A systematic use of such HTS current leads is made for the LHC magnets [85] and most superconducting magnets built recently. The operation of the magnet in the so-called persistent mode where no current is permanently flowing in the lead can also reduce losses. To operate the magnet in the persistent mode, the superconducting coil terminals are shorted by a length of superconducting wire. During the charging of the magnet by the power supply this wire is heated and is not in a superconducting state. When the current has reached its nominal value, the heater is switched

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off and the current flows from the input to the output terminal of the coil, by-passing the current leads. The current power supplied can then be turned off while the current and field stay constant in the coils. The current flows in the coil in a closed circuit and decays slowly with time at a rate of 1 × 10−3 to 1 × 10−4 per day which requires a new charging every few days. In the persistent mode of operation, the current flows very rarely in the leads which can be built with a smaller diameter thereby reducing the heat leak by conduction. This is the solution retained for the ESRF superconducting wiggler.

References [1] Foster, G. W., ‘Experience with Permanent Magnets in the Fermilab 8 GeV Line and Recycler Ring’, EPAC98 Conference, Stockholm. [2] Chavanne, J., O. Chubar, P. Elleaume and P. Van Vaerenbergh, ‘Non Linear Numerical Simulation of Permanent Magnets’, Presented at the European Particle Accelerator Conference, Vienna, June 2000. [3] Bizen, T. et al., Nucl. Instrum. Meth. A467, 185 (2001). [4] Marechal, X., PhD Thesis, Universite Joseph Fourier-Grenoble I, 1992. [5] Gradshtein, I. S. and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press, New York, 1980). [6] Halbach, K., Nucl. Instrum. Meth. 187, 109–17 (1981). [7] Hezel, T. et al., J. Synchrotron Rad. 5, 448–50 (1998). [8] Binns, K. J., P. J. Lawrenson and C. W. Trowbridge, The Analytical and Numerical Solution of Electric and Magnetic Fields (J. Wiley, 1992. ISBN 0 471 92460 1). [9] Proc. of the Roxie Users Meeting and Workshop, Editor S. Russenschuck, Report CERN 99-01. [10] The Los Alamos Accelerator Code Group keeps a list of software available for performing field computation. It can be consulted at ‘http://www-laacg.atdiv.lanl.gov/electromag.html’. [11] Tortschanoff, T. CERN Report LEP-MA/84-20. Intermag Conference, Hamburg, 9–13 April 1984. [12] The computer Code POISSON is based on the paper by Winslow, A. M., J. Comp. Phys. 2, 149–72, (1967). POISSON is maintained and distributed by the Los Alamos Accelerator Code Group, see ‘http://www-laacg.atdiv.lanl.gov/services.html’. [13] ANSYS, Engineering Analysis System, Swanson Analysis System Inc. see ‘http://www.ansys. com/’. [14] MAGNET, Infolytica Corporation, see ‘http://www.infolytica.com’. [15] FLUX3D, Cedrat S. A., see ‘http://www.cedrat-grenoble.fr/’. [16] MAXWEL, Ansoft Corporation, see ‘http://www.ansoft.com/’. [17] TOSCA, Vector Fields Limited, Oxford, England, see ‘http://www.vector-fields.co.uk/index.html’. [18] Zhou, P.-B., Numerical Analysis of Electromagnetic Fields (Springer-Verlag, Berlin, 1993). [19] Urankar, L. K., IEEE Trans. Mag. MAG-16, 1283–8 (1980); Urankar, L. K., IEEE Trans. Mag. MAG-18, 911–17 (1982); Urankar, L. K., IEEE Trans. Mag. MAG-18, 1860–7 (1982). [20] Ciric, I. R., IEEE Trans. Mag. 28, 1064–7 (1992). [21] Elleaume, P., O. Chubar and J. Chavanne, ‘Computing 3D magnetics fields from insertion devices’, Proceedings of the PAC97 Conference, May 1997, pp. 3509–11. [22] Chubar, O., P. Elleaume and J. Chavanne, ‘A 3D magnetostatics computer code for insertion devices’, J. Synchrotron Rad. 5, 481– 4 (1998). [23] Mathematica, Wolfram Research Incorporation, see ‘http://www.wolfram.com/’. [24] Trowbridge, C. W. Finite Elements in Electrical and Magnetic Field Problems, Ch. 10, M. V. K. Chari and P. P. Silvester (ed.) (John Wiley, 1980). [25] Radia is available from ‘http://www.esrf.fr/machine/groups/insertion devices/Codes/software.html’. [26] Trowbridge, B., ‘Integral Equations Revisited’, International Compumag Society Newsletter 2(3), 6 (1995).

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[27] Quimby, D. C. and A. L. Pindroh, ‘High Field Strength Wedged-Pole Hybrid Undulator’, Rev. Sci. Instr. 58, 339–45 (1987). [28] Chavanne, J., P. Elleaume and P. Van Vaerenbergh, ‘Segmented High Quality Undulators’, Proc. 1995 Particle Accelerator Conference, p. 1319. [29] Humphries, D., J. Akre, E. Hoyer, S. Marks, Y. Minamihara, P. Pipersky, D. Plate and R. Schlueter, ‘Design of End Magnetic Structures for the Advanced Light Source Wigglers’, Proc. 1995 Particle Accelerator Conference, p. 1447. [30] Chang, L. H., Ch. Wang, C. H. Chang and T. C. Fan, ‘Passive End Pole Compensation Scheme for a 1.8 T Wiggler’, Proc. 1995 Particle Accelerator Conference, p. 1450. [31] Chavanne, J., P. Elleaume and P. Van Vaerenbergh, Proc. 1999 Particle Accelerator Conference, p. 2665. [32] Gottschalk, S. C., D. C. Quimby and K. E. Robinson, Proc. 1999 Particle Accelerator Conference, p. 2674. [33] Chavanne, J., P. Elleaume and P. Van Vaerenbergh, J. Synchrotron Rad. 3, 93–6 (1996). [34] Chavanne, J., P. Elleaume and P. Van Vaerenbergh, J. Synchrotron Rad. 5, 196–201 (1998). [35] CERN Accelerator School, Magnetic Measurement and Alignment, Report CERN 92-05. [36] Walker, R. in Synchrotron Radiation Sources, A Primer, Ch. 7, H. Winick (ed.) (World Scientific, 1994). [37] Seitz, F., Phys. Rev. 79, 372 (1950); Goldberg, C. and R. E. Devis, Phys. Rev. 94, 1121 (1954). [38] Borovikov, V. M. et al., J. Synchrotron Rad. 5, 382–5 (1998). [39] Chavanne, J., P. Van Vaerenbergh and P. Elleaume, ‘A 3 Tesla Asymmetric Permanent Magnet Wiggler’, Nucl. Instrum. Meth. A421, 352–60 (1999). [40] Warren, R. P. Nucl. Instrum. Meth. A272, 257–63 (1988). [41] Temnykh, A. Proc. of the PAC97 Conference, p. 321 [42] Chavanne, J., C. Penel, P. Elleaume and P. Van Vaerenbergh, ESRF Internal Report ESRF/MACH ID 97/30. [43] Fortgang, C. ESRF Internal Report ESRF/MACH-ID95/26 [44] Chavanne, J., P. Elleaume and P. Van Vaerenbergh, SRI97 Conference August 1997, Himeji, Japan. [45] Bartington Instruments, see ‘http://www.bartington.com/’. [46] Cox, A. D. and B. P. Youngman, SPIE Proceedings 582, 91–7. [47] Ryynanen, M., J. Synchrotron Rad. 5, 468–70 (1998). [48] Young, W. C., Roark’s Formulas for Stress and Strain (Mc Graw Hill, New York, 1989). [49] Hoyer, E. et al., Nucl. Instrum. Meth. 208, 79–90 (1983). [50] Wang, Ch., M. C. Lin, C. H. Chang, L. H. Chang, H. H. Chen, T. C. Fan, K. T. Hsu, J. Y. Hsu, C. S. Hwang, K. T. Pan, ‘Conceptual Design for the SRRC Elliptically Polarizing Undulator EPU5.6. Part II: Magnetic Loading and Structure Deformation’, Proc. 1996 European Particle Accelerator Conference, p. 2570. [51] Bizen, T., Y. Hiramatsu, Y. Miyahara, A. Nakamura and T. Shimada, Proc. 1998 European Particle Accelerator Conference, p. 2240. [52] Hsieh, H., S. Krinsky, A. Luccio, C. Pellegrini and A. Van Steenbergen, Nucl. Instrum. Meth. A208, 79–90 (1983). [53] Gudat, W., J. Pfluegher, J. Chatzipetros and W. Peatman, Nucl. Instrum. Meth. A246, 50–3 (1986). [54] Yamamoto, S., T. Shioya, M. Hara, H. Kitamura, X. Zhang, T. Mochizuki, H. Sugiyama and M. Ando, Rev. Sci. Instrum. 63, 400 (1992). [55] Kitamura, H. SRI97, J. Synchrotron Rad. 5, 184–8 (1998). [56] Hara, T., M. Yabashi, T. Tanaka, T. Bizen, S. Goto, X. M. Marechal, T. Seike, K. Tamasaku, T. Ishikawa and H. Kitamura, ‘The Brightest X-ray Source: A Very Long Undulator at SPring-8’, Rev. Sci. Instrum. 73, 1125–8 (2002). [57] Stefan, P. M. et al., J. Synchrotron Rad. 5, 417–19 (1998). See also Stefan, P. M. et al., Nucl. Instrum. Meth. A412, 161 (1998).

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[58] Chavanne, J., P. Elleaume and P. Van Vaerenbergh, Proc. of the 1999 Particle Accelerator Conference, p. 2662. See also Chavanne, J., P. Van Vaerenbergh and P. Elleaume, Proc. of the EPAC 2000 Conference. [59] Tanabe, T., H. Kitamura and P. Stefan, Proc. 1997 Particle Accelerator Conference, p. 3515. [60] Carr, R., Nucl. Inst. Meth. A306, 391 (1991). [61] Bachrach, R. Z. et al., Nucl. Instrum. Meth. A208, 117–25 (1983). [62] Isoyama, G., S. Yamamoto, T. Shioya, H. Ohkuma, S. Sasaki, T. Mitsuhashi, T. Yamakawa and H. Kita-mura, Rev. Sci. Instrum. 60, 1863 (1989). [63] Gluskin, E., D. Frachon, P. M. Ivanov, J. Maines, E. A. Medvedko, E. Trakhtenberg, L. R. Turner, I. Vasserman, G. I. Erg, Yu. A. Evtushenko, N. G. Gavrilov, G. N. Kulipanov, A. S. Medvedko, S. P. Petrov, V. M. Popik, N. A. Vinokurov, A. Friedman, S. Krinsky, G. Rakowsky and O. Singh, ‘The Elliptical Multipole Wiggler Project’, Proc. 1995 Particle Accelerator Conference, p. 1426. [64] Nahon, L. et al., Nucl. Instrum. Meth. A396, 237–50 (1997). [65] Klein, R., J. Bahrdt, D. Herzog and G. Ulm, ‘The PTB Electromagnetic Undulator fro Bessy II’, J. Synchrotron Rad. 5, 451–2 (1998). [66] Walker, R. P., D. Bulfone, B. Diviacco, W. Jark, P. Michelini, L. Tosi and R. Visintini, G. Ingold, F. Schäfers, M. Scheer, G. Wüstefeld, M. Eriksson and S. Werin, Proc. 1997 Particle Accelerator Conference, p. 3527. See also Walker, R. P., R. Bracco, D. Bulfone, B. Diviacco, P. Michelini, N. Pangos, M. Vento, R. Visintini and D. Zangrando, Proc. 1998 European Particle Accelerator Conference, p. 2255. [67] Chavanne, J., P. Elleaume and P. Van Vaerenbergh, ‘A Novel Fast Switching Linear/Helical Undulator’, EPAC98 Conference, Stockholm. [68] Green, M. A., W. R. Winter, M. Thikim, C. A. Baumann, M. V. Fisher, G. C. Rogers, D. E. Eisert, W. S. Trzeciak and R. A. Bosch. Proc. 1999 Particle Accelerator Conference, p. 2659. [69] Wilson, M. N., Superconducting Magnets (Oxford Science Publications, Clarendon Press, Oxford, 1983). [70] Borovikov, V. M. et al., J. Synchrotron Rad. 5, 440–2 (1998). [71] Ando, A. et al., J. Synchrotron Rad. 5, 360–2 (1998). [72] Leblanc, G., E. Wallen and M. Eriksson, ‘Evaluation of the Max Wiggler’, Presented at the European Particle Accelerator Conference, Paris, June 2002. [73] Elias, L. R. and J. M. Madey, Rev. Sci. Instrum. 50(11), 1335 (1979). [74] Ingold, G., I. Ben-Zvi, L. Solomon and M. Woodle, Nucl. Instrum. Meth. A375, 451–5 (1996). [75] Caspi, S., R. Schlueter and R. Tatchyn, Proc. of the 1995 Particle Accelerator Conference, pp. 1441–3. [76] Fritz, M., T. Heze, M. Homscheidt, H. O. Moser, R. Rossmanith, T. Schneider, H. Backe, S. Dambach, F. Hagenbuck, K.-H. Kaiser, W. Lauth, A. Steinhof and T. Walcher, ‘First Experiment with a 100 Period Superconductive Undulator with a Period length of 3.8 mm’, Proc. 1998 European Particle Accelerator Conference, p. 2234. [77] Rossmanith, R. and H. O. Moser, A. Geisler, A. Hobl, D. Krischel and M. Schillo, ‘Superconductive 14 mm Period Undulator for Single Pass Accelerators (FELs) and Storage Rings’, Presented at the European Particle Accelerator Conference, Paris, June 2002. [78] Huang, Y. C., H. C. Wang, R. H. Pantell, J. Feinstein and J. Harris, Nucl. Instrum. Meth. A318, 765–71 (1992). [79] Huang, Y. C., J. Schmerge, J. Harris, G. P. Gallerano, R. H. Pantell and J. Feinstein, Nucl. Instrum. Meth. A341, 431–5 (1994). [80] Ohmi, K., N. Ikeda and S. Ishi, ‘Design of a Short-period Superconducting Undulator at KEK-PF’, KEK Report 98-8, June 1988. [81] Chang, C. H., C. S. Hwang, C. H. Wang, F. Z. Hsiao and T. C. Fan, ‘Magnetic Design for a High Field Strength Staggered Undulator’, Presented at the European Particle Accelerator Conference, Vienna, June 2000. [82] Stampfer, M. and P. Elleaume, ‘A 4 T Superconducting Wiggler for the ESRF’, Proc. of the EPAC94, pp. 2316–18 (World Scientific Pub. Co, London).

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[83] Hwang, C. S., C. H. Chang, J. R. Chen, C. C. Kuo, G. H. Luo, Y. J. Hsu and C. T. Chen, Presented at the European Particle Accelerator Conference, Vienna, June 2000. [84] Blum, E. B. et al., ‘A Superconducting Wiggler Magnet for the NSLS X-ray Ring’, PAC97, 1997 Particle Accelerator Conference, Vancouver, B. C. Canada, May 12–16, 1997. [85] Ballarino, A. and A. Ijspeert, ‘Expected adavantages and disadvantages of High-Tc current leads for the Large Hadron Collider’, ICEC 16, 1147–50 (1996).

6

Polarizing undulators and wigglers Hideo Onuki

1

Introduction

Circularly polarized radiation is recognized as being very useful in a broad range of scientific disciplines, including solid state physics, chemistry, biology, medicine, etc. Recently, its use has been extended to the VUV and X-ray regions. This chapter deals with insertion devices (IDs) that can produce circular polarization, or, more generally, that can switch the polarization between two arbitrary states, so-called polarizing IDs. Table 6.1 is a summary of the polarizing IDs that produce circular or variable polarization. The following sections describe the operating principles and the properties of each of the IDs in Table 6.1. Chapter 3 contains a general description of undulator radiation.

2 2.1

Undulators Undulators with crossed and retarded magnetic fields

A helical magnetic field is made with crossed and retarded magnetic fields, given by

 2π 2π B = iBx0 sin y + kBz0 sin y−α λu λu

(1)

where λu is the wavelength of the magnetic field, Bx0 , Bz0 the peak magnetic field in the x- and z-axes, α the phase difference between the Bx and Bz , where the Bx and Bz are Table 6.1 Classification of polarizing IDs Classification Undulator (UR) Crossed and retarded magnetic fields

Interference effect Wiggler (WR) Elliptical magnetic field Circle orbit

Devices Double helix UR Crossed and overlapped UR Planar UR Tandem UR Electromagnet UR Crossed UR Permanent magnet WR Electromagnet WR Asymmetric WR

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the x and z components of the field, and i and k the unit vectors along the x- and z-axes, respectively. When we take Bx0 = Bz0 it is apparent that we can obtain two helical magnetic fields for α = π/2 (right-handed) and α = 3π/2 (left-handed), and more generally, an elliptical magnetic field for arbitrary values of α, Bx0 , and Bz0 . Let us investigate the orbit of relativistic electrons running along the y-axis by means of a computer simulation. For simplification, it is assumed that each magnetic field distribution (Bx,z ) is the same as shown in Figure 6.1, where λu = 5.6 cm and Bx0 = Bz0 = 2.5 kG. The distribution of Bx , Bz shown in Figure 6.1 satisfies 

∞

Bx,z (y) dy = 0

(2)

0

and 

∞ y

0

Bx,z (y  ) dy  dy = 0

(3)

0

whose conditions represent the constraints of zero deflection and zero displacement of the electron beam after passing through the system. The results of the computer simulation for a 600 MeV electron are shown in Figure 6.2, where each curve represents the trajectory of the electron orbit projected on the x–z plane for a special value of α. With respect to the polarization of synchrotron radiation emitted from the electron moving along the orbits indicated in Figure 6.2, the direction of polarization is identical to the direction of acceleration the electron. Therefore, we can obtain orthogonal linear polarizations for α = 0 and π, right- and left-handed circular polarizations for α = π/2 and 3π/2. In general we have elliptical polarizations for arbitrary values of α.

3

u = 5.6 cm

Magnetic field strength (kG)

2

1

0

10

20

30

40 (cm)

–1

–2

–3

Figure 6.1 Magnetic field distribution in each axis used in computer simulations.

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–15

15

0

15

–15

–15

15

–15

15



–15

15

0

–15

/4

15

0

–15

0

–15

0

15

–15

15

15

–15

–15

/2

15

0

–15

15

5/4

–15

15

3/4

15

0

–15

0

15

3/2

–15

0

–15

15

7/4

Figure 6.2 Calculated trajectories of electrons projected on the x–z plane for some special values of α. The units of each axis are in micrometers.

The wavelength of the nth harmonic radiation observed on the y-axis from the ellipticalorbit electrons, λn , is given by

Kz2 Kx2 λu λn = 1+ + (4) 2nγ 2 2 2 where γ is the electron energy in the rest mass unit, and Kx and Kz the horizontal (x-axis) and vertical (z-axis) deflection parameters: Kx = 0.0934Bx0 [kG]λu [cm]

(5)

Kz = 0.0934Bz0 [kG]λu [cm]

(6)

Synchrotron radiation is emitted along the direction of motion of the electrons with a small opening angle θ ≈ 1/γ . Hence, for K < 1 (weak-field case), the pitch angle of the helical orbit is smaller than 1/γ , and the radiation is emitted into an angle 1/γ . For K > 1 (strong-field case), the radiation is emitted into a cone of half-angle K/γ (Figure 6.3) [1]. Unlike the planar undulator (Bx0 = 0 in Eqn (1)) or the elliptical undulator (Bx0 < Bz0 ), the helical magnetic field produces only fundamental harmonic in the forward direction. There are several methods of producing the magnetic field expressed by Eqn (1). The following sections discuss these methods.

2.1.1

Double helix undulators

It is well known that bifilar coil windings carrying current in opposite directions (Figure 6.4) produce a helical magnetic field along axis. This type of undulator had been fabricated

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217

K=2 ≈ K/ K = 0.5 K = 0.707 K=1

Figure 6.3 Spatial power distribution for synchrotron radiation emitted by electrons in helical orbit for several values of the magnetic field parameter K. (Reproduced with permission from [1].)

e–

y

Figure 6.4 Double-helix undulator.

for measuring electron (or positron) beam polarization by a Russian group. The undulator, which was composed of a double copper spiral with intense water cooling, was installed in the straight section of storage ring VEPP-2M in 1980 [2]. The main parameters of this undulator are: undulator period λu , 2.4 cm; number of periods N = 10 with a current of 5 kA in the helical conductor, and the magnetic field on the axis, B is 1.3 kG. In the energy range of the storage ring, from 500 to 700 MeV, the radiation from the undulator on the first harmonic was in the soft X-ray range from 8 to 13 nm. The degree of polarization was analyzed with a 45◦ Bragg-reflector type polarimeter by Gluskin et al. In general, the strength of the magnetic field in the double helix undulator primarily depends on the aperture radius of the helixes. Figure 6.5 indicates the relationship between the magnetic field B0 on the axis and the radius r0 [3]. In the FEL experiment using an electron beam at the linear accelerator at Stanford University, a superconducting coil had already been used in the helical undulator to increase the magnetic field before the experiment by the Russian group [4].

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102 3 ·104 105 A

104

u B 0 (cm · kG)

10 3 ·103

103

1

10–1

1

2

3

4 5 6 2 r0 X= u

7

8

Figure 6.5 Relationship between magnetic field B0 and radius r0 [3].

In the double helix undulator in Figure 6.4, the polarization is fixed and cannot be modulated. Alferov, Bashmakov and Bessonov proposed a new type of undulator in which two bifilar helical coils are overlapped and coaxial, the directions of spirals opposite to each other, and the pitches of the spirals are the same (Figure 6.6) [5]. If just the first coil or the second coil is supplied with electric current, right-handed or left-handed circular polarization can be obtained, enabling very rapid switching between the two states. If the currents are supplied to both coils simultaneously, an elliptical magnetic field is formed on the undulator axis. In this case the undulator generates polarized radiation of any ellipticity by mechanically varying the relative phase shift between the two coils. 2.1.2

Crossed and overlapped undulators

Onuki proposed that crossed and retarded magnetic fields identical to Eqn (1) could be realized by two crossed and overlapped planar magnet arrays [6]. These magnet arrays can be effected by either permanent magnetic arrays or by electromagnet arrays [6]. Either type can generate polarized radiation of any ellipticity by changing the phase shift between two magnetic field distributions. In the case of an undulator utilizing permanent magnet blocks, the state of polarization can be switched by mechanically changing the relative phase between two planar undulators, as shown in Figure 6.7. The Onuki-type polarizing undulator was developed in

Polarizing undulators and wigglers 4

3

219

2 1

y

u 5

6

Figure 6.6 Universal double-helix undulator [5].

z

B

A

y A

x

B

u

Figure 6.7 Schematic view of crossed and overlapped type polarizing undulator proposed by Onuki [6].

the Electrotechnical Laboratory (the present National Institute of Applied Industrial Science and Technology) and successfully produced the world’s first polarized radiation of arbitrary ellipticity [7]. Two polarizing undulators of the same type have been installed in the electron storage ring, TERAS, and in the compact storage ring, NIJI-II (Table 6.2). Figure 6.8 is an overview of PU-2 (in Table 6.2), which was inserted in the straight section between the two bending magnets of NIJI-II. In both cases, the magnet arrays of the undulator were placed outside the storage ring vacuum duct. The vacuum duct must have a considerably large cross section, especially horizontally, for the multiturn injection of electron beams. To satisfy this requirement, the undulator was fitted around a vacuum duct with a square cross section with its 8.1 cm inner diagonal in the case of NIJI-II (outer diagonal, 8.9 cm) set horizontally. Consequently, the direction of each magnetic field in the undulator was inclined from the horizontal (vertical) plane by 45◦ , as shown in Figure 6.8. A modulation frequency of the polarization of about 3 Hz was achieved by use of a motor-driven crank.

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H. Onuki Table 6.2 Main parameters of polarizing undulators in the ETL

Overall length (cm) Block size of magnet (cm3 ) Magnet period (cm) Number of periods Magnet materials Peak magnet fields (kG) HF∗ configuration PF∗ configuration K parameters HF∗ configuration PF∗ configuration Modulation frequency (Hz) Undulator gap (cm) Storage ring Electron energy (MeV)

PU-1

PU-2

38 2 × 2 × 5.7 8 4 Nd–Fe–B alloy

131.15 2.15 × 2.15 × 6.3 8.6 15 Nd–Fe–B alloy

1.5–0.28 2.1–0.40

1.35–0.35 1.91–0.50

1.08–0.21 1.53–0.30 3 5.8–20 TERAS 800 (max.)

1.08–0.28 1.53–0.4 3 6.4–20.4 NIJI-II 600 (max.)

Note ∗ ∗

HF: Helical field. PF: Plane field.

Figure 6.8 Overview of polarizing undulator PU-2 installed in the storage ring NIJI-II.

The polarization characteristics of the radiation from the undulator were studied in the UV region using a polarimeter [8]. The state of the polarization can be represented by the Stokes parameters, S0 , S1 , S2 and S3 . In this case, the degree of circular polarization is specified by Pc (= S3 /S0 ). Figure 6.9 shows the Pc s and the spectra of the first harmonic radiation in the helical field configuration with E = 227 MeV, K = 0.57 in the undulator PU-2. The

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1.0

0.9 Measurement Cal. with emittance Cal. without emittance Measurement

Pc

0.8

0.7

0.6

0.5

260

280

300

320

340

360

380

400

Wavelength (nm)

Figure 6.9 Spectra and degrees of polarization of the first harmonic radiation in the helical field configuration with E = 227 MeV and K = 0.57. The dots indicate the measured degrees of circular polarization Pc and radiation spectra. The solid and dashed lines indicate the calculated spectra with and without beam emittance effect, respectively [8].

dots indicate the measured spectra and degrees of the polarization. The solid lines indicate the calculated spectra and degrees of polarization with the beam emittance effect, and the dashed lines represent those without the beam emittance effect. The method of calculation is the same as that previously reported in detail [9]. Accurate electron-beam profiles have been experimentally determined and modeled by the Monte Carlo method. The calculation was performed by tracing about 100 representative electrons whose distribution was assumed to be Gaussian and neglecting the angular divergence and energy spread of the electron beam. The calculated Pc s without the emittance have sharp dips at 266.5, 307.5, 332, 361 and 395 nm, as shown in Figure 6.9. These dips are smoothed in the calculation of the Pc s including the emittance. The measured spectra are normalized at the first harmonic peak for comparison with the calculated spectrum with the emittance. Figure 6.9 also shows that the degree of circular polarization Pc is higher than 95% between 270 and 320 nm at the first harmonic, and, at the peak wavelength, Pc is almost 100% within measurement uncertainty (about a few percent). We note in conclusion that we can expect almost 100% circular polarization on the axis from a helical field undulator. The dependence of the polarization ellipse on the observing direction was studied in detail in [10]. Figure 6.10 illustrates the experimental results showing the dependence of the polarization ellipse of the first harmonic radiation (400 nm) on the observing direction in the undulator (PU-1). It should be noted that each ellipse is slightly distorted by the birefringence of the silica glass window through which the undulator radiation passes. The features of the polarization ellipses in the horizontal and vertical axes in Figure 6.10(a) and (c) are qualitatively the same as those that were theoretically predicted [5]. Figure 6.10(b) shows that only linear polarization can be observed for all observing directions in the case of plane field configuration. The radiation is polarized nearly in the vertical plane for small values of γ θ, while for larger values the plane of polarization rotates. These characteristics are very close to the theoretical calculation of Kitamura [11]. Both PU-1 and

222

H. Onuki Z

(a)  = 1  2

0.8 0.4 0

X



–0.4 –0.8 (b)

=

0.8 0.4 0

X



–0.4 –0.8 (c)  = 3  2

0.8 0.4 0

X



–0.4 –0.8 1.2

0.8

0.4

0 

–0.4

–0.8 –1.2

Figure 6.10 Dependence of the polarization ellipse of the first harmonic radiation on the observing direction at E = 230 MeV and K = 1.0 [10].

PU-2 polarizing undulators had little influence on the positions of the electron beam and light beam, and they did not disturb experiments in the other beamlines during the polarization modulation with a frequency of 3 Hz. 2.1.3

Planar undulators

Elleaume proposed the use of a planar type undulator to obtain the crossed and retarded magnetic fields of Eqn (1) [12]. The planar undulator has the advantage that the large horizontal

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223

aperture required for electron beam injection can easily be accommodated. The planar undulator consists of two magnet arrays symmetrically positioned above and below a flat vacuum duct in the electron storage ring. The upper magnet array produces a horizontal sinusoidal magnetic field while the lower magnet array produces a vertical sinusoidal magnetic field of an identical spatial period. The polarization is controlled by longitudinally displacing one magnet array with respect to the other. One of the examples proposed by Elleaume is shown in Figure 6.11. In comparison with conventional planar undulators, this example requires precise alignment of the undulator with respect to the electron beam. The concept of a planar-type polarizing undulator was realized by constructing a tandem-type linear/helical undulator, Helios, in which two planar-type undulators were used in tandem [13]. Since 1993, several of these devices have been in use at the ESRF. In Sincrotrone Trieste, Diviacco and Walker proposed a similar planar-type undulator that produces higher magnetic field strength, as shown in Figure 6.12 [14]. This undulator consists of two halves, one a set of permanent-magnet blocks with alternating vertical and horizontal magnetization assembled in a single magnetic array, the other half being symmetrical with respect to the horizontal midplane except for an inversion of the direction of magnetization. Each magnet array produces vertical and horizontal field components. The improved symmetry of the magnetism results in there being no longitudinal field component on the axis. The helicity remains perfectly circular, independent of gap, and the handedness is invariant, that is, the device has fixed circular polarization. A more advanced design was proposed and built by Sasaki et al. [15–17]. In their proposed planar device, APPLE-I, each array consists of two adjacent conventional Halbach-type magnet arrays as shown in Figure 6.13. The lower-front and upper-back arrays are fixed in space with 45◦ tilted magnetization. The lower-back and upper-front arrays with a magnetization direction of 135◦ can be longitudinally shifted to switch the polarization. As the gap between the upper and lower arrays is changed, the ellipticity of the polarization varies somewhat.

z y x

u 2

Figure 6.11 Planar-type polarizing undulator proposed by Elleaume [12].

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H. Onuki

z y x

u

Figure 6.12 Schematic view of the planar circularly polarized undulator proposed by Diviacco and Walker. (Reproduced with permission from [14].) D

e–

z D

y

u x

Figure 6.13 Schematic view of planar polarizing undulator proposed by Sasaki et al. (Reproduced with permission from [15].)

The photon energy of the fundamental varies when the polarization state is switched. Such a device was installed in the low-energy ring, JSR, at JAERI in Japan, and was successful in producing various polarizations [17]. During the processes of shifting the arrays and changing the undulator gap, no noticeable change in radiation axis was observed [17]. A similar

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225

device was constructed at SSRL, and the polarization in the X-ray region was measured using a multi-layer polarimeter. The undulator consists of 26 periods, each of which is 6.5 cm long. It generates X-rays in the range of 500–1000 eV when SPEAR is operated at 3 GeV. The variation of Stokes parameters with row phase φρ (upper-left and lower-right move one way while upper-right and lower-left move in the opposite direction) is shown in Figure 6.14. The measurement was around a photon energy 708 eV. S0 , S1 , S2 , and S3 were normalized by S0 . S2 represents linear polarization at ±45◦ , and is negligible for all φρ . The circular polarization shows a broad peak at φρ = 0.16λu (λu : period of the magnetic field) with a maximum of S3 = 0.98. The maximum of vertical linear polarization S1 was −0.90 at φρ = 0, while the maximum of horizontal polarization was 0.98 at φρ = 0.48λu . The time required for the mechanical switching between the right- and left-circular polarizations was about 5 s. A modified Sasaki type undulator which incorporates the design of a magnetic block shape to get a large good-field region and better field performance in tuning the polarization was proposed [19] after the prototype had been fabricated, installed in the SRRC (in Taiwan), and successfully tested [20]. In the new design, the dimensions of each magnet block were changed into 40 mm(width) × 31 mm(height) with a protrusion of 0.5 mm at the edge closest to the beam axis. A different type of the planar undulator was proposed by Kimura, Kamada, Hama, Maréchael, Tanaka and Kitamura [21]. Each magnetic array in their undulator consists of three lanes (Figure 6.15). The center lanes and side lanes above and below the electron beam provide vertical and horizontal magnetic fields. The phase between the horizontal and vertical fields can be changed by shifting the side lanes. The undulator has an advantage over other planar undulators in that the photon energy of the undulator radiation can be changed while maintaining the degree of circular polarization. Such a device was constructed and installed in the 0.75 GeV storage ring UVSOR at the Institute for Molecular Science. Another was installed in the 0.7 GeV racetrack-type compact ring HiSOR at the Hiroshima Synchrotron Radiation Center [22].

Stokes parameters

1

0 S0 S1 S2 S3

–1 0

0.1

0.2 0.3 Row phase

0.4

0.5

Figure 6.14 Stokes parameters as a function of row phase. (Reproduced with permission from [18].)

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Helical configuration

Linear configuration

Figure 6.15 Schematic drawings of the polarizing undulator installed in the UVSOR and HiSOR storage rings [21,22].

2.1.4

Tandem-type undulators

In general, experiments in circular dichroism (CD) or magnetic circular dichroism (MCD) require fast cycling of the polarization handedness of circularly polarized radiation for the AC amplification of small CD or MCD signals. The fast cycling can be achieved by changing the electron beam orbit in a tandem undulator. There are two methods for fast switching. One switching method was proposed by Elleaume [13]. His device, Helios, consists of two planar-type undulators in a tandem configuration and chicane magnets, as shown in Figure 6.16. Each planar undulator consists of two magnet arrays. The upper (lower) magnet array generates the horizontal (vertical) field. The vertical magnetic fields of both undulators are identical whereas the horizontal fields are reversed, resulting in opposite circular polarization between the upstream and downstream undulators. The chicane consists of magnet blocks placed at the extremities of each undulator segment. The chicane enables the electron beam to propagate at different angles in the two undulators. The angle between the two radiation directions of the two undulators is between 200 and 300 µrad in the 6 GeV storage ring. The chicanes dissociate the two radiation beams with opposite polarization generated by the upstream and downstream undulator sections. If both beams are reflected and focused on the sample in the experimental station by a mirror and a chopper wheel is placed in front of the mirror, fast periodic flipping of the polarization state of the X-ray beam can be produced. The polarization can be switched from circular to linear by changing the longitudinal position of the upper magnet arrays in the upstream and downstream undulators. The other type of fast switching in a tandem-type undulator, which does not require the use of a chopper wheel in its beamline, was proposed by Hara et al. [23]. Their method uses kicker magnets. At SPring8, they use five kicker magnets to make and switch two

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Phasing Gap

Figure 6.16 Magnet design of Helios. The chicane magnets are shaded. (Reproduced with permission from [13].)

Orbit A First undulator

Second undulator

Off-axis radiation ≅ 300 µrad On-axis radiation

Electron beam

Kicker

Right-hand circular polarization

Left-hand circular polarization

Off-axis radiation

Orbit B

Figure 6.17 Tandem undulator proposed by Hara et al. [23].

local orbit bumps of the electron beam (orbit A and B in Figure 6.17). The first and second planar undulators, which are of the same type as in Figure 6.15, are set to the left- and right-handed circular polarizations, respectively. The electron beam on orbit A generates off-axis radiation (right-handed circular) at the first undulator, and the radiation is stopped at the front-end absorber, while the beamline-axis radiation (left-handed circular) at the second undulator goes to the experimental station. In the case of orbit B, the right-handed circular radiation is emitted on the axis at the first undulator. The kick angle of 300 µrad is enough to dissociate the two undulator beams. The kicker magnets are driven by a quasi-square wave. The expected switching frequency is 10 Hz, which is determined by the capacity of the power source for the kickers. This tandem undulator is being installed in the beamline at SPring8. Although we can expect high-frequency switching of the handedness of the polarization of the tandem undulators in both Figures 6.16 and 6.17, these configurations sacrifice monochromatic flux because only half the total period of the undulator is used in the experiments. Since the distances from the first and second undulators to the experimental station are slightly different, the radiation intensity is slightly different between the righthanded and left-handed circular polarizations. We must consider this difference in intensity in the measurement of CD or MCD spectra. 2.1.5

Electromagnet undulators with crossed and retarded magnetic fields

Recently, electromagnet undulators that are capable of fast switching for the helicity of the magnetic field were proposed and are being built. One of these is a polarizing undulator

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proposed by Nahon et al. [24,25]. Although the configuration of the magnet arrays is the same as the crossed and overlapped undulator proposed by Onuki, electromagnets are used to produce two identical vertical and horizontal magnetic fields. Polarization switching is implemented by a simple inversion of the polarity of one of the power supplies driving the magnetic fields. Such a device was constructed and installed in Super-ACO (Figure 6.18). Their polarizing undulator, OPHÉLIE, consists of two identical ten-period magnet arrays (each 25 cm long), which are orthogonal to each other, and which have succeeded in producing undulator radiation. The second type of electromagnet undulator was proposed by Chavanne et al. [26]. In this polarizing undulator, the horizontal magnetic field is produced with an array of permanent magnets similar to that shown in Figure 6.11, whereas the vertical magnetic field is produced with an array of electromagnets. Such a device is under construction at the ESRF. In the magnet design, both horizontal and vertical fields can be varied between 0 and 0.21 T. It is expected that any state of polarization can be generated at a repetition rate between DC and 20 Hz. The APS and BINP (Budker Institute of Nuclear Physics) joint group proposed that both vertical and horizontal fields were made by electromagnets whose undulator is similar to the Nahon type, but is actually a planar type [27]. The cross section of both arrays is shown in Figure 6.19. The important feature of the undulator is to change the helicity of the circular polarization with a frequency of 10 Hz, as well as to change the direction of linear polarization by 90◦ .

Figure 6.18 Electromagnet polarizing undulator installed in the Super ACO composed of two crossed undulators with 10 periods of 25 cm. The only mechanical motion is a longitudinal translation of the vertical undulator [24]. (Courtesy of L. Nahon.)

Polarizing undulators and wigglers

229

Horizontal field pole cross section

Horizontal coils

Vertical coils

Vacuum chamber

Vertical field pole cross section

Figure 6.19 Planar type electromagnet polarizing undulator with crossed and retarded magnetic fields [27].

z y

e–

x Variable phase delay (electron path length modulator)

Figure 6.20 Schematic view of the crossed undulator. The shaded block is the modulator [29].

2.2

Polarizing undulators with interference effect (crossed undulators)

Moissev, Nikitin, and Fedosov [28], and Kim [29] independently proposed a crossed undulator, as shown in Figure 6.20, which utilizes an interference phenomenon. As illustrated in Figure 6.20, two undulators oriented at 90◦ with respect to each other are separated axially. In the first undulator, electrons oscillate in the x-direction, emitting linearly polarized radiation in the x-direction. In the second undulator, the radiation is polarized in the z-direction.

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The phase φ between the radiations from the two undulators is given by
 2π Lt φ= −L λ β

(7)

where λ is the radiation wavelength, β the electron speed/light speed, Lt the length of the electron trajectory from the beginning of the first undulator to the beginning of the second undulator, and L the straight distance between the beginning of the first and second undulators [29]. We can observe polarized radiation of any ellipticity by changing φ. The polarization can be modulated mechanically by changing the distance between the two undulators. More conveniently, the polarization can be modulated electromagnetically by introducing a short magnet, called the modulator, between the two undulators. The idea of using a modulator was proposed by Kim [29]. The optical wavetrain emitted from the second undulator is separated in time from the one emitted from the first one. An optical device such as a monochromator can be used to enable the second wavetrain to interfere with the first one. The monochromator stretches the wavetrain in the axial direction so that the wavetrains from the two undulators overlap in time after passing the monochromator. This type of undulator was constructed and installed in the electron storage ring BESSY [30,31]. A modulator with a three-pole electromagnetic structure is located between two undulator segments and controls the state of the polarization. The polarization is measured by a polarimeter consisting of two reflection units that rotate independently around a common axis. According to experimental results [30,31], the degree of circular polarization at a photon energy of 30 eV is about 50%. The radiation from each undulator is linearly polarized up to about 90%. The advantage of the crossed undulator is the rapid switching between the right- and left-states of circular polarization. It is anticipated that the modulator enables modulation of the polarization with frequencies up to 10 Hz. The degree of circular polarization depends on the size of the electron beam and angular divergence. It seems likely that satisfactory performance by a crossed undulator requires a low-emittance electron beam. We have no optical device that can perfectly overlap the two optical wavetrains from the undulators in time. Hence, it is basically difficult for the degree of circular polarization to reach 100%.

3 3.1

Wigglers Elliptical (magnetic field) wigglers

An elliptically polarized magnetic field can efficiently generate radiation with a high degree of circular polarization at higher harmonics. This type of insertion device was independently proposed by Bessonov [32,33], and by Yamamoto and Kitamura [34]. The elliptical wiggler constructed in the TRISTAN Accumulation Ring [36], which is structurally analogous to the crossed and overlapped undulator proposed by Onuki (Figure 6.7), consists of a strong vertical magnetic field of a longer period and a weak horizontal field of the same period, as shown in Figure 6.21. The strong vertical field produces broadband wiggler radiation, and the relatively weak horizontal field phase shifted a quarter period alternately bends the electron trajectory up, then down. Perhaps an easier way to understand radiation from an elliptical wiggler is to approximate the electron trajectory by a three-dimensional sinusoidal curve, in which the adjacent half-periods lie on different planes, as illustrated in Figure 6.22. This has

Electron beam

Figure 6.21 Schematic illustration of an elliptical wiggler. The electrons move on the surface of the virtual elliptical cylindroid. (Reproduced with permission from [34].)

A

B A

Figure 6.22 Schematic of the electron trajectory in the elliptical wiggler. Radiations from A and B have the same polarization with the same direction of rotation. (Reproduced with permission from [35].)

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been described elsewhere by Kim [35]. Locally “off-axis” elliptically polarized light (B) in the device midplane adds to like-handedness elliptically polarized light (A) from subsequent half-periods also directed in the device midplane. Figure 6.23 indicates the Kx dependence of the axis ratio, χ , of polarization ellipse, and the spectral function, Gn , which is proportional to the radiation power, when Kx is varied from 0 to 2.0 with Kz fixed at 2.0 [34]. The higher the harmonic number, the better the degree of polarization. When the electron orbit is strongly distorted, the radiation power is emphasized. For example, in the case of Kx = 1, the 7th harmonic radiation has a value for χ as high as 0.74 while its power becomes a tenth of the fundamental. The first elliptical wiggler (with λu = 16 cm, N = 21, Ky ≤ 15 and Kx ≤ 3) was installed in the straight section of the 6.5 GeV TRISTAN accumulation ring, and succeeded in producing elliptically polarized intense X-rays [36]. The direction of the polarization ellipse in this device can be reversed mechanically by translating the horizontal magnet arrays, which requires several minutes because the vertical gap must first be opened in order to reduce the strong magnetic force acting on the horizontal magnet blocks. Recently, a planar type of elliptical wiggler was designed with permanent magnet blocks by Maréchal et al. [37], and built and installed in the SPring8 storage ring [38]. This wiggler, with

(a)

1.0

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.8 n =1



0.6 7

0.4

5

3

17

0.2

Kz = 2.0

0 (b)

1 n =1 10–1

3

Gn (Kx, Kz )

5 7

10–2

10–3

10–4 0

11 13 15 17

Kz = 2.0

0.2

0.4

0.6

0.8

1.0 Kx

9

1.2

1.4

1.6

1.8

2.0

Figure 6.23 Dependence of the axis ratio, χ (a), and the spectral function, Gn (Kx , Kz ) (b), on the value of Kx with Kz fixed at 2.0. (Reproduced with permission from [34].)

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a 12 cm period, has a critical energy of 50 KeV at a 2 cm gap. It will provide high-brilliance elliptically polarized hard X-rays in the 100–300 KeV range. An elliptical wiggler can also be constructed with electromagnets. At the Advanced Light Source (ALS) at Lawrence Berkeley Laboratory, a new design proposed that the device features vertical and horizontal magnetic structures of 14 and 14 21 periods, respectively [39]. The vertical structure is designed with a hybrid permanent magnet producing a peak field of 2.0 T. The horizontal structure is an iron core electromagnetic design, shifted longitudinally by 1/4 period, and has a maximum peak of 0.095 T at a frequency up to 1 Hz. An analogous elliptical wiggler was jointly designed, built, and tested by the APS, NSLS, and the Budker Institute of Nuclear Physics (Novosibirsk) [40,41]. The full-size wiggler was installed in the APS and consists of planar arrays of magnets and poles to produce the vertical magnetic field and an electromagnet to generate the weaker horizontal field. The horizontal electromagnetic field can be operated at a frequency of 10 Hz, at a peak current of 1 kA. This current generates a horizontal field of 0.0997 T (Kx = 1.5). The peak vertical field is 0.983 T at the minimum gap of 2.4 cm (Kz = 14.7). Experimental tests showed that helicity switching occurred as was expected [40]. A planar type of electromagnetic elliptical wiggler in which both the vertical and horizontal magnetic fields are produced by electromagnets was designed, constructed, and installed in the 2.0 GeV synchrotron light source Elettra, and tested [42,43]. The device was designed to provide a source of circularly polarized light in the VUV/soft X-ray region with a variable helicity of up to a 100 Hz switching rate in the horizontal field.

3.2

Asymmetric wiggler

It can be understood that the linear polarization of the radiation from the usual planar undulator or wiggler employing a sinusoidal magnetic field is caused by combining the circular polarization in a half-period section of a positive pole with the opposite circular polarization in the adjacent half-period of a negative pole. Therefore, an asymmetry of the magnetic field u

Bz

Gap

Air space

Figure 6.24 Schematic of a magnet array in the asymmetric wiggler at the HASYLAB, and its magnetic field distribution. (Reproduced with permission from [46].)

234

H. Onuki 1.0

Degree of circular polarisation

E= 110564 eV 0.8

0.6

0.4

0.2

0.0

0

0.05

0.1 0.15 ver (mrad)

0.2

0.25

Figure 6.25 Degree of circular polarization at 11.564 keV as a function of the vertical angle. The full line represents the calculated result. A representative error bar is also shown. (Reproduced with permission from [46].)

can produce quasi-circular polarized radiation. The asymmetry can be built by arranging the magnets of opposite poles to have different strengths and lengths, so as to produce a net zero integrated field over the length of the wiggler. This so-called asymmetric wiggler was proposed by Goulon et al. [44], and such a wiggler was installed in the Super-ACO [45], HASYLAB [46], and ESRF [47]. At HASYLAB [47], the asymmetric magnetic structure is arranged as shown in Figure 6.24. The entire magnet structure resembles an array of one-period wigglers and air space composed in each module where the weak poles are formed by the half-poles. The period is 24 cm with an air space of 8 cm, which can easily be varied. A ten-period asymmetric wiggler was installed. At a gap of 3.4 cm, a peak field of 0.82 T is achieved and the K parameter reaches 12.7. The degree of circular polarization of the radiation from the asymmetric wiggler was tested. Figure 6.25 indicates the degree of circular polarization of 11.564 keV photons below the orbital plane as a function of the vertical angle. Magnetic circular dichroism (MCD) of the Pt L3 edge of a Pt–Fe alloy was used for the first measurement because the MCD of the sample was precisely known from earlier measurements on a bending magnet SR. The handedness of circular polarizations above and below the orbital plane is opposite to each other. Basically, perfect circular polarization cannot be obtained from an asymmetric wiggler.

References [1] Kincaid, B. M., J. Appl. Phys. 48, 2684 (1977). [2] Keserashvily, G. Ya., A. P. Lyssenko, V. M. Horev and Yu. M. Shatunov, Proc. All Union on Synchrotron Radiation, SR-82, Novosibirsk, 109 (1983). [3] Kondo, J., Electrotechnical Laboratory Survey Report, No. 200, p. 18 (1979).

Polarizing undulators and wigglers [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

[21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]

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Elias, L. R. and J. M. Madey, Rev. Sci. Instrum. 50, 1335 (1979). Alferov, D. F., Yu. A. Bashmakov and E. G. Bessonov, Sov. J. Tech. Phys. 21(11), 1408 (1976). Onuki, H., Nucl. Instrum. Meth. A246, 94 (1986). Onuki, H., N. Saito and T. Saito, Appl. Phys. Lett. 52, 173 (1988). Yuri, M., K. Yagi, T. Yamada and H. Onuki, J. Electron Spectr. Related Phenom. 80, 425 (1996). Yagi, K., M. Yuri, S. Sugiyama and H. Onuki, Rev. Sci. Instrum. 66, 1993 (1995). Yagi, K., H. Onuki, S. Sugiyama and T. Yamazaki, Rev. Sci. Instrum. 63, 399 (1992). Kitamura, H., Jpn. J. Appl. Phys. 19, L185 (1980). Elleaume, P., Nucl. Instrum. Meth. A291, 371 (1990). Elleaume, P., J. Synchrotron Rad. 1, 19 (1994). Chavanne, J., E. Chinchio, M. Diot, P. Elleaume, D. Frachon, X. Maréchal, C. Mariaggi and F. Revol, Rev. Sci. Instrum. 63, 317 (1992). Diviacco, B. and R. P. Walker, Nucl. Instrum. Meth. A292, 517 (1990). Sasaki, S., K. Miyata and T. Takada, Jpn. J. Appl. Phys. 31, L1794 (1992). Sasaki, S., K. Kakuno, T. Takada, T. Shimada, K. Yanagida and Y. Miyahara, Nucl. Instrum. Meth. A331, 763 (1993). Sasaki, S., T. Shimada, K. Yanagida, H. Kobayashi and Y. Miyahara, Nucl. Instrum. Meth. A347, 87 (1994). Carr, R., J. B. Kortright, M. Rice and S. Lidia, Rev. Sci. Instrum. 66, 1862 (1995). Fan, T. C., C. S. Hwang, C. H. Chang, Ch. Wang and J. R. Chen, Proc. EPAC’98 Conference, pp. 2219–21 (1998). Hwang, C. S., C. H. Chang, H. P. Chang, K. T. Pan, Jenny Chen, C. H. Kuo, G. Y. Hsiung, J. R. Chen, K. T. Hsu, Ch. Wang, G. H. Luo, D. J. Wang, C. C. Kuo and K. K. Lin, Proc. EPAC’98 Conference, pp. 2225–7 (1998). Kimura, S., M. Kamada, H. Hama, X. M. Maréchal, T. Tanaka and H. Kitamura, J. Electron Spectr. Related Phenom. 80, 437 (1996). Hiraya, A., K. Yoshida, S. Yagi, M. Taniguchi, S. Kimura, H. Hama, T. Takayama and D. Amano, J. Synchrotron Rad. 5, 445 (1998). Hara, T., T. Tanaka, X.-M. Maréchal, K. Kumagai and H. Kitamura, J. Synchrotron Rad. 5, 426 (1998). Nahon, L., M. Corlier, P. Peaupardin, F. Marteau, O. Marcouillé, P. Brunelle, C. Alcaraz and P. Thiry, Nucl. Instrum. Meth. A396, 237 (1997). Nahon, L., M. Corlier, P. Peaupardin, M. Marteau, O. Marcouillé and C. Alcaraz, J. Synchrotron Rad. 5, 428 (1998). Chavanne, J., P. Elleaume and P. Van Vaerenbergh, Proc. EPAC’98 Conference, pp. 317–19 (1998). Gluskin, E., J. Synchrotron Rad. 5, 189 (1998). Moissev, M., M. Nikitin and N. Fedosov, Sov. Phys. J. 21, 332 (1978). Kim, K.-J., Nucl. Instrum. Meth. 219, 425 (1984). Bahrdt, J., A. Gaupp, W. Gudat, M. Mast, K. Molter, W. B. Peatman, M. Scheer, Th. Schroeter and Ch. Wang, Rev. Sci. Instrum. 63, 339 (1992). Bahrdt, J., A. Gaupp, W. Gudat, M. Mast, K. Molter, W. B. Peatman, M. Scheer, Th. Schroeter and Ch. Wang, SR News 5, 12 (1992). Bessonov, E. G., Proc. 6th All-Union Conference on the Physics of Far Ultraviolet Radiation and the Interaction of Radiation with Matter (Moscow, 1982). Bessonov, E. G. and E. G. Gaskevich, Sov. Phys. Lebedev-Inst. Reports No. 7, p. 16 (Allerton Press Inc., 1985). Yamamoto, S. and H. Kitamura, Jpn. J. Appl. Phys. 26, L1613 (1987); Kitamura, H., Synchrotron Radiation News 5, 14 (1992). Kim, K.-J., SPIE 1345, 116 (1990). Yamamoto, S., H. Kawata, H. Kitamura, M. Ando, N. Sakai and N. Shiotani, Phys. Rev. Lett. 62, 2672 (1989). Maréchal, X. M., T. Tanaka and H. Kitamura, Rev. Sci. Instrum. 66, 1937 (1995).

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[38] Maréchal, X. M., T. Hara, T. Tanabe, T. Tanaka and H. Kitamura, J. Synchrotron Rad. 5, 431 (1998). [39] Hoyer, E., J. Akre, D. Humphries, S. Marks, Y. Minamihara, P. Pipersky, D. Plate and R. Schlueter, Rev. Sci. Instrum. 66, 1895 (1995). [40] Gluskin, E. et al., Proc. PAC 95 (Dallas, Texas, 1995). [41] Singh, O., S. Krinsky, P. M. Ivanov and E. A. Medvedko, Rev. Sci. Instrum. 67 (1996) (CD ROM). [42] Walker, R. P., D. Bulfone, B. Diviacco, W. Jark, P. Michelini, L. Tosi, R. Visintini, G. Ingold, F. Schäfers, M. Scheer, G. Wüstefeld, M. Eriksson and S. Werin, Proc. EPAC’97 Conference, pp. 3527–9 (1997). [43] Tosi, L., B. Diviacco, R. P. Walker and D. Zangrando, Proc. EPAC’98 Conference, pp. 1353–5 (1998). [44] Goulen, J., P. Elleaume and D. Rauox, Nucl. Instrum. Meth. A254, 192 (1987). [45] Barthés, M., A. Daël, P. Elleaume, C. Evesque, J. Goulon, G. Krill, C. Laffon, L. Leclerc, J. Michaut and D. Rauox, Proc. 10th Int. Conf. Magnet Technology (Leningrad, 1991). [46] Pflüger, J. and G. Heintze, Nucl. Instrum. Meth. A289, 300 (1990). [47] Chavanne, J. and P. Elleaume, Rev. Sci. Instrum. 66, 1868 (1995).

7

Exotic insertion devices Shigemi Sasaki

1

Introduction

It is not very easy to define what ‘exotic’ insertion devices really are. For example, the first permanent magnet insertion device for generating circularly polarized radiation was considered an exotic device when it was proposed [1]. The planar devices for circular polarization were also considered exotic when they were proposed [2–5]. However, these devices have already been used in many synchrotron radiation facilities and are no longer called exotic. Therefore, in this chapter, devices that have some interesting features but are not yet used widely are discussed.

2 2.1

Quasi-periodic undulator Introduction

Conventional undulators have periodic magnetic structures that generate radiation with higher harmonics. In general, the higher harmonics of radiation are harmful in experiments and are usually removed by means of instruments in an optical beamline, such as a total reflection mirror, gas filter, etc. These conventional techniques to get rid of the higher harmonics, however, may not be appropriate for some beamlines. A concept of quasi-periodic undulator (QPU) that never generates rational harmonics was proposed in order to avoid the problem of higher harmonics [6,7]. The idea originally came from an analogy between the diffraction property of crystals and the radiation property of undulators. It is well known that a quasi-periodic lattice causes sharp diffraction peaks without translational symmetry in a reciprocal space. Also, we recognize the formal equivalence between the equation of X-ray scattering by matter and that of synchrotron radiation. The X-ray intensity diffracted by a one-dimensional scatterer is [8]   I (q) = 

∞

−∞

ρ(r)e

−2π iqr

2  dr 

(1)

where ρ(r) is the electron density of the scatterer. On the other hand, the spectral angular intensity distribution of synchrotron radiation is [9] d2 I (ω) e2 = dω d. 16π 2 ε0 c

2    ∞ n × {[  ˙ n − β(t)] × β(t)}   eiω[t−n·r (t)−c] dt    2   −∞ [1 − n · β(t)]

(2)

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S. Sasaki

As seen above, the Eqns (1) and (2) have the same form. Therefore, one can recognize that the radiation from a QPU can be treated in the same manner as the diffraction from a onedimensional (1D) quasi-crystal. The idea of a QPU is to let electrons draw a quasi-periodic trajectory in an undulator in order to produce the irrational harmonics of radiation.

2.2

Principles – Creation of quasi-periodicity

One of the simplest ways of creating the 1D quasi-periodic lattice is to use the projection from the two-dimensional (2D) square lattice. The projection of lattice points in a window with a certain width on to the irrationally inclined line gives the 1D quasi-periodic lattice [6]. Figure 7.1 shows how to create a 1D quasi-periodic lattice with two kinds of lattice points (open and full circles) corresponding to the sites of electron scatterers (usually corresponding to the magnetic poles) in positive and negative directions. Here, we draw the line AA through the origin with an irrational gradient of tan α with respect to the x-axis. In order to create a 1D quasi-periodic lattice, we next define the window AA B B inclined with the slope of tan α, and we project the lattice points contained in the window onto the inclined axis AA (hereafter we refer to this axis as R ). As shown in Figure 7.1, we intuitively find that the points projected onto R have coordinates of (R , 0) with two kinds of inter-site

y

B⬘

R⊥ W/cos

W

R|| A⬘

Window d

B a

a

d⬘ 

Negative matter

x

O A

Positive matter

Figure 7.1 Creation of 1D quasi-periodic lattice with two kinds of scattering centres from a 2D square lattice. Open circles represent a positive centre and full circles a negative one.    The lattice points √ in the window AA B B are projected onto AA . The slope, tan α, is taken to be 1/ 5.

Exotic insertion devices

239

distances, d = a sin α and d  = a cos α having a ratio of d 1 = d tan α

(3)

The points are aligned in a quasi-periodic fashion defined by 5 4 tan α  Ri = ia cos α + a(sin α − cos α) i+1 1 + tan α

(4)

for the ith point, where )z* stands for the greatest √ integer less than z. In this example, we selected an irrational inclination of tan α = 1/ 5. We can redefine the coordinate along the inclined line as 5 4 tan α zˆ m = m + (tan α − 1) m+1 (5) 1 + tan α where m is an integer. The first term in this equation corresponds to a periodic component of spacing between the lattice points and the second term increases the distance by a factor of (tan α − 1) quasi-periodically. Hence the distance between any two consecutive positions (zm − zm−1 ) takes a value of 1 or tan α, forming a quasiperiodic array, where zm = zˆ m d  2.3

Modified creation theory of quasi-periodicity

In the previous section, the 2D square lattice was used for creating the 1D quasi-periodic lattice. In this case, the parameter we can change √ is tan α only, and the appropriate values of tan α are limited. For example, tan α = 1/ 5 is a good choice for eliminating the third harmonic in the brilliance spectrum, but it is not a good choice in the flux spectrum. The non-negligible amount of flux at the positions of rational harmonics appears when opening up the aperture. This problem may be solved by reducing the inclination (corresponding to a smaller value of tan α). However, this is not realistic because the ratio d  /d becomes too great. In order to increase the choices of 1D quasi-periodic lattices for various needs, an additional degree of freedom was introduced [10]. As shown in Figure 7.2, a 2D rectangular lattice with lattice parameters of a and b was introduced for creating a new 1D quasi-periodic lattice. As we did in the previous section, we first draw a window, AA B B, inclined with a slope of tan α against the horizontal axis. Next, we project the lattice points in the window onto the inclined axis AA (hereafter referred to as the z-axis). From a simple consideration, we obtain the position of the mth quasi-periodic lattice point along the z-axis as follows: zm = zˆ m

d r tan α

(6)

and 4

5

tan α zˆ m = m + (r tan α − 1) m+1 r + tan α

(7)

240

S. Sasaki B⬘

A⬘

d⬘ B

d b a



A

Figure 7.2 Creation of a 1D quasi-periodic lattice from a 2D rectangular lattice. In this example, √ the slope, tan α, is taken to be 1/ 5 and the ratio, r(= b/a), is taken to be 1.5. A 1D quasi-periodic lattice is produced on the AA line.

where r = b/a. As we see in Figure 7.2, we have two inter-site distances d = b sin α and d  = a cos α. Therefore, d  /d = 1/r tan α. This new scheme allows us to select a smaller inclination, tan α with the value of r larger than 1 in a practical range of the ratio, d  /d. 2.4

Magnetic structure of QPU

Hereafter, in this chapter, the orthogonal set of axes Ox, Oy and Oz is used instead of Ox, Oz and Os, which is used in the other chapters. Ox and Oy are the horizontal and vertical transverse axes while Oz is the longitudinal axis. Since the x–y–z coordinate system is used in all figures and references in this chapter, this system is also used in sentences and equations in the rest of this chapter in order to avoid confusions. One of the simplest ways to construct a QPU is to place the magnetic poles at the theoretical positions along the undulator axis. This simple scheme can be applied either to the permanent magnet system or to the electromagnetic system. A prototype QPU was constructed at the Japan Atomic Energy Research Institute (JAERI) and its performance was tested at the Electro-Technical Laboratory (ETL) [11,12]. Figure 7.3 shows the schematic drawing of the QPU prototype. The arrows in the magnet blocks represent the directions of magnetization. The magnet blocks isolated from their neighbours are thinner by a factor of 0.7 to reduce the strength of the on-axis magnetic field at the same magnitude as the other non-isolated

Exotic insertion devices

241

d × 0.35 d × 0.7 d y Gap

z

d⬘ d d⬘ d⬘ d d⬘ d⬘ d d⬘ d⬘ d⬘ d d⬘ d⬘ d d⬘ d⬘ d d⬘ d⬘ d d⬘ d⬘ d⬘ d d⬘

Figure 7.3 Schematic drawing of QPU magnetic structure prototype.



Figure 7.4 Schematic drawing of ESRF-QPU with new scheme. The horizontally magnetized blocks at certain positions predicted by the QPU theory with the displacement δ.

regular magnet blocks. The magnets at both ends are thinner by a factor of√0.65 in order to compensate for the net steering of the electron beam. In this undulator, 1/ 5 was selected as the value of tan α, and d = 25 mm.

2.5

New magnetic structure of QPU

A new magnet arrangement for a QPU is based on a conventional undulator design with four blocks per period, leaving the longitudinal positions of magnet blocks unaltered and introducing the required phase deviations in some other way. This can be achieved by changing the vertical position of the horizontally magnetized blocks or removing these blocks [13]. At the European Synchrotron Radiation Facility (ESRF), a QPU with the new scheme was applied to the conventional 46 mm period undulator and tested [14]. In this scheme, a new parameter δ was introduced. This parameter stands for the recess from the undulator surface. The recess is applied to the horizontally magnetized block at the quasi-periodic position as shown in Figure 7.4. When δ equals zero, the structure is that of the conventional periodic undulator. The advantages of these undulators with the new scheme are that there is no resulting angular deviation of the trajectory, and that they are much more compact longitudinally compared to the original QPU design. This allows a larger flux per unit length of undulator. The modified creation theory of quasi-periodicity can also be applied to the new magnetic structure. At the Sincrotrone Trieste (ELETTRA), an elliptical quasi-periodic undulator (EQPU) with the new scheme was designed and successfully constructed. A schematic drawing of the 125 mm period EQPU is shown in Figure 7.5 [15]. In this design, the dimensions of magnet blocks were carefully chosen and the horizontally magnetized blocks at the modified theoretical positions were removed in order to achieve the optimum magnetic field variation.

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Figure 7.5 Schematic drawing of ELETTRA-EQPU with new scheme. The horizontally magnetized blocks at certain positions predicted by the modified QPU theory are removed. (Reprinted with permission [15].)

Angular flux (photons/sec/0.1% bw/mrad2)

700 × 1015

Conventional Quasi-periodic A Quasi-periodic B

600 500

H=1

400 300 200 H=3

100

H=5

0 2

3

4

5

6 7 8 9 Photon energy (keV)

10

11

12 13 keV

Figure 7.6 Spectral angular flux density of ESRF-QPU. The radiation spectra from an original QPU and a conventional undulator are also shown for comparison. All devices were assumed to be of the same length. Broken line is for a conventional undulator, solid line for an original QPU, and dashed line for a new QPU. (Reprinted with permission [14].)

2.6

Radiation spectra

Figure 7.6 shows an example of the angular flux density of the ESRF QPU. In this figure, the radiation spectra of a conventional undulator and of a QPU of the original configuration are also shown for comparison. All three undulators were assumed to be of the same length (1656 mm). The photon energy of their fundamental was around 2.4 keV for a gap of 20 mm and an electron energy of 6 GeV. The radiation intensity of the original QPU is the smallest due to a smaller number of poles of the same length. The suppression of the rational harmonics is significant for both the conventional QPU and the new QPU. Figures 7.7 and 7.8 show the flux spectra of the ELETTRA EQPU through a large aperture (0.7 × 0.7 mrad2 ) for the linear and circular polarization modes, respectively [15]. The

Exotic insertion devices

243

1.0 0.9 Normalized flux

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

0

1

2

3

 /1

4

5

6

7

Figure 7.7 Flux spectrum of ELETTRA-EQPU (solid line) in the linear polarization mode. Broken line shows the radiation spectrum of a conventional undulator. The peak intensity is normalized at the position of fundamental. (Reprinted with permission [15].) 1.0 0.9 Normalized flux

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

0

1

2

3

 /1

4

5

6

7

Figure 7.8 Flux spectrum of ELETTRA-EQPU in the circular polarization mode. Broken line shows the radiation spectrum of a conventional elliptically polarizing undulator. (Reprinted with permission [15].)

radiation from a conventional EPU is also shown for comparison. The flux, at integral multiples of the fundamental, is significantly reduced. In particular the intensity of the second, third and fourth harmonics are below 10% of the fundamental peak. It should also be noted that the performance remains good in the circular and elliptical polarization modes.

3 3.1

Figure-8 undulator Principles and magnetic structure

There are many different ways to reduce the higher harmonics from undulator radiation. It is possible to eliminate the contamination of higher harmonics by using a combination of

244

S. Sasaki y

(a)

(b)

y

B

B

A C

u

x

A

z

C D

D (c)

(d) A

y

x

A

B u D

C

B

x

D

C z

Figure 7.9 Electron trajectory in a figure-8 undulator projected on the (a) x–y, (b) y–z or (c) z–x plane and (d) the relation between relative velocities, βx and βy . The period of βy is twice as long as that of βx reprinted from Tanaka et al., Nucl. Instrum. Meth. in Phys. Res. A 364, 369 (1995), with permission from Elsevier Science.

a QPU and a monochromator; however, it is not possible to eliminate the on-axis power load problem in this scheme. The figure-8 undulator was designed and constructed at SPring8 in order to solve the problem of on-axis power density in the linear polarization mode. In Figure 7.9, the electron orbit projected on the x–y, y–z or z–x planes in the figure-8 undulator is shown. The electron velocity projected on βx –βy is also shown. As in a helical device, an electron orbit always has the finite deflection angle of the pointing direction against the undulator axis. This is the reason for the lower on-axis power density with fewer harmonics. The magnetic field components to achieve the electron orbit in Figure 7.9 are described by the following equations [16]:
 2π Bx = −Bx0 sin z (8) 2λu and


2π By = −By0 sin z λu

 (9)

where Bx0,y0 are the peak magnetic fields. The horizontal magnetic field has twice the period length of the vertical magnetic field. The schematic drawing of the magnetic structure of the figure-8 undulator is shown in Figure 7.10.

Figure 7.10 Schematic drawing of a figure-8 undulator reprinted from Tanaka et al., Nucl. Instrum. Meth. in Phys. Res. A 364, 369 (1995), with permission from Elsevier Science.

×1017 (a)

10 Planar K = 4.72

(b)

Photon flux density (photons/sec/mrad2/0.1% bw)

5

1.7 ×1017

0 1.5

1.2 ×1017 1st 1.5th 2nd

1.0

0.5

figure-8 Kx = Ky = 3.34

0.5th

0

0.0 0

2

4

20 40 60 80 Photon energy (keV)

6

100

Figure 7.11 Spectral on-axis flux densities of (a) a planar undulator and (b) a figure-8 undulator. The photon flux densities (photons/sec/mrad2 /0.1% bw) at the fundamental are 1.7 × 1017 and 1.2 × 1017 , respectively reprinted from Tanaka et al., Nucl. Instrum. Meth. in Phys. Res. A 364, 369 (1995), with permission from Elsevier Science.

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S. Sasaki

3.2

Radiation spectra

The comparison between the radiation spectrum of a planar undulator and that of a figure-8 undulator is made for high K values (K = 4.72 for planar undulator and Kx = Ky = 3.34 for figure-8 case) as shown in Figure 7.11. For the calculation of spectra, 8 GeV, 100 mA, and zero emittance were assumed. The period length of the undulator was assumed to be 10 cm and the number of periods 44. The intensity of the fundamental radiation of the figure-8 undulator is about two-thirds that of the planar undulator; in addition, some half-odd-integer harmonics exist in the spectrum. However, this device has the advantage of having a much lower intensity in the higher harmonics as opposed to the characteristics of the planar undulator.

4 4.1

The devices with reduced harmonic intensity Undulator with a non-sinusoidal magnetic field

In the Kurchatov Institute (KSRS), the non-sinusoidal magnetic field undulator was proposed in order to reduce the on-axis power density [17]. It consists of three magnetic poles in a half period, and the central pole in each period has a relatively weak magnetic field value compared to the other two poles. Figure 7.12 shows the computed magnetic field map with the pole structure in a half period and magnetic field [18]. Since there is no pole at the centre of a half period (this is different from the structure originally proposed in [17], which has three poles in a half period), there is a valley at the centre of the half period. As a result of this non-sinusoidal field, the maximum output power is directed off-axis, at the position of the maximum magnetic field. According to calculations for the 30 cm period device, the fundamental peak intensity decreases by only 30% compared with an ordinary undulator. On the other hand, the suppression of the on-axis power density is significant, smaller than 1/3 of an ordinary case. An advantage of this scheme is that an ordinary electromagnetic undulator can be adapted easily to this structure. A disadvantage is that it is not so easy to construct a shorter period undulator as two poles are required instead of one in a half period. 4.2

Crossed EPU

It is planned for many elliptically polarizing undulators (EPUs) to be installed in the storage rings of third generation light sources. Some facilities are planning to install a double EPU in the same straight section with a chicane magnet system in order to achieve a fast switching circular polarization helicity. Such a configuration can be easily modified to a crossed EPU system by replacing a central chicane magnet with a modulator [19]. Assuming the first EPU is set in a right-handed (‘RH’) circular polarization mode, and the second is set in a left-handed (‘LH’) mode, if an electron passed through such a system, the radiation polarization vector would be written as  1  ε = √ ε− + eiφ ε+ 2

(10)

Here ε− (ε+ ) is the complex orthogonal unit vector of a left- (right-) handed circular polarization. The combination of a RH circular polarization and a LH polarization with the phase difference φ results in a linear polarization. The polarization direction depends on the phase φ. The phase φ can be changed by modifying the path length of the electron orbit between

Exotic insertion devices

0

5

247

150

By (z), kGauss

4 3 2 1 0

0

50

100

150

Z,mm

Figure 7.12 The magnetic structure with a computed magnetic field map and magnetic field distribution in a half period. (Reprinted with permission [18].)

the two undulators, and an electromagnetic modulator can be used for the rapid change of phase. As described elsewhere [15,19], the degree of polarization is largely affected by the electron beam divergence, electron energy, and K value. Therefore, this scheme is available only for low energy third generation sources.

4.3

PERA undulator

A new scheme has been studied [20] for generating linearly polarized radiation with fewer higher harmonics and lower on-axis power density at a relatively high K value. The new undulator named ‘PERA’ consists of two different period lengths of magnetic arrays, and has a slightly better performance than the figure-8 undulator with respect to the brightness and the angular distribution of radiation.

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S. Sasaki

Figure 7.13 Schematic drawing of a PERA undulator. Central rows generate vertical magnetic field, while outer rows generate horizontal field.

Figure 7.13 shows a schematic drawing of PERA. The magnetic structure of this device is similar to the figure-8 undulator, but with a different combination of period lengths. In the middle rows, which generate a vertical field, one of every three longitudinally magnetized blocks has a reduced height and an opposite direction of magnetization. Outer rows generate a horizontal magnetic field on the undulator axis. The magnetic field of this device is shown in Figure 7.14. The electron trajectory in this device is a flatter version of the figure-8 undulator, but in the vertical direction. The calculated on-axis flux density of a PERA undulator is shown in Figure 7.15 assuming the magnetic field shown in Figure 7.14 and 2 GeV, 100 mA, zero emittance of the electron beam. It should be noted that the direction of linear polarization is vertical. An advantage of this device is that it generates a slightly higher flux density with a lower power density. A disadvantage is that the structure is more complicated than that of a figure-8 undulator.

5 5.1

Some other exotic undulators Exotic circular devices with harmonics

A novel insertion device for generating the circularly polarized radiation with a wide available energy range was proposed at SPring8 [21]. It consists of two helical undulators with two different period lengths, one of which has a period length three times longer than the other. Figure 7.16 shows a schematic drawing of the device. Unlike the ordinary helical undulator, this device generates higher harmonics with a high degree of circular polarization. This new feature of radiation property may be attractive for some synchrotron radiation users, although the figure of merit Mb , does not exceed that for utilizing higher harmonics of a conventional helical undulator in an elliptical mode. The

Exotic insertion devices

249

0.8

(a)

0.6 0.4

By (T)

0.2 0.0 –0.2 –0.4 –0.6 –0.8 –1000

–500

0 z (mm)

500

1000

–500

0 z (mm)

500

1000

0.6

(b)

0.4

Bx (T)

0.2 0.0 –0.2 –0.4 –0.6 –1000

Figure 7.14 Magnetic field distribution of PERA undulator (a) vertical field, (b) horizontal field.

figure of merit is defined as Mb = BPc2 , where B is the spectral brightness, and Pc is the degree of circular polarization. One of the interesting features of this type of device is that the irrational higher harmonics with a high degree of circular polarization √ appear if an irrational ratio of two period length combinations (for example, λA /λB = 2) is chosen [22]. This additional feature may be useful for filtering out the higher harmonic contamination as we do in the utilization of QPU.

5.2

APPLE-8 undulator

This device consists of magnet rows with two different period lengths and an APPLE type motion mechanism [23,24]. By changing the magnet row phase, various types of radiation are

S. Sasaki Angular flux density (photons/sec/mrad2/0.1%bw)

250

7.0E+14 6.0E+14 5.0E+14 4.0E+14 3.0E+14 2.0E+14 1.0E+14 0.0E+00 0

20

40

60

80

100

Photon energy (eV)

Figure 7.15 On-axis flux density spectrum of PERA undulator.

u

Helical undulator 2

u/3 Helical undulator 1

Figure 7.16 Schematic drawing of a helical device with higher harmonics reprinted from Tanaka and Kitamura, J. Synchrotron Rad., 4, figure 1, 193 (1997), with permission from the International Union of Crystallography.

generated. In the case of APPLE-8, the horizontal magnetic field is twice the period length of the vertical magnetic field which is equivalent to the figure-8 undulator when the magnet row phase is zero. Figure 7.17 shows a schematic drawing of APPLE-8 in a finite magnet row phase.

Exotic insertion devices

251

Figure 7.17 Schematic drawing of an APPLE-8 undulator.

In a certain finite magnet row phase, this device is capable of generating a magnetic field as follows: -

. 2π 2π Bx = B0 cos z + cos z (11) λ1 λ2 and By = B0





. 2π 2π sin z − sin z λ1 λ2

(12)

where λ1 = 2λ2 . An electron beam draws a pseudo-helical trajectory in the field above, thereby generating a circularly polarized fundamental radiation with very small higher harmonics. The advantages of this device are the capability of variable polarization, like the APPLE undulator, and the low on-axis power density even in the linear mode, like the figure-8 undulator. A disadvantage is that the structure is too complicated.

5.3

Microwave undulator

Usually, even in the exotic undulators described above, the magnetic field is used for wiggling the electron beam. However, it is the electric field which is used in the microwave undulator, which consists of a resonator (RF cavities). The standing wave in the resonator is used for generating the periodic undulation of the electron beam [25]. A microwave undulator prototype was developed in the High Energy Accelerator Research Organization (KEK), and tested in its linac with an electron energy of 0.1–0.5 GeV. The undulator radiation in the visible wavelength region from the 5.5 cm period undulator was observed. The

252

S. Sasaki

advantage of this device is that it is capable of rapidly changing the linear polarization direction.

5.4

Plasma undulator

Unlike the ordinary permanent magnet undulators, a micro-pitched plasma undulator was proposed at JAERI [26]. The plasma undulator consists of an array of hundreds of discharged and pinched slender columns of plasma. Two types of undulating forces on the electron beam can be considered. One is a periodic magnetic field generated by the electric currents flowing through the array of the small radius plasma columns and the other is a periodic electrostatic field generated by the interaction of the relativistic electron beam with the rippled density of the plasma. In [26], the period length of the undulator was considered to be of the order of 1 mm for both types of undulation methods, and the K value was calculated to be of the order of 0.1. The concept of the latter case (the electric field type) was further developed to achieve a shorter period length by using a laser interference and resonant ionization technique [27]. Figure 7.18 shows a schematic view of the laser-plasma micro-undulator. This scheme is based on two processes: interference of two laser beams to generate optical fringes and resonant ionization of vapour atoms. The pitch of the ripple is controlled by changing the interference angle and the laser wavelength, while the density of the ripple is controlled by the vapour density and the laser intensity. The period length of 10–100 µm can be achieved with this new scheme.

Water-cooled vessel Quartz sensor Relativistic electron beam Plasma micro-undulator Collimated vapour beam Laser beam dump Tunable laser

Aperture plate

Deflection magnet

Vacuum pump Crucible

Figure 7.18 Schematic view of the laser-plasma micro-undulator reprinted from Ikehata et al., Nucl. Instrum. Meth. in Phys. Res. A 383, 608 (1996), with permission from Elsevier Science.

Exotic insertion devices

6

253

Conclusions

There are many other exotic devices which have not been brought out here such as the asymmetric wiggler, the variable period undulator, the strong focusing undulator, etc. Readers should be aware that the exotic insertion devices listed here are only some examples, and some important exotic devices may be missing from this chapter. The in-vacuum undulator, mini-pole undulator and superconducting undulator have not been described here intentionally because they are conventional devices with additional special technology to extend their operational limits. Some of the devices described in this chapter will be considered conventional devices in the near future.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

Onuki, H., Nucl. Instrum. Meth. A246, 94 (1986). Elleaume, P., Nucl. Instrum. Meth. A291, 371 (1990). Diviacco, B. and R. P. Walker, Nucl. Instrum. Meth. A292, 517 (1990). Sasaki, S., K. Miyata and T. Takada, Jpn. J. Appl. Phys. 31, L1794 (1992). Sasaki, S., Nucl. Instrum. Meth. A347, 83 (1994). Hashimoto, S. and S. Sasaki, Nucl. Instrum. Meth. A361, 611 (1995). Sasaki, S., H. Kobayashi, M. Takao, Y. Miyahara and S. Hashimoto, Rev. Sci. Instrum. 66, 1953 (1995). Guinier, A., X-ray Diffraction (translated by P. Lorrian) (Freeman, San Francisco, 1963). Jackson, J. D., Classical Electrodynamics, 2nd edn, p. 670 (Wiley, New York, 1975). Sasaki, S., Sincrotrone Trieste Internal Report ST/M-TN-98/24 (1998). Kobayashi, H., K. Ohashi, S. Sasaki, T. Shimada, M. Takao, S. Hashimoto and Y. Miyahara, JAERI-Tech, 96-014 (1996) (in Japanese). Kawai, M., M. Yokoyama, K. Yamada, N. Sei, T. Yamazaki, T. Shimada, M. Takao, Y. Miyahara, S. Hashimoto, S. Sasaki, T. Koseki, K. Shinoe, Y. Kamiya and H. Kobayashi, Proceedings of European Particle Accelerator Conference, EPAC96, p. 2549, Barcelona, Spain, 1996. Diviacco, B. and R. P. Walker, Sincrotrone Trieste Internal Report ST/M-TN-97/11. Chavanne, J., P. Elleaume and P. Van Vaerenbergh, Proceedings of European Particle Accelerator Conference, EPAC98, p. 2213, Stockholm, Sweden, 1998. Diviacco, B., R. Bracco, D. Millo, D. Zangrando and R. P. Walker, Proceedings of European Particle Accelerator Conference, EPAC98, p. 2216, Stockholm, Sweden, 1998. Tanaka, T. and H. Kitamura, Nucl. Instrum. Meth. A364, 368 (1995). Khlebnikov, A. S., N. V. Smolyakov, S. V. Tolmachev and O. V. Chubar, Proceedings of European Particle Accelerator Conference, EPAC96, p. 2558, Barcelona, Spain, 1996. Khlebnikov, A. S., private communication. Sasaki, S., Proceedings of Particle Accelerator Conference, PAC97, p. 802, Vancouver, B. C., Canada, 1997. Sasaki, S., B. Diviacco and R. P. Walker, Sincrotrone Trieste Internal Report, ST/M-TN-98/8 (1998). Tanaka, T. and H. Kitamura, J. Synchrotron Rad. 4, 193 (1997). Sasaki, S., N. Nakamura, Y. Kamiya, M. Yokoyama, M. Kawai, H. Kobayashi and K. Kakuno, J. Electron Spectroscopy and Related Phenom. 92, 315 (1998). Sasaki, S., B. Diviacco and R. P. Walker, Proceedings of European Particle Accelerator Conference, EPAC98, p. 2237, Stockholm, Sweden, 1998.

254 [24] [25] [26] [27]

S. Sasaki Sasaki, S., Nucl. Instrum. Meth. A347, 83 (1994). Shintake, T., K. Huke, J. Tanaka, I. Sato and I. Kumabe, Jpn. J. Appl. Phys. 22, 844 (1983). Suzuki, Y., Nucl. Instrum. Meth. A331, 684 (1993). Ikehata, T., Y. Suzuki, R. Nagai, Y. Sadamoto, N. Y. Sato and H. Mase, Nucl. Instrum. Meth. A383, 605 (1996).

8

Free electron lasers Marie-Emmanuelle Couprie

1

Introduction

Undulators and wigglers are widely used all over the world as a source of synchrotron radiation covering the spectral range from the infrared to the hard X-ray. However, historically, they were first built for Free Electron Lasers (FEL). The development of FELs followed the pioneering ideas [1] and experiment led by J. M. J. Madey in 1977, in the infrared, at Stanford on a linear accelerator [2]. The second FEL oscillation was then achieved in Orsay, on the storage ring ACO, in the visible range in 1983 [3]. Since then, a large number of simple and advanced undulators have been built and integrated into FELs. FELs facilities provide a fully coherent tunable light in a wide spectral range for scientific applications in various domains. Moreover, an FEL can be considered as a “complex” insertion device in a synchrotron radiation facility of intermediate energy, allowing pump-probe two-color applications to be performed. As illustrated in Figure 8.1, FEL oscillation results from the interaction of an optical wave with a relativistic electron beam circulating in the periodic permanent magnetic field of an undulator (period λ0 and peak magnetic field B0 along the vertical direction y). The relativistic particles are transversely accelerated and emit synchrotron radiation, at the resonant wavelength λr and its harmonics: λr =

λ0 (1 + K 2 /2) 2γ 2

0

Figure 8.1 Principle of the Free Electron Laser.

(1)

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M.-E. Couprie

where the deflection parameter K = 0.94λ0 [cm] B0 [T] and γ is the normalized energy of the electrons. The interaction between the optical wave and the electron bunch occurs along the undulator progression. Generally, an optical resonator, the length of which is adapted to the recurrence of the electron bunches, allows the radiation to be stored and the interaction to take place at each passage. The optical wave and the charged particles exchange energy, leading, under given circumstances, to a modulation of the electronic density at the wavelength of light (microbunching), phasing the emission and reinforcing the coherence of the produced radiation. An additional second order energy exchange between the optical wave and the electron beam leads to the amplification of the stored radiation whose intensity increases non-linearly until saturation is reached (the gain of the system becomes equal to the cavity losses missing; meanwhile the spectral and temporal widths narrow. Through the system constituted by the relativistic electron beam in the undulator, coherent harmonics can be produced from an external laser source or from the FEL itself. The coherent light can also be produced in one single pass, in the Self Amplified Spontaneous Emission (SASE) mode. By changing the deflection parameter K or the electron energy, one changes the resonant wavelength. As a result, the FEL is intrinsically a tunable source of radiation. The undulator period is typically a few centimeters long, therefore, the higher the electron energy, the shorter the wavelength. As shown in Figure 8.2, FELs on low energy accelerators (MeV range) operate in the microwave and far infrared ranges; FELs on intermediate energy accelerators (50 MeV) cover the mid-infrared and UV ranges, and systems on higher energy accelerators (100 MeV–GeV) reach the UV, VUV and X-rays ranges. As the gain is higher for low beam energies and long wavelengths, and since the optics are more performant in the infrared than in the UV and VUV, the FEL was developed faster in the infrared (the first FEL oscillation was achieved in the UV in 1988 in Novosibirsk [4]) and short wavelength was generated by coherent harmonics at 100 nm on Super-ACO in 1990 [5]. Very recently, saturation was observed in the SASE regime on the TESLA-TTF experiment at 80 nm [6]. Besides, one generally distinguishes the Compton-type FEL from the Raman-type FEL. In a Compton FEL, the amplification process is the result of the interaction of the radiation field with each individual electron. In a Raman-type FEL, the amplification process is the result of the interaction of the radiation field with a collective mode of the electron plasma in which Coulomb interaction between the electrons plays a major role. The transition between the Raman and the Compton regimes is set by the dimensionless parameter µ which is the product of the relativistic plasma frequency and the interaction time Ti : µ = γ −3/2 ωp Ti where ωp is the plasma frequency and γ mc2 the electron energy. µ 1 (µ  1) corresponds to the FELIX UVSOR Santa-Barbara ElsaStanford TESLA FELI NiJI-4 Clio S-ACO 1

0

Van de Graff induction Linac microtron

mm

IR

100 RF Linac

visible IR

1000 MeV RF Linac storage ring

UV visible

XUV

X

Figure 8.2 Range of photon energies accessible by various types of electron accelerators.

Free electron lasers

257

+

Macropulse

Micropulse

FEL pulse growth

Figure 8.3 Schema of a LINAC-driven FEL.

Compton (Raman) regime. Raman FELs are based on very low energy electrons (a few MeV) and essentially cover the millimeter to far infrared range while the Compton FELs are based on higher energies (MeV to GeV) and cover the infrared to the X-ray range. The most popular type of accelerator used in the infrared is the conventional RF linear accelerator (LINAC), as illustrated in Figure 8.3. The electron beam, produced by a cathode, consists of a series of several microsecond pulses, emitted at a repetition rate ranging between 1 and 100 Hz. Following the passage of the beam through the RF accelerating structure, the macropulse is bunched into a few thousands of picosecond micropulses, with a spacing given by the RF field wavelength (typically 0.3–1 ns). A superconducting RF LINAC provides longer macropulses (typically 1 ms). Recent use of a photocathode produces very short micropulses (in the femtosecond range). The electron beam goes to a beam dump, and a “new” bunch interacts at each passage with the FEL. Several user facilities are currently operating in the infrared, exploiting the high average power associated with the wide tunability for scientific applications. Average FEL power as high as a few hundred watts in the infrared has been recently obtained at Jefferson Lab. (USA) [7] and JAERI (Japan) [8]. The FEL IR Demo of Jefferson Lab has recently been operated in the Energy Recovery Linac mode, leading to kW average power [7]. In order to insure a proper synchronization between the optical pulse which reflects back and forth between the mirrors and the succesive bunches, the distance between the mirrors should be an integer of the electron bunch distance. A storage ring is a common accelerator for hosting an FEL (see Figure 8.4). In this case, the storage ring FEL (SRFEL) reproduces the pulsed megahertz structure (although the filling should be limited to a few bunches in order to avoid inter-bunch longitudinal instabilities). Synchrotron radiation and the FEL are naturally synchronized, and can be combined for pump-probe two-color experiments for the study of the dynamics of excited species [9]. Unlike LINACs, one may operate a large circulating current in the storage ring (few hundred milliamperes) which implies a large circulating power (1 GW). The power of a storage ring FEL is nevertheless limited by the electron energy spread induced by the electron beam. The polarization characteristics result from the undulator. Planar undulators creating a vertical field mainly provide horizontal, linearly polarized FELs. Helical undulators, as in the first experiment by J. M. J. Madey, create circularly polarized FELs. Recent developments of permanent magnet schemes for undulators for the production of adjustable polarized synchrotron radiation also benefit the SRFELs as on UVSOR [10] and ELETTRA [10].

258

M.-E. Couprie

∆ R S

RF

∆t UV FEL

Figure 8.4 Schema of a storage ring FEL. Lc w0

Z0 t

Figure 8.5 Optical resonator.

Various types of FEL configurations can be distinguished. The most conventional configuration is the oscillator. The FEL is generated from the spontaneous emission of the undulator, with a multipass interaction between the electromagnetic wave stored in the optical cavity and the electron beam. Such a scheme is generally used for intermediate and low gains (see Figure 8.1). The Master Oscillator Power Amplifier, a variant of this simple oscillator configuration, first operated in Stanford [11], is composed, after a first oscillator, of a second undulator allowing the FEL radiation to be amplified, while limiting the power load on the optics. Usual optical cavities generally consist of two curved mirrors of radius of curvature Rc (Figure 8.5). The cavity length Lc is imposed by the synchronism condition. The FEL is a Gaussian beam defined by the modes of the optical cavity parameters, determining the spatial coherence of the source. In the infrared, cavities are relatively short (of the order of 5 m), broad band metallic mirrors are employed and the FEL is extracted by a hole coupling or a Brewster plate. The FEL beam is at the diffraction limit, the product of the dimensions by the angular divergence is of the order of the wavelength, the mode size increases in the infrared, and the gap of the undulator should not be too small in order to avoid diffraction

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259

losses. For short wavelength FELs, the undulator increases in length, and the optical resonator reaches several tens of meters; some ring resonators have also been proposed [12]. Besides alignment difficulties, long optical cavities allow a reduction of the power density on the front mirror. The optical mode reaches its minimum size w0 at the position of the waist. For a symmetrical optical cavity, w0 is [13]
w0 =

λ 2π

1/2 [Lc (Lc − 2Rc )]

πw02 Z0 = λ

1/4

(2)

The Rayleigh length Z0 is defined as the longitudinal distance for which the cross section of the mode is twice that of the waist’s. The mode diameter w(s) and divergence θ at a distance s are then given by w2 (s) = w02

s2 1+ 2 Z0

and

θ=

λ πw0

(3)

The optical resonator is stable when 2Rc ≥ Lc . The limit of stability corresponds to the confocal cavity (Rc = Lc ) and the concentric cavity (Rc = Lc /2) [12]. The choice of the mirror radii results from a compromise between the minimization of the optical mode in the optical klystron and stability considerations [13]. The VEPP 3 FEL has been operated with a confocal cavity [14]. Coherent Harmonic Generation is the second configuration which has been used. Coherent harmonics are produced by single pass interaction of the electron beam in the undulator with the beam of an external laser (see Figure 8.6). The microbunching induced by the electromagnetic wave occurs on the harmonics of the resonance wavelength [15]. Coherent harmonics can also be generated from the FEL itself, provided its peak power is high enough, avoiding the difficulties associated with possible drifts of the external laser with respect to the electron beam. The coherence of the harmonics can also be reinforced with the help of a specific undulator, whose fundamental wavelength should be adapted to the harmonic to be amplified (the magnetic field can be reduced or the period can be subharmonic). In the Self Amplified Spontaneous Emission configuration (SASE), the gain is sufficiently important to provide a coherent wave at the end of the undulator in one single pass (see Figure 8.7) [16]. It generally requires very long undulators, but avoids optical problems related to the mirrors, especially in the X-ray range. After the description of the physics of FEL and the interaction process, the status and state-of-the-art of the FEL sources is presented. 0 /3



1–10 m

Figure 8.6 Coherent harmonic generation.

260

M.-E. Couprie 10–100 m



Figure 8.7 SASE configuration.

2

Physics of FELs

The FEL interaction is based on the fact that the optical wave travelling parallel in the same direction as an electron bunch inside the undulator can be amplified (or absorbed). The FEL starts from “noise”, which is here referred to as the spontaneous emission or synchrotron radiation given by the undulator radiation. With an optical cavity, the first electron bunch passing in the undulator (see Figure 8.1) emits radiation. This optical wave is reflected by the front mirror, the rear mirror, and then amplified in the interaction region (the undulator) to the detriment of the kinetic energy of the second electron bunch. The radiation results from the sum of the amplified emission from the first electron bunch and the spontaneous emission of the second electron bunch. The light is then reflected by the two mirrors, and generates the stimulated emission in the undulator. More precisely, the electromagnetic field induces an energy modulation in the electron bunch; it is then transformed into a density modulation with a period λr , leading to an emission in phase of the different microbunches, reinforcing the coherence of the obtained radiation. Clearly, electron–photon interaction requires a right spatio–temporal overlap between the optical wave and the electron bunch, and the optical cavity length should then be a sub-multiple of the interval between two consecutive bunches. The laser effect can be obtained only if the gain per pass is greater than the cavity losses. As the power builds up, a simultaneous spectral and longitudinal narrowing takes place, which is ultimately limited by the Fourier limit. As a result, the radiation pulse of the FEL is spectrally much narrower and shorter than the spontaneous emission, making it quite attractive. The multipass and the amplification continue until the FEL equilibrium regime is reached for which the gain equals the losses, this gain reduction at high power of the laser beam is called saturation. This section describes the bunching phenomenon, the light amplification and presents the different FEL configurations. 2.1

Bunching of the electron bunch

 The The radiation field can be represented by an optical wave of transverse electric field E. electron velocity is written as v. According to the Lorentz equation, the energy variation -γ of an electron interacting with the radiation can be expressed, as [17] e -γ = − 2 mc

 Interaction

E · v dt

(4)

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261

where e and m are the electron charge and mass and c the speed of light. Both the electric field E and the electron velocity v must be expressed at the position of the electron at time t. As the optical field propagates along the undulator, the interaction mainly takes place with the electrons’ transverse velocity. In a first approximation, the optical wave of pulsation ω can be represented by a plane wave: E = E sin(ωt − ks + <0 ) x

(5)

with k = ω/c, the time origin being set at the entrance of the electron in the undulator, the phase <0 the phase of the monochromatic wave with respect to each single electron, s the longitudinal position of the electron at time t and x a unit vector oriented in the horizontal plane perpendicular to the magnetic field and to the main electron beam axis. For a planar undulator, making use of the results of Chapter 3, Section 3, namely v = (cK/γ ) cos(2πs/λ0 ) x , one transforms Eqn (4) into  eKE -γ = − B0 E cos(ω0 t) cos(ωt − kv0 t + <0 ) (6) γ mc Interaction where v0 is the average value of the longitudinal velocity of the electrons and ω0 = 2π/λ0 . This leads to a beating wave with two frequencies . = ω0 ± (ω − kv0 ). The integration of the sum term in (6) is equal to zero whatever ω is. The energy exchange comes from the difference frequency term, and is only significant for . ∼ = 0, which corresponds to a narrow wavelength range λ around the resonant energy γr2 = (λ0 /2λ)(1 + (K 2 /2)). A detailed quantitative approach [18] gives -γ ≈ −

eKENλ0 sin δ [J (ξ ) − J (ξ )] cos(<0 ) 0 1 2γ mc2 δ

with

ξ=

K2 4 + 2K 2

and

δ = πN

λr − λ λ

(7)

where N is the number of periods of the undulator, Jn (ξ ) the Bessel function of order n and of variable ξ . The sign of -γ depends on the phase <0 between the electron and the optical wave. If one electron is accelerated (-γ > 0), another electron located longitudinally one-half wavelength ahead or behind is decelerated by the same amount (-γ < 0). The longitudinal distribution of the bunch is much greater than the wavelength, therefore, the phase <0 is uniformly distributed between 0 and 2π. As a result, there is no variation of the electron energy average over the electrons in the bunch. There is, nevertheless, an increase of the rms energy spread which can be derived from (7) as 1 -γ  = 2 2




eKNλ0 2γ mc2

2 E

2

2 2 sin δ [J0 (ξ ) − J1 (ξ )] δ2

(8)

where E 2  is the average of the square of the electric field over the electron beam. The electrons after propagation are then accelerated or decelerated by energy enhancement or loss, leading to a spatial modulation. The electrons are bunched around a phase <0 multiple of 2π, the electronic density is then modulated with a period equal to the resonant wavelength and sets into coherence the elementary oscillators. Identically, a modulation is generated with a planar undulator on the harmonics of the incident light leading to the process of “Coherent Harmonic Generation” [15].

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2.2

Small signal gain

2.2.1

Madey’s theorem

Near resonance, the optical wave and the electrons exchange energy, the electrons gather around positions for which the energy variation δγ mc2 keeps a constant sign. The modulation depends on the electric field of the wave, and the second order energy exchange -γ2 , averaged over phases, is then evaluated with a bunched electron distribution, either directly or using Madey’s theorems, as follows [18]:  d<  2αm2 c4 I -γ 2  = d. θ=0 e λ2 E 2 

and

-γ2  =

1 ∂ -γ 2  2 ∂γ

(9)

 with α the fine structure constant, I the electron beam current, d
mcI -γ2   eε0 E 2 dS

(10)

where ε0 is the vacuum permeability and the radiation field is integrated over the transverse coordinates. Depending on the sign of (λ−λr ), the optical wave is either absorbed to the benefit of a gain of kinetic energy of the electrons (corresponding to the inverse FEL configuration for laser-assisted beam acceleration) or is amplified to the detriment of the kinetic energy of the particles, leading to the FEL effect. 2.2.2

Small signal gain in the planar undulator case

In the case of a planar undulator of N periods, period length λ0 , peak on-axis magnetic field B0 , the gain can be expressed as

2 sin ∂ δ Gund (δ) = π2 r0 γ −3 λ20 N 3 K 2 Ff ρe [J1 (ξ ) − J0 (ξ )]2 (11) ∂δ δ2 with ρe the electronic density, r0 the electron classical radius, K the deflection parameter and ξ and δ given by Eqn (7). The filling factor Ff represents the overlap between the laser transverse modes and the transverse dimensions of the electron beam σx and σz in the undulator. Ff can be approximated at equilibrium, considering the waist w0 of the gaussian eigenmode TEM 00 of the cavity, according to Ff = 

1 

1 + (w0 /2σx )2

1 + (w0 /2σy )2

 · fc (w0 , σx , σy )

(12)

Free electron lasers

263

The correction factor fc takes into account the dynamical change of an electron with respect to the laser waist along the insertion device [13,19], and is close to 1 when w0 /σi  1. The choice of the beam transverse size and of the radii of curvature of the mirrors results from a compromise between the electronic density term, Ff and the stability of the optical resonator, leading generally to a “round” electron beam. For large σx and σz , Ff is close to 1 and the electronic density is reduced. For an optical resonator very close to the stability limit, Ff is of the order of 0.5–0.6. Besides, the FEL starts from the spontaneous emission of the undulator, which does not correspond to the eigenmode TEM 00 of the cavity and the transverse structure of this radiation evolves during the propagation. The TEM00 mode is only established after a given number of passes, leading, in some cases, to a reduced filling factor at start-up [19]. Maximizing Eqn (11) over the electron energy, one obtains (in SI units) Gund = 1.5 × 10−14 γ −3 λ20 N 3 K 2 Ff ρe [J1 (ξ ) − J0 (ξ )]2

(13)

The gain depends on the beam energy, the number of periods N and the length of the undulator. For high gain systems, higher order terms should be considered in the energy exchange, leading to exponential regimes. The gain itself depends on the electric field of the wave. The first UV storage ring FELs implemented on short straight sections are low gain, low signal systems, and an “optical klystron” allowing artificial enhancement of the gain, is generally employed [20]. 2.2.3

Small signal gain in the case of the optical klystron

An optical klystron is composed of two identical sections of an undulator, separated by a dispersive section, creating a wide wiggle of magnetic field (cf. Figure 8.8). The radiation emitted by two undulators interferes (cf. Figure 8.9), and the spectrum exhibits a fine structure analogous to that of the Young slits, the envelop being given by a single undulator radiation. The spectral repartition is given by   d<  d<  (opt. klystron) = 2 (undulator)[1 + f cos αk0 ] (14) d. θ=0 d. θ =0 where αk0 = 2π(N + Nd )(λR /λ)(γR2 /γ 2 ). The interference order (N + Nd ) represents the phase advance in terms of periods of the optical wave, with respect to the electrons in the Dispersive section Modulator Radiator

Modulation in energy

Density modulation

Coherent emission

Figure 8.8 Principle of the optical klystron.

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M.-E. Couprie

SE (au)

2

1

0

Gain (au)

2

0

–2

180

200  (nm)

220

Figure 8.9 On-axis spectrum observed on the Super-ACO FEL as a function of the wavelength (upper curve). Gain deduced from the spontaneous emission (SE) (lower curve).

dispersive section. The fringe contrast or the so-called “modulation rate” f , defined as the ratio (Imax − Imin )/(Imax + Imin ) mainly depends on the beam energy spread σγ /γ , fγ as fγ = e

−8π2 (N +Nd )2



σγ γ

2

(15)

The effects of magnetic field inhomogeneities, beam angular spread, and transverse position of the bunch with respect to the optical wave are generally negligible. From the first Madey’s theorem, the gain is proportional to the derivative of the spontaneous emission vs energy, or quasi-equivalently vs wavelength (see Figure 8.9(b)): the higher the strength of the dispersive section, the higher the gain. Maximization of the gain G0k with respect to the interference order Nd is given by (16). G0k =

f · L2k0 (N + Nd ) N 3 λ20

Gund

with

4π(N + Nd )σγ =1 γ

and

f = 0.61

(16)

The typical energy spread is in the 5×10−4 –2×10−3 range and corresponds to an optimum value of (N + Nd ) between 40 and 160. The gain enhancement is limited by the energy spread. If the straight section is long enough to accomodate up to 100–200 periods, there

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265

is no advantage in using an optical klystron over a simple undulator. However, it is almost never the case for storage ring driven FELs and the gain enhancement reached by using an optical klystron instead of an undulator of same total length is typically 3–7. Besides, unlike the undulator case where bunching and radiation take place simultaneously over the whole length of the undulator, the processes are quite separated for an optical klystron. The electron entering the first undulator undergoes the energy modulation due to the interaction with the electromagnetic wave. The dispersive section delays the electrons so that the energy modulation is transformed into a density modulation. The bunched electrons at the entrance of the second undulator radiate a coherent light. This process for the spontaneous emission recalls the microwave klystrons, and is at the origin of the term “optical klystron” where the undulators are equivalent to the resonant cavities, and the dispersive section to the drift space. 2.2.4

Small signal gain in the case of the helical undulator

Even though the first FEL oscillator, operated at Stanford in 1977 used a superconducting helical undulator, permanent magnet planar undulators are nowadays more extensively used for FELs because their technology is simpler and well mastered. There is nevertheless a recent interest in the use of helical undulators for two reasons. First, for a helical undulator emitting at the same wavelength as the previous planar undulator, λr = (λ0 /2γ 2 )(1 + K 2 ), the gain is enhanced by 1/(J1 (ξ ) − J0 (ξ ))2 . As a result, for a sufficiently large K value, the small signal gain is doubled. Second, the on-axis spectrum emission is limited to the fundamental wavelength, preventing mirror degradation produced by the planar undulator harmonics. This has been experimentally demonstrated at UVSOR where a helical undulator was installed in 1996 for a SRFEL [10]). The use of helical undulators is of major importance for storage ring FELs where the deflection parameter K tends to be large and the spontaneous emission over the harmonics is an important source of damage for the mirrors of the cavity. The segmentation offered by the optical klystron solution facilitates the design of long insertion devices for future SRFELs. 2.3

Saturation and power

Depending on the accelerator type, the FEL saturation can result from two different processes: homogeneous broadening on single pass accelerators and inhomogeneous broadening on recirculating machines. 2.3.1

FEL on single pass accelerators (homogeneous broadening)

When the FEL power increases, a considerable number of electrons are trapped in the ponderomotive potential of the wave and undulator field. The equivalent “pendulum” oscillates around its equilibrium position, and according to its position, transfers kinetic energy to the FEL. For a strong FEL power and a single pass accelerator (general case of LINAC FELs), the energy of the electrons is considerably reduced in the undulator, so that electrons do not remain resonant from the entrance to the exit of the undulator [17,21]. The amplification mainly occurs in the front part of the undulator. At the undulator’s exit, the microbunching can become nonlinear (so-called “over-bunching”). The magnetic field or the period of the undulator can then be slightly adapted (tapered undulators) [22] in order to keep the resonance condition all along the undulator. In the case where the electron performs several oscillations in phase space (corresponding to coherent synchrotron oscillations in the longitudinal phase

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M.-E. Couprie

space), the light intensity is spatially modulated and sidebands appear in the spectral line of the FEL which broadens. On LINAC-based devices, the FEL peak power Ppeak and average power P  are directly related to the energy contained in the electron beam: Ppeak =

E·I 4eN

P  = D · Ppeak

(17)

where N is the number of periods of the undulator, I the stored current, E the beam energy, D the duty cycle and e the electron charge. Average power is in the 1–100 W range, whereas peak power is of the order of 10 MW. 2.3.2

FEL on multipass accelerators (inhomogeneous broadening)

For circulating accelerators such as storage rings, the FEL induced modification of the electron energy is kept on many turns, leading to a specific saturation process. The interaction between the optical wave and the electron bunch induces a broadening of the beam energy distribution, strongly limiting the FEL power on storage ring FELs where the damping requires a high number of turns. As the FEL is not Fourier transformed at start-up or after a perturbation on its dynamical evolution, one should consider, generally, time and wavelength dependant distribution. Let us then consider a FEL micropulse at the wavelength λ and at the longitudinal position τ with respect to the synchronous electron. The optical resonator is adjusted on the tuning -T = Tph − Tel , with Tel the revolution time of the synchronous electron, Tph the round trip time of the photons in the optical resonator. The evolution of the optical wave electric field En at the nth pass is En+1 (τ − -T , λ) = En (τ, λ)r 2 [1 + g(τ, λ)] + es (τ, λ)

(18)

with g the gain in amplitude at passage turn n, r the mirror reflectivity and es (τ, λ) the spontaneous emission being the initial source of the FEL. The gain can be expressed as follows in the optical klystron case: 
2 λ − λr −8π2 (N +Nd )2 (σγ /γ ) −τ 2 /2σ 2 g(τ, λ) = g0 e (19) ·e · sin 2π -λk0 with -λk0 is the spectral width of a fringe of the optical klystron, λr the resonant wavelength, σ the bunch length and σγ /γ the energy spread of the electron beam. Equation (19) shows the dependence on the energy spread, through the modulation rate, of the optical klystron (first term), on the position of the FEL pulse with respect to the electron bunch longitudinal profile (second term) and the fringe modulation (third term). The evolution per passage turn of the energy spread σγ /γ , from a “0” state without FEL, results from the Fokker–Planck equation [23], as

2
2
2

σγ σγ 2 σγ σγ 2T0 + In = − − (20) γ n+1 τs γ n γ n γ 0 with τs the synchrotron damping time, In the FEL intensity at passage turn n (corresponding to the integration of the square of the optical wave electric field En2 over λ and τ ). With

Free electron lasers

267

the dimensionless total laser intensity Ilas and energy spread Q, defined by (Q = Ilas = 1) at equilibrium with FEL, it becomes Qn+1

2T0 = Qn + (In − Qn ) τs

with

Q=

σγ2 − σγ2off σγ2eq − σγ2

(21)

with a beam energy spread which is σγ /γoff without FEL and σγ /γeq at equilibrium. The FEL interaction leads to an increase of the electron energy spread, which is called “bunch heating”. As the energy spread and the bunch length are proportional on storage rings, the FEL also induces a bunch lengthening. The FEL intensity growth when the FEL develops leads to an increase of energy spread and correlated bunch lengthening and thus to a reduction of the electronic density, and to a reduction of the gain towards the level of saturation. From the FEL heating, the average laser power can be derived, which can be expressed as
 σγ P ≈ ηc Ps (22) γ with (σγ /γ ) the relative energy spread when the laser operates and ηc the efficiency of the mirrors (i.e. the transmission/total losses ratio). Ps = 6.04 · 10−9 (γ 4 I /ρ0 ) is the total synchrotron radiation power (in Watts) generated over the whole ring in the bending magnets, where ρ0 is the dipoles’ radius of curvature (in meters), I the average beam current (in Ampere), γ the normalized beam energy. This expression is know as the “Renieri Limit” [23,24]. The power is limited to the power emitted by synchrotron radiation Ps because of the recirculation. For small perturbations around the equilibrium state, the evolution of the laser intensity can be expressed as
 dIlas G−L (23) + is = Ilas dt Tel with L the cavity losses, is the spontaneous intensity and G the intensity gain per pass. Combined with (21), it leads to a system of equations that can be linearized into second order differential equation, the general solution of which is a damped oscillator with the synchrotron damping time τs and frequency:  Fr =

1 √

π 2

1 τr τs

(24)

where τr is the laser risetime τr ∼ Tph /(L ln(G−L)). Consequently, any perturbation around this laser frequency inducing a gain instability can lead to an important oscillation of the laser intensity, periodically stable or unstable, giving a pulsed macrotemporal structure in the millisecond range [24]. More detailled analysis shows that this pulsed structure systematically occurs for specific detunings. Spectrally, the laser line narrows with respect to the spontaneous emission since the gain is proportional to the electronic density, but it also lengthens at each pass because of the spontaneous emission and the slippage (shift between the photons emitted at the beginning and at the end of the undulator). Spectrally, the solutions of the amplifying system installed in the optical cavity, without taking into account either the spontaneous emission or the

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M.-E. Couprie

saturation processes, lead to the “super-modes” [23]. These first models are not always very satisfying because the equilibrium situation is not systematically reached. Following this general presentation on fundamental principles of the FEL, the state-of-theart of Compton FEL performances will be presented in more detail in the next section.

3

Status of FEL facilities

3.1

Infrared

3.1.1

Spectral range

FELs are excellent sources in the infrared range. Table 8.1 gives a list of the FEL facilities presently operating in the world and covering the infrared domain of the spectrum. They mostly use RF LINACs for the mid infrared and electrostatic low energy accelerator for the millimeter to the far infrared range. Longer wavelengths are now reached with a modification of the undulator (such as on CLIO [25]) or with an energy change of the accelerator. A number of user facilities have been established which combine several undulators and FELs operated by the same accelerator using different transport lines for an extension towards longer wavelengths. This is the case for FELIX in Netherlands [26] (300 µm), the SCAFEL in Stanford (100 µm) [27] or FELI in Japan (Tsuda) which covers the very wide spectral range from 0.28 to 60 µm for user applications [28] (see Figure 8.10). On the SCAFEL, one part of a single long macropulse from the superconductiong LINAC goes to a first undulator, and the second part is re-accelerated and feeds a second FEL after another undulator. In addition, Table 8.1 FELs in the mid-infrared Name

Location

Wavelength

Acc. type

E (MeV)

USCB CREOL FIREFLY/SCAFEL LEENA ILE/ILT Univ. Tokyo FELIX ISIR ELSA LISA ENEA CIRFEL IHEP CLIO LANL Darmstadt Vanderbilt FELI, IFEL Jefferson Lab JAERI Univ. Twente Weizmann Inst. KAERI LEBRA

CA (USA) Florida (USA) Stanford (USA) Himeji (Japan) Osaka (Japan) Japan Netherlands Osaka (Japan) Bruyères (Fr) Frascati Frascati (Italy) Grumman (USA) Beijing (China) Orsay (France) USA Germany Tenessee (USA) Tsuda (Japan) Virginia (USA) Japan Netherlands Israël Korea Japan

30 µm–2.5 mm 355 µm 80–200 µm 65–75 µm 47 43 5–110 µm 21–126 µm 18–24 15 0.6–3.5 mm 8–20 10 3–53 4–8, 16 7 2.1–9.8 µm 0.278–140 µm 4.8 µm 24 µm 200 µm 3 mm 10 mm 1.5 µm

Electrostatic Electrostatic RF LINAC RF LINAC RF LINAC RF LINAC RF LINAC RF LINAC RF LINAC RF LINAC Microtron RF LINAC RF LINAC RF LINAC RF LINAC RF LINAC RF LINAC RF LINAC RF LINAC RF LINAC LINAC Electrostatic Electrostatic RF LINAC

6 1.3 4–45 5.4 8 13 25 17 18 25 — 9–14 30 21–50 17.4 40 43 33–170 38.5 15.8

20–40

Free electron lasers 30 MeV 5–22 µm

80 MeV 1–6 µm

0.23–1.2 µm

Und. 1

Und. 2

Und. 3

269

165 MeV Und. 4 20–60 µm 33 MeV

Figure 8.10 The FELI falicity covering the wavelength range from 60 µm to possibly 230 nm.

Figure 8.11 Schema of the NIJI4 FEL facility. An optical klystron is used to operate a VUV FEL at 212 nm. Production of X-rays using Compton backscattering. Micropulses Few ns

Macropulses

Few ms–µs

Few µs

Figure 8.12 Temporal structure of LINAC FELs.

several storage ring driven FELs such as NIJI4 (Tsukuba, Japon) [29] (see Figure 8.11) and Duke University are considering operating in the infrared in order to produce hard X-rays or Gamma Rays by Compton backscattering of the electron beam with the laser light in the optical cavity. 3.1.2

Temporal structure and spectral line

The FEL temporal structure reproduces the electron beam’s temporal structure. RF LINAC based devices (cf. Table 8.1) present macropulses (on the microsecond scale) constituted by series of picosecond micropulses (see Figure 8.12 and 8.13). The pulse duration is limited by the electron bunch and N λ, the advance of the photons emitted at the beginning of the undulator with respect to those emitted at the end. Femtosecond pulses could be achieved [30]. 3.1.3

Power

The average power produced by an RF LINAC FEL is typically a few watts. In 1998, average power amounting to several hundred watts was produced using superconducting LINACs at

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M.-E. Couprie

∆t (10–ns)

OPO

CLIO micropulses SCAFEL micropulses

n = 12 ps Nd-Yag n = 9 ns

Chemical laser

Non-linear optics CO2 laser

n = 6 µs

CLIO macropulses SCAFEL macropulses

n = 3 ms

1 10 000

10 1000

100 µm 100 cm–1

Figure 8.13 Temporal structure of LINAC driven FELs compared to other conventional laser sources in the infrared. The vertical axis is the pulse duration. LINAC driven FELs have a double temporal structure made of micropulses and macropulses. On conventional (superconducting) LINACs, the macropulses are a few µs (ms) long. The macropulses are composed of a train of a few thousands of micropulses which are a few ps long and spaced at the radiofrequency of the LINAC. OPO stands for Optical Parametric Oscillator.

JAERI (24 µm) in Japan and at Jefferson Laboratory (5 µm) in USA making this FEL useful for industrial applications such as metal processing, chemical vapor deposition and polymer surface modification [31]. With the energy recovery on the LINAC, kW average power has been obtained at Jefferson Lab [7]. 3.1.4 Comparison with alternative sources Alternative sources can also be used in the infrared. Chemical lasers, CO2 gas lasers and methanol gas lasers can provide high peak power but are not tunable. Recently developed Optical Parametric Oscillators (OPO) are tunable in the near infrared but offer smaller peak powers than FELs. Synchrotron radiation from bending magnets or from specific magnetic devices cover the whole infrared to far infrared domain for Fourier Transform spectroscopy; however, it is less powerful and do not offer the picosecond (femtosecond) time structure available from LINAC driven FELs. Figure 8.13 presents the time structure as a function of wavelength reached by the normal and superconducting LINAC driven FELs as compared to other sources. Among infrared sources, FELs appear to be an excellent powerful coherent radiation for scientific and medical applications. 3.2 3.2.1

Visible and UV range Spectral range and power

The visible and UV range are currently covered by five storage rings (cf. Table 8.2) and one LINAC driven FEL (cf. Table 8.3). In the UV (cf. Figure 8.14) six SRFELs and one LINAC

Location

Orsay (France) Novosibirsk (Russia) Tsukuba (Japan)

Orsay (France) N. C. (USA) Okazaki (Japan) Tsukuba (Japan) Dortmund (DE) Trieste (IT)

France

FEL

Presently shut down ACO VEPP3 [4] TERAS [34]

Operational FELs Super-ACO [35] DUKE [36] UVSOR [37] NIJI-4 [38] DELTA [39] ELETTRA [40]

Proposed SOLEIL [41] 1993

1989 1996 1992 1992 1999 2000

1983 1988 1991

First lasing date

1.5–2.1

0.6–0.8 1 0.5 0.24 0.5–1.5 1

0.16–0.25 0.35 0.24

Energy (GeV)

350–100

690–345 413–194 488–238 595–212 432 350–190

460–650 690–240 598

Spectral range (nm)

few. W

300 100-few 1000 1–few 0.01 few 500

2 <6

Output power (mW)

10

100

100 10 10

50 >6

I (mA)

>5

2–3

1 1

2–12

3–4 0.5

τ (h)

> 10

2 > 10 0.7 5–10 5–10 15

0.3 10

Gain (%)

5/280

10/110

12–60/120 6–50/ 15/178 /50

/37 200/125

FWHM width (ps)/rep. rate (ns)

Table 8.2 Summary of the characteristics of the various storage ring FEL sources. I is the electron beam current, τ is the lifetime of the current (and FEL) decay

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M.-E. Couprie

Table 8.3 LINAC driven FEL sources in the UV range FEL

Location

Presently shut down BNL (ATF) USA Los Alamos USA Operational FELs Argonne USA FELI Japan

LINAC

Configuration

λ (nm)

E (MeV)

P 

Photo-inj.

Oscillator Oscillator

500 375

230 15

0.7 mW

Photo-inj. Thermo-ionic gun

SASE Oscillator

vis.-UV 353–278 -λ/λ = 0.08% 400

210 165

6 mW

250

72 160

DUVFEL USA Photo-inj. In construction or project SPARC Italy Photo-inj. Jeff. lab USA

GHC, SASE SASE Oscillator

L:Los Alamos N:NIJI4 S:S.-ACO

De:DELTA Du:DUKE F:FELI 1000

kW

U:UVSOR V:VEPP3 E : ELETTRA

U E

S

E

Power (mW)

230

Du

100 E

S

10 S D E Du U U 1 N V 200

V

F

V De

V

N

De

L 300

400

500

600

700

Wavelength (nm)

Figure 8.14 Average power and wavelength tunability reached by various FEL in the visible and UV range of the spectrum.

driven source are presently in operation. Figure 8.14 shows the FEL power vs the wavelength in the visible–UV spectral range. The FEL power can be increased by operating at high energy with transparent mirrors (see Figure 8.15). More than 1 W has been obtained at UVSOR [37]. High power, good reliability and stability has made it possible to operate the Super-ACO FEL as a user facility [9,32]. The VUV range is reached on the DUKE and ELETTRA’s FELs. Some of the rings (DUKE, NIJI4 and DELTA) are optimized and dedicated to FEL operation, while others are synchrotron radiation facilities (UVSOR, Super-ACO), where the FEL is part of the activity and benefits from the scientific environment for user applications combining the FEL and synchrotron radiation. The FEL implemented on the third generation storage ring Elettra achieved the laser oscillation at 350 and 190 nm in two years after the begining of the project. The FELs on third generation light sources (Elettra, SOLEIL) offer new prospects for average power greater than 1 W in the UV, below 200 nm and even below, in the oscillator

Free electron lasers

273

350

Output power (mW)

300 500 MHz, T = 0.1%

250 200 150 100 50 0 20

100 MHz, T = 0.01%

500 MHz, T = 0.01% 30

40

50 I (mA)

60

70

80

Figure 8.15 Average output power delivered by the Super-ACO FEL for different type of RF cavities and transmission T of the output mirror of the cavity. The higher gain reached with the 500 MHz cavity allows the use of a mirror with larger transmission. As a result a higher output power is observed.

configuration. These features makes UV/VUV SRFELs unique tools for user applications either alone or in combined with the synchrotron radaition for pump/probe experiments. The FEL operation is more difficult in the UV range than in the infrared range because the accessible gains are significantly reduced, requiring very efficient electron beams and long undulators and because, generally, the mirrors offer smaller reflectivities R. The contributions of the useless losses, such as absorption and scattering, increase in the UV. As the latter are significantly greater in the UV, super-polished substrates should be employed (roughness of the order of 1 Å) [33]. Multilayer mirrors can be produced with some particular oxides and fluorides, or with Al. In addition, the FEL UV mirrors are generally submitted to a hostile environment: residual gas from an imperfect ultra-vacuum and FEL power and harmonic content of undulator radiation, which can be particularly harmful in the case of SRFELs operating with a planar undulator. The resistance of the optics depends on the material type and on the deposition technique. Mirrors can be kept in these conditions during more than 100 hours of lasing. The mirror degradation can be further reduced (factor 20) by using a helical undulator, as succesfully demonstrated on the UVSOR FEL [10]. The degradation remains more important for shorter wavelengths as the FEL wavelength is closer to the absorption bands of the material. Even in the case of LINAC driven FELs, the mirrors can be damaged, either by the FEL power itself, or because of the Bremstrahlung. Craters have been observed on FELI mirrors.

3.2.2

Temporal structure and spectral line

3.2.2.1 GENERAL BEHAVIOR VS TUNING

The UV LINAC driven FELs exhibit the same temporal structure as the IR FELs. SRFELs present a double temporal structure. Due to the bunch recirculation, an SRFEL has a pulsed structure at high frequency (cf. Figure 8.16). Apart from this microtemporal structure, SRFELs can also be pulsed at the millisecond range (cf. Table 8.4), showing a kind of oscillations of relaxation between the laser intensity

274

M.-E. Couprie (a)

(b)

FEL 2–60 ps FWHM

Synchrotron radiation 25–120 ps FWHM

100 ns–1 µs

Figure 8.16 (a) Microtemporal structure of the FEL and pulses of spontaneous emission as presents the record of a double sweep streak camera observed on Super ACO. The total vertical scale is 1.7 ns. The horizontal scale shows the variation of the time struture over a 2 µs total time. The upper traces correspond to the electron bunch profile observed through synchrotron radiation. The lower trace is the FEL trace which is nine times shorter than the electron bunch; (b) presents a schema of the time structure observed through a simple linear scale with the typical pulse lengths and repetition rate. Table 8.4 SRFEL temporal structure FEL

RMS electron bunch length (ps)

RMS FEL pulse width (ps)

Repetition rate (ns)

FEL risetime (µs)

Natural frequency (Hz)

Super-ACO UVSOR VEPP3 NIJI-4 DUKE ELETTRA

90–300 57–130 900

5–25 6 90

80 200

350 125

47

500

33 7–33

3–12 6–20

120 178 125 50 358 110

and the FEL induced energy spread growth damped in the ring. The natural FEL frequency depends on the laser risetime τo and on the synchrotron damping time τs . The SRFEL macrotemporal structure strongly depends on the tuning conditions (i.e. synchronization between the electron bunches stored in the ring and the light pulses bouncing between the two mirrors) or on the beam stability (a coherent motion or rapid perturbations usually drive the FEL into its pulsed regime, as generally observed in the first SRFEL experiments). The FEL usually exhibits a well reproducible behavior vs detuning. For example, the Super-ACO FEL presents five zones of tuning (Figure 8.17) [42]. In zone 3 centered around

I = 80 mA

Super-ACO cw

100 P (mW)

e 80

FEL 200 ps

40

cw

cw

Periodic pulse Zone 1 234

0

5

0

–50

50

Periodic pulse

UVSOR cw

cw Planar undulator

1.2 P (mW)

 L (µm)

460 nm 1 Helical undulator

270 nm 0.4

–14

Intensity (au)

15

0

14

 L (µm)

NIJI4

I = 10 mA

10

5 20 ms 0 300

200

100

0

100

200 L (µm)

Detuning length (µm)

Figure 8.17 FEL average power as a function of the longitudinal detuning of the cavity observed on Super-ACO (350 nm), UVSOR (460 and 270 nm) and NIJI4 (234 nm).

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M.-E. Couprie

perfect tuning, the laser power is maximal, the spectral and temporal widths are minimal, the FEL is “cw” but the micropulse can present some jitter, intensity fluctuations and spectral drifts. In the two adjacent slightly detuned zones (2 and 4), the FEL is pulsed in the ms range, with a slightly wider temporal and spectral pulse, and a lower FEL power. For larger detuning (zones 1 and 5), the FEL is again “cw”, the power is reduced and the spectral and temporal distributions are wider, but the FEL is much more stable in intensity and position, so that the first user applications in biology have been carried out under these conditions [32]. On UVSOR with a planar undulator, the central region is so thin that a unique pulsed central region appears, corresponding to zones 2 and 4, and it is surrounded with two cw zones with larger detuning. The detuning curve was then modified to a 5 zones curve after the installation of the new helical undulator, providing a higher gain. The FEL is pulsed for perfect synchronism both on ACO and NIJI4 [43, 38]. On LINAC driven FELs, the pulses are shorter around the perfect tuning, whereas the spectral width is generally wider. The spectral width can be reduced by detuning the FEL and lengthening the pulse duration. The FEL can also be operated in the Q-switched mode when the gain is periodically suppressed to allow the energy spread, modified by the FEL interaction, to relax towards its initial value, corresponding to the maximum gain of the system. The gain can be modified by a change in the synchronism condition (RF frequency jump, as on Super-ACO and UVSOR) or in the transverse overlap (transverse kick of the electron beam, as on ACO and DUKE). The gain is established during one hundred microseconds and suppressed during a few tens of hundreds of milliseconds. The FEL starts from a non-perturbed situation, following a first giant intense macropulse, and a better pulse to pulse stability. 3.2.2.2 TEMPORAL WIDTH OF THE FEL MICROPULSE

The SRFEL reproduces the microtemporal structure of bunches at a high repetition rate (MHz). The gain being proportional to the electronic density, the FEL micropulse builds up for the maximum gain at perfect tuning, or roughly for a gain corresponding to the cavity losses for a detuned FEL. The temporal pulse narrows by a factor ranging between 5 and 40, providing a better source for spectroscopic applications. A detuned FEL pulse is larger. Short pulses (5–15 ps FWHM) are delivered on UVSOR and Super-ACO with a harmonic cavity, or 6 ps at DUKE ELETTRA with high voltages on the RF cavities. The FEL pulse position can be very unstable for perfect synchronization – the jitter detunes the FEL. The FEL micropulse temporal width is minimal for perfect tuning in zone 3 (cf. Figure 8.18(a)). The behavior can be asymmetrical between zones because of the shape of the bunch distribution. The pulse duration seems to be rather independent from the gap of the dispersive section. The FEL pulse duration gets smaller when the stored current decreases (cf. Figure 8.18(b)) [44]. A single micropulse (on Super-ACO or UVSOR) can differ from a Gaussian distribution and can present internal sub-structure (see Figure 8.19). It can either be a stable distribution, or present internal pulses which simultaneously evolve, or which drift inside the distribution, depending on the detuning, with a complex interplay between the FEL and the longitudinal distribution of the electron bunch. 3.2.2.3 SPECTRAL WIDTH OF THE FEL LINE

In the spectral domain, the natural relative width of an FEL is of the order of 0.01%, ranging between the width of the undulator line (1/N, i.e. 10% for 10 periods) and the Fourier limit (0.001%–0.0001% depending on the FEL pulse duration). A narrow spectral width was

Free electron lasers (b)

60

2.4

50 40

∆ las (Å)

RMS las(ps)

(a)

277

30 20

2 1.6

10 1.2

0

1

2 3 4 Detuning zone

5

20

10 0 10 Cavity tuning (µm)

20

Figure 8.18 Time and spectral structure observed on the SUPER-ACO FEL as a function of the cavity tuning. (a) RMS FEL pulse duration for several values of the dispersion parameter Nd of the optical klystron (• : Nd = 79,
Nd = 43,  Nd = 41). (b) FWHM of the spectral width of the laser pulse. (b)

40 30

2 1.5

∆ las (Å)

RMS las(ps)

(a)

20 10 0 0

5

10 15 20 25 30 35 I (mA/bunch)

1 0.5 0

5

10

15 20 25 I (mA/bunch)

30

35

Figure 8.19 Time and spectral structure observed on the UVSOR FEL as a function of the ring current. (a) RMS pulse duration. (b) FWHM of the spectral width of the laser pulse.

obtained by inserting an etalon in the optical cavity on VEPP 3 [45] (cf. Figure 8.20). Storage ring FELs generally present a high spectral resolution, roughly given by  1 -λ = λ π

λ N σl

(25)

N, the undulator number of periods should be replaced by (N + N d) for an optical klystron. The UVSOR FEL line can present some internal sub-pulses, (see Figure 8.21). The line width, of 0.11 nm at relatively high current, decreases with I (cf. Figure 8.21(a)). The line width gets larger with detuning (cf. Figure 8.18), according to simulations. 3.2.2.4 CORRELATION BETWEEN SPECTRAL AND TEMPORAL WIDTH

In the past (1996), precise studies performed on Super-ACO and UVSOR showed that the FEL did not reach the Fourier limit (see Figure 8.22(a)), given by c-τ -λ/λ2 = 0.44 in the case of Gaussian distributions. Experimentally, on the UVSOR FEL, c-τ -λ/λ2 = 4–39 in Qswitched mode, and 4.3–16 in natural regime. On the Super-ACO FEL, c-τ -λ/λ2 = 5.5–24 in the “cw” regime, and shorter lines were measured in the presence of the longitudinal feedback system, such as 15 ps FWHM. Recent operation with the 5th harmonic cavity,

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0.1 Å 0.017 Å

Figure 8.20 Ultrasmall laser linewidth observed at 630 nm on VEPP-3 using an intra-cavity etalon.

(a)

(b)

(c)

300 ps

–100

(d)

0 100 Time (ps)

8

6

4 2 Wavelength (Å)

1 ms

500 ps

10 ms

350 ps

(e)

10 ms

Figure 8.21 Time structure observed on the UVSOR FEL in the Q-switched mode (curves a and b) and on the SuperACO FEL (curves c, d and e); a and b present the temporal and spectral structure of the laser pulse showing an important substructure; c, d and e present several samples of the time structure observed in different conditions. It may vary from a single stable pulse (curve c) to multiple pulses (d and e) with an important evolution of the profile with time (curve d).

together with improvements on the feedback system, allowed the FEL to operate very close to the Fourier limit by a factor 1.3 above (see Figures 8.22 and 8.23) [46]. The DUKE FEL is also operarated at the Fourier limit. LINAC driven FEL sources on the other hand, can provide femtosecond pulses, but with a wider spectral line. They are generally operated at the Fourier limit (cf. Figure 8.24), insuring a good temporal coherence.

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279

FEL micropulse

800 ps FB off

FB off

100 ms

FB on

800 ps

1Å

FB on

100 ms 3538 Å Time

Figure 8.22 Stabilizing effect of a longitudinal feedback observed on the SuperACO FEL. The curve on the left (right) presents the time structure (spectral profile) with feedback on or off. The feedback strongly reduces the time jitter of the pulse and results in a narrower spectral profile. 1.2

Zone 3 Zone 5

∆las (Å)

1.0 0.8 0.6 0.4

Fo

0.2

u r ie

r li m it

0 0

20

40

60 las (ps)

80

100

120

Figure 8.23 FWHM spectral width vs RMS laser pulse length measured on Super-ACO at 350 nm with one main RF cavity in natural regime (• zone 5 and
zone 3). x with an additional harmonic cavity allowed the operation close to the Fourier limit.

3.2.3

Transverse structure

The transverse mode structure is mainly determined by the choice of the optical resonator. Generally, a TEM00 mode is provided for FEL users. Different transverse modes can be observed by misaligning the optical axis with respect to the undulator one (cf. Figure 8.25) [19]. Table 8.5 gives the parameters of the optical resonators used by various SRFELs. 3.2.4

Comparison with other UV sources

Different conventional laser sources are available in the UV: Nd:Yag, excimer, Ar+ with UV lines or doubled visible line are fixed wavelength lasers (Table 8.6). Tunable sources are provided by dye lasers, possibly associated with up-frequency conversion schemes, by optical pumping of a Yag or excimer laser, parametric amplifiers with frequency conversion, third and fourth harmonic of ultra-short Titanium Sapphire (Ti:Sa) laser and high order harmonics by focussing an ultra-short Ti:Sa laser in a gas cell. Usual tunable conventional lasers (dyes and ns and ps OPA) are optimum from 350 to 200 nm. Systems such as four-wave mixing

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10–1

Line width

TESLA 10–3

10–5

CLIO Los Alamos Stanford @DUKE UVSOR Super-ACO VEPP3 VEPP3 (étalon)

Pulse duration (ps)

10–7 103

VEPP3

102 UVSOR Stanford DUKE Super–ACO Los Alamos CLIO

101 100

TESLA 10

–1

1

10

100 1000 104 Wavelength (nm)

105

Figure 8.24 Relative spectral width and pulse duration of FELs (LINAC driven sources are underlined). Table 8.5 Parameters of the optical cavity for several storage ring FELs

ACO SuperACO UVSOR VEPP3 NIJI-4 DELTA DUKE ELETTRA

Lc cavity (m) length

λ (nm)

5.50 18 13.3 18.7 14.8 14.40 53.73 32.4

630 350 270 250 352 488 170 350–220

Losses (%)

1 0.3 0.8

2

Z0 (m) Rayleigh length

Divergence θTEMOO (µrad)

w0 (µm) waist at the center

Rc (m) Rc1 , Rc2

0.36 3.1 2.5 2.3 2.85 0.28 2.3 4.6

742 193 308 179 198 501 155 122

270 578 470 446 565 310 350 570

2.80 10 8, 6 10, 10 8.5, 8.5 7.47 28.0 17.5

and harmonic generation in gases can provide shorter wavelengths, but with the drawback of increased complexity and very low average power. The advantage of the FEL is that it covers the whole UV range, provided in situ mirror changes with the new optical cavity design including a mirror transfer chamber. Synchrotron radiation is more widely tunable, but the desired spectral range have to be selected with a monochromator, leading to a lower number of photons on the sample. Low repetition rate sources present a high peak power whereas high repetition rate sources have a high average power. FELs can offer short pulses for temporal studies, at a high repetition rate, serving as an intermediary between the conventional laser sources and specific Nd–Yag and Ti:Sa oscillators.

Average power

Range Pulse duration Repetition rate Energy per pulse Peak power

OPA

UV IR–UV 50 ps–50 ns 200 fs–5 ns 1 Hz – a few kHz MHz Specific dye 1 µJ–100 mJ MW (ns and ps dyes > 275 nm), otherwise, lower max: W (ns)

Dye

mW

UV 100 fs 76 MHz 0.1 mJ to few µJ

Ti : Sa

Four wave mixing

10–100 µW

> 10 µJ

VUV

Harmonic generation in gas

W

Several tens of µJ MW

UV–VUV ps

SRFEL expected on DUKE-Elettra

Table 8.6 Comparison between several coherent sources of radiation in the UV–VUV range of the spectrum

kW

UV–VUV, X-ray ps-hundreds fs

LINAC FEL

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Figure 8.25 Various transverse modes of the FEL observed on Super ACO as one displaces the trajectory of the electron beam in the optical klystron away from the axis of the optical cavity. From left to right one observes the TEM00, TEM01, TEM23 modes of the cavity.

FELs operate at relatively high energy per pulse, slightly smaller than those of nanosecond or picosecond lasers’ with a repetition rate smaller by four or five orders of magnitude. The peak power of future UV FELs could reach several megawatts, equivalent to the peak power of excimer lasers or nanosecond and picosecond dye lasers operated above 275 nm. It could be greater than the peak power presently delivered by tunable sources below 275 nm. It is smaller than the peak power of fixed wavelength sources as Nd–Yags, fentosecond excimers, and harmonic generation in gases. The expected average power in the watt range for SRFELs corresponds to the highest values delivered by nanosecond dye lasers in the UV; excimers however provide higher values at some well defined wavelengths. The high power (kW) extracted at Jefferson Lab in the UV is very high, and first industrial applications have already been prepared. The average power of SRFELs is much higher than that of synchrotron radiation. As the FEL directly oscillates in the UV, spatial profiles are narrower than those obtained by frequency conversion from visible or IR lasers. FELs can then be focussed to very small spots, because they are transversally diffraction limited. Synchrotron radiation from third generation light sources can be at the diffraction limit, but the wavefront is largely distorted by the monochromator needed to reach the 0.01% spectral resolution of the FEL. The results presented above still do not portray the ultimate performances of FELs in the UV, since all the parameters of these experiments have not been fully optimized for the FEL. In particular, the Super-ACO FEL (Figure 8.25) operates on a second generation storage ring, with a 3 m straight section of limiting the gain, and the possible extension to short wavelength. A user program at DUKE FEL is now starting, and oscillation further in the UV range at 194 nm has been acheived in 1999 [36]. Besides, projects under development such as at ELETTRA, a third generation synchrotron radiation facility, leads us to expect a high quality FEL which will deliver coherent radiation down to 200 nm of several watts, which can be combined with the existing synchrotron radiation beam lines, for pump probe, two-color experiments.

3.3

VUV and X-ray

At present, the shortest wavelength in the oscillator configuration is 190 nm at ELETTRA [40]. VUV and X-ray FELs are mainly based on coherent harmonic generation and on self amplified spontaneous emission (SASE).

Free electron lasers 3.3.1

283

Coherent harmonic generation

The main advantage of coherent harmonic generation (CHG) over the oscillator mode is the absence of an optical cavity, which makes it very attractive as a source of coherent radiation in the VUV. The pioneering experiments of CHG were carried out on ACO and the best performance was reached on Super-ACO using a doubled mode-locked Yag laser in 1987, the third (177 nm) and fifth harmonic (106 nm) of which were observed with respectively 1.5 × 107 and 1 × 105 coherent photons per pulse [5]. In these experiments, the performances were somewhat reduced due to the alignment imperfections and time jitter between the laser and the electron beam pulses. The new generation of synchrotron source are known to offer a much higher beam stability and combined with the recently available amplified Ti : Sa laser pulsed at high repetition rate, one can expect CHG at shorter wavelengths and higher flux. In this context, the peak power circulating in the optical cavity of a storage ring oscillator operated in a Q-switched mode can be sufficient to produce the bunching on the harmonics. In such a case, no additional external laser is needed. The synchronization and the alignment between the optical wave and the electron beam are straightforward, and a high repetition rate can be expected beyond that available from the Ti:Sa laser. Nevertheless, the use of an external laser insures the short wavelength operation, without limitations due to the mirrors of the optical resonator. A variant of CHG at high gain [47] uses a short electron bunch produced in a LINAC with a photoinjector passing through an optical klystron followed by a second undulator, acting as a “radiator”. An additional shifter displaces the position of the electron micro-pulses with respect to the light, so that the “heated” electrons pulses, having generated coherent radiation in the optical klystron, are replaced by fresh pulses in the second “radiator”. This enhances the coherence of the radiation produced. 35 MW have been produced on the second harmonic of a CO2 laser [47].

3.3.2

Self amplified spontaneous emission

A number of FEL oscillators have observed gains greater than 100% per passage, large enough to reduce the specifications for the reflectivity of the mirrors but far from enough to reach saturation and admit a complete removal of the optical cavity. To fully eliminate the need of any mirror, one must produce coherent radiation in a single pass in the SASE mode. The operation of a SASE FEL sets new challenges in terms of accelerator and undulator technology. A typical SASE experiment for short wavelengths includes an RF photo-injector, an emittance correction, a pulse compressor, a LINAC and a very long undulator. The undulator, 10–100 m long, produces spontaneous emission amplification of several orders of magnitude. It requires additional focussing throughout its length. SASE physics presents various areas of research, such as parametric gain dependance, start-up noise, saturation, mode profile control, coherent synchrotron oscillations. The good agreement between experimental results and numerical simulations lead us to believe that the theory is sufficiently well understood. First, SASE has been demonstrated at saturation in the millimeter range [49] or in the mid-infrared [50], at start-up in the infrared on CLIO [51], at Stanford [52], and very recently in a UCLA/Los Alamos [48] and at Brookhaven at 1 µm [53]. A single pass gain of 105 has recently been observed at 12 µm using a 2 m long undulator installed on an 18 MeV LINAC equipped with a photo-injector [48]. The microbunching phenomenon has been directly observed at CESTA [54]. Intermediate target experiments are also being carried out at Brookhaven [53] and saturation was achieved at 400 nm. The interest is so great that a few groups have started some major experimental developments. At Argonne National Laboratory a SASE FEL has

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M.-E. Couprie

achieved saturation in the UV range (530 and 385 nm), intended to test the feasibility and separate focussing for the undulator [55]. Another very ambitious experiment at DESY which makes use of the newly developed high gradient superconducting cavities built for the TESLA Test Facility (Hamburg) has demonstrated saturation at the shortest wavelength in the 125–80 nm spectral range in Phase 1 and the Phase 2 aiming saturation at 6 nm is under development [56]. Two first user experiments have already been carried out. There are projects at SLAC [57] (Stanford), SPring8 (Japan) [58] and DESY for SASE based X-ray sources at 0.1 nm and shorter. More modest projects are also proposed at BESSY, in Italy. The undulator length required for such facilities is around 100 m per beamline. The peak brilliance from a SASE type FEL is predicted to be 108 to 109 above what is available from the best storage ring based synchrotron sources (see Figure 8.27). Consequently, SASE driven X-ray sources are considered to be the best candidates for the next generation of light sources. The various projects of FEL based VUV and X-ray sources are presented in Table 8.7. The TESLA-TTF project [56] seems to be close to producing very attractive coherent radiation in a new spectral range for the FEL, with femtosecond pulses, at an average power of 3 W. This radiation seems to be a unique tool for new experimental opportunities because of the very high brilliance (Figures 8.26 and 8.27). Nevertheless, the natural spectral resolution of the SASE is expected to be around 0.1–1%, somewhat larger than FEL oscillators the temporal and spectral distribution of the SASE radiation intrinsically presents some internal substructure, known as spikes. SASE based FELs seem very attractive for the X-ray region, whereas the UV–VUV domain can be covered by FELs in the oscillator mode or in the coherent harmonic generation mode. First user experiments are already carried out in the VUV on the ELETTRA, DUKE and TESLA-TTF Phase 1’s FELs. SASE sources generate great hopes for projects such as the TESLA-TTF for the short wavelength operation, and an extensive R&D programme on SASE sources is under way. Energy Recovery Linac based devices are another alternative for fourth generation light sources, and FELs can also be incorporated, as proposed on the English 4GLS project. Table 8.7 FEL sources in the VUV and X-ray range (PI stands for photo-injector) FEL

Location

Accelerator

Presently shut down TESLA-TTF Germany LINAC Phase1 Super-ACO France SR In operation DUKE USA AS ELETTRA Italy AS LEULT USA LINAC Under construction and project TESLA-TTF Germany LINAC Phase 2 TESLA Germany LINAC, PI VISA USA LINAC, PI BESSY Germany LINAC, PI SPARC Italy LINAC, PI SPARX Italy LINAC, PI FERMI Italy LINAC, PI LCLS USA LINAC, PI

Type

λ(nm)

E (GeV)

Pulse

SASE

103–80

0.25

500 fs

GHC

100

O O SASE

194 190 532, 385

0.7–1.1 0.9–1.3 0.44

6 ps 6 ps 1 ps

SASE

6

1

370 fs

SASE GHC, SASE SASE SASE SASE, GHC GHC, SASE SASE

0.1 75 62–1 >1 10–0.1 40–1.2 0.4–0.15

20

50 fs 2 ps 100 fs 100 fs 100 fs 100 fs 300 fs

∼ 200 ps

0.3–2.25 0.15 2 1–3 14.35

Free electron lasers 3.3.3

285

Comparison with VUV and X-ray sources

Experiments can be performed in the VUV with photons produced by harmonic generation in gases, four-wave mixing or synchrotron radiation. In the X-ray range, new lasers operate at specific wavelengths. As X-ray tubes are limited by their low intensity, synchrotron radiation seems to be the most convenient source, because of its large tunability, high flux and brilliance, and well as good stability, even though it has to be monochromatized. Prospects offered by FEL type sources seem to be very attractive compared to synchrotron radiation from insertion devices. It includes both the harmonic generation scheme and the SASE, which will provide very high average (Figure 8.26) and peak brillance (cf. Figure 8.27), together with short pulses (sub-picosecond) much shorter than the electron bunches in a storage ring. Synchrotron radiation still has the advantage of serving many users simultaneously, with a large tunability associated with a very stable beam, whereas SASE sources still have to demonstrate their operation in the X-ray range and pursue the technological developments required to make a stable, routinely operated user facility. Difficulties with the light transport in the beamline are expected due to tremendous power densities. Nevertheless, SASE-FELs appear to be promising candidates for the next generation light sources.

Average brilliance (photons/sec/mm2/mrad2/0.1% bw)

1E25 3rd gener.SRFEL TTF-FEL

SLAC-LCLS

1E22 1E19

3rd gener.harm. of SRFEL

ALS U5 Bessyll-U125

1E16 1E13

High order harmonics (20 Hz)

1E10 1E7 1

10

100 Energy (eV)

1000

10 000

Figure 8.26 Average brilliance as a function of photon energy of several sources of radiation including conventional undulator synchrotron sources, SASE sources, storage ring FELs and harmonic generation.

Table 8.8 X-rays and Gamma-rays produced by Compton backscattering from an FEL FEL

CLIO UVSOR NIJI-4 S.ACO DUKE

Reference

[59] [60] [29] [61] [62]

Electrons (MeV)

50 600 263 800 500

FEL

CBS

λ (µm)

P intra cavity peak

E

nb. ph/s

3.5–7 0.46, 0.27 0.5 0.35

7 GW 8 kW 4 kW 45 kW

7–12 keV 14–25 MeV 3 Mev 35 Mev 12 Mev

5 E6 2 E6 2.5 E5 2 E6

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M.-E. Couprie

Peak brilliance (photons/sec/mm2/mrad2/0.1% bw)

SLAC-LCLS 1E32 1E29 1E26

TTF-FEL 3rd gener.SRFEL 3rd gener.harm. of SRFEL

High order harmonics X-ray laser (LULI) X-ray laser (Livermore)

1E23 Bessyll-U125 ALS U5

1E20

TTF-FEL (spont.) Laser plasma sources Al

Fe

1000

10 000

1E17 1

10

100 Energy (eV)

Figure 8.27 Peak brilliance as a function of photon energy of several sources of radiation including conventional undulator synchrotron sources, SASE sources, storage ring FELs and harmonic generation.

3.3.4

Compton backscattering

X-ray radiation or gamma rays can easily be produced with a FEL, by Compton backscattering (CBS) on a relativistic electron when a radiation pulse reflected by the downstream mirror of the cavity interacts with an electron bunch in a head-on collision. The photons are backscattered in the forward direction and the wavelength λcbs is Doppler shifted with respect to the laser wavelength λL according to λcbs =

λL 4γ 2

(26)

As the electron bunches and the FEL are naturally synchronized and transversally overlapped, such radiation is easy to produce. These sources extend the wavelength range much beyond that accessible with synchrotron radiation sources. Table 8.8 presents the energy and flux of Compton backscattered photons produced by various visible or infrared oscillators. The angular distribution is similar to that of the undulator with a very small deflection parameter. The small spectral width of the FEL confers on the produced radiation a good spectral resolution. The radiation is tunable by simple modification of the FEL wavelength. It is also possible to produce these gamma rays by focussing an external laser on the electron beam and several such gamma ray sources are in operation at NSNL, ESRF (100 MeV on the GRAAL experiment) and SPring8. For the production of gamma rays, the main advantage of the FEL source over an external laser is its high power and tunability.

4

Conclusion

Free electron lasers appear to be very attractive sources of coherent radiation covering a very large spectral range, currently ranging from the millimeter to the VUV range. SASE saturation in the UV at LEULT and in the VUV at TESLA-TTF Phase 1 sets a major step towards the X-ray range. Various skills, from the accelerator physics, insertion device technology, optics

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287

and FEL physics itself have been gathered. Shortly after their development in the infrared range, FELs have been utilized as coherent sources for user applications, providing very original results in various scientific fields, leading for instance, to an annual FEL users workshop. Since 1994, Storage Ring Free Electron Lasers are also being exploited by the scientific community, and, in particular, for two color applications combing both FEL and synchrotron radiation, naturally synchronized, both tunable, and polarized. Implemented in a synchrotron radiation facility, they provide an additional purely coherent source for scientific applications. For developing a highly reliable FEL source for users, all the parameters of the experiments should be carefully designed and the dynamics of the FEL should be precisely analysed and controlled. The two approaches, on LINAC and Storage Ring based devices, are complementary. New directions are now opened by ERL base light sources. Very exciting prospects are now viewed with the FEL projects under development, where very efficient beam parameters and insertion devices will be employed. This certainly constitutes one path towards the next generation of light sources.

References [1] Madey, J. M. J., H. A. Schwettman and W. M. Fairbank, IEEE Trans. Nucl. Sci. 20, 980–3 (1973); Madey, J. M. J., J. Appl. Phys. 42, 1906 (1971); Madey, J. M. J., Nuovo Cimento 50B, 64 (1979). [2] Madey, J. M. J., Phys. Rev. Lett. 36, 717 (1976); Deacon, D. A. G., L. R. Elias, J. M. J. Madey, G. L. Ramian, H. A. Schwettman and T. I. Smith, Phys. Rev. Lett. 38, 892–4 (1977). [3] Billardon, M., P. Elleaume, J. M. Ortéga, C. Bazin, M. Bergher, M. Velghe, Y. Petroff, D. A. Deacon, K. E. Robinson and J. M. J. Madey, Phys. Rev. Lett. 51, 1652 (1983); Billardon, M., P. Elleaume, J. M. Ortéga, C. Bazin, M. Bergher, M. E. Couprie, R. Prazeres, M. Velghe and Y. Petroff, Europhys. Lett. 3(6), 689–94 (1987); Elleaume, P., J. M. Ortéga, M. Billardon, C. Bazin, M. Bergher, M. Velghe and Y. Petoff, J. Phys. 45, 989 (1984); Deacon, D. A. G., M. Billardon, P. Elleaume, J. M. Ortéga, K. E. Robinson, C. Bazin, M. Bergher, M. Velghe, J. M. J. Madey and Y. Petroff, Appl. Phys. B 34, 207–19 (1984); Elleaume, P., “Laser à électrons libres sur l’anneau de collisions d’Orsay,” Thèse de Doctorat d’Etat, Univ. Paris-Sud, Orsay, France, 1984. [4] Kulipanov, G. N., I. V. Pinaev, V. M. Popik, A. N. Skrinsky, A. S. Sokolov and N. A. Vinokurov, Nucl. Instrum. Meth. A296, 1–3 (1990). [5] Prazeres, R., J. M. Ortéga, C. Bazin, M. Bergher, M. Billardon, M. E. Couprie, H. Fang, M. Velghe and Y. Petroff, Europhys. Lett. 4(7), 817–22 (1987); Prazeres, R., Y. Lapierre and J. M. Ortéga, Nucl. Instrum Meth. A259, 184–91 (1987); Couprie, M. E., M. Billardon, M. Velghe, C. Bazin, J. M. Ortéga, R. Prazeres and Y. Petroff, Nucl. Instrum. Meth. A296, 13–19 (1990); Couprie, M. E., M. Velghe, R. Prazeres, D. Jaroszynski, M. Billardon, Phys. Rev. A. 44(2), 1301–15 (1991). [6] Ayvazyan, V. et al., Phys. Rev. Lett. 88(10) (2002). [7] Neil, G., Nucl. Instrum. Meth. A358, 159–62 (1995); Benson, S. et al., Nucl. Instrum. Meth. A429, 27–32 (1999); Neil, G. et al., Phys. Rev. Lett. 84(4), 662–5 (2000). [8] Minehara, E. J., M. Suwamura, R. Nagai, N. Kikuzawa, N. Nishimori and M. Sugimoto, Nucl. Instrum. Meth. A429, 9–11 (1999). [9] Marsi, M., M. E. Couprie, L. Nahon, D. Garzella, A. Delboulbé, T. Hara, R. Bakker, G. Indlekofer, M. Billardon and A. Taleb-Ibrahimi, Appl. Phys. Lett. 70(7), 895–97 (1997); Nahon, L., E. Renault, M. E. Couprie, F. Mérola, P. Dumas, M. Marsi, A. Taleb-Ibrahimi, D. Nutarelli, R. Roux and M. Billardon, Nucl. Instrum. Meth. A429, 489–96 (1999); From the operation of an SRFEL to a user facility, Hosaka, M., S. Koda, M. Katoh, J. Yamazaki, K. Hayashi, K. Takashima, T. Gejo and H. Hama, Nucl. Instrum. Meth. A483(1–2), 146–51. [10] Hama, H., Nucl. Instrum. Meth. A375, 57–61 (1996); Walker, R. et al., Nucl. Instrum. Meth. A429, 179 (1999).

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Part II

Applications

9

Impact of insertion devices on macromolecular crystallography Soichi Wakatsuki

1

Introduction

With the advent of the third generation synchrotron radiation facilities, possibilities of macromolecular crystallography experiments have gone through dramatic changes in the last few years. This is largely owing to insertion devices, particularly undulators, for which the third generation sources are optimised. The high brilliance X-ray beams from undulator beamlines have very low emittance, which enables efficient collection of high quality data sets from weakly scattering micro-crystals of medium to large unit cell dimensions. Stability and tunability of undulator beams are critical for rapid phase determination of protein structures using anomalous signals. The introduction of cryogenic freezing techniques of protein crystals was critical for collecting high quality data without severe radiation damage to the protein crystals. In addition, the development of fast and large area CCD detectors has helped in the efficient use of high brilliance from insertion devices. Taken together, it is now conceivable to determine crystallographic structures of proteins and their complexes en masse in a concerted way, an endeavour called structural genomics. Insertion devices are critical components of such structural genomics projects, which are being vigorously pursued in the United States, Europe, Japan and other countries. The first multiwavelength anomalous diffraction (MAD) experiments conducted using an undulator at a third generation synchrotron source was on N-Cadherin, D1 by Shapiro et al. [1]. The experiment was conducted on ID10, the Troika beam line, at the ESRF. The optics was a rather simple set-up consisting of a Bragg diamond monochromator and a mirror for harmonics rejection, without focusing. Data were collected on a MAR IP detector. For the MAD data collection on Yb LIII -absorption edge, the U46 undulator was tuned to give highest flux at different energies. Beamlines exploiting insertion devices have been constructed and in operation in other third generation synchrotron facilities – APS, ALS, and SPring8. While many beamlines use traditional double crystal monochromator/mirror optics, diamond optics was proposed for beam splitting of the undulator beam on third generation sources. There are already a few beamlines that are based on the diamond optics: Hyogo Prefecture Beamline and BL45XU at SPring8, ID10 and ID14 at the ESRF. The original design of a multi-station beamline using diamond optics was proposed by Nielsen [2] and later materialised as the Troika beamline, ID10 at the ESRF, which is mainly used for physics, materials science and surface science. The success of ID10 triggered the design concept of ID14, Quadriga, the first multi-station beamline dedicated to protein crystallography on a third generation synchrotron facility [3]. In contrast to the ID10 experiment on N-Cadherin, on the straight section of ID14 of the ESRF the undulator gaps are not changed during MAD experiments since there are three

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side stations upstream and any change of undulator spectrum will affect the intensities of the side stations. A number of successful MAD experiments have been conducted using different absorption edges on ID14/EH4 without changing the undulator gaps. The use of high brilliance will continue to be essential for protein crystallographic studies in the next decades. Structural genomics [4] projects around the world are expected to produce many thousands of proteins for crystallographic investigations and their structures will be determined in a co-ordinated manner. High throughput beamlines are one of the key components of the structural genomics. For example, National Institute of Health, USA, is not only supporting molecular biology groups in establishing expression and purification systems, but is also going to constructing a new sector at APS dedicated to structural genomics and target-specific therapeutics. While structural genomics projects will require high throughput stations with MAD capability, the remaining molecular and cellular biology projects will demand stations with even higher specifications. Projects in this field will shift more and more towards complex structures such as protein–nucleic acid and protein–protein complexes, and membrane proteins. Many of these complexes are by nature transient and therefore difficult to keep in stable complexed forms. It is often extremely difficult to crystallise them as well. Crystals of large complexes tend to have large unit cell dimensions, thus, diffract weakly and suffer radiation damage upon X-ray radiation. All these difficulties in turn make it absolutely important to have ample beam time on high-brilliance beamlines. For these extremely challenging projects, each research group will require a large amount of beam time to search for best crystals, either native or heavy atom derivatives, and collect many data sets since many of them may not lead to accurate enough phases. Since crystal growth cannot be predicted very reliably, it is also essential for these groups to have a quick access to the high-brilliance beamlines. This will only be possible if the synchrotron sources are able to provide enough beamlines for both structural genomics type users and challenging structural biology projects on complexes. The new possibilities of tackling far more difficult biological problems combined with the phenomenal speed of data collection achievable on the third generation synchrotron radiation facilities triggered a new wave of developments to maximise the efficiency of synchrotron data collection. The new developments cover a wide range of aspects of protein crystallography, such as new schemes of beam time access, standardisation of end-station instrumentation and on-line computation for rapid structure solution. High throughput of data collection and the genome sequencing projects have persuaded structural biologists to move into a new pursuit: exploring the phase space of protein structures. This chapter discusses first the requirements and examples of undulators for macromolecular crystallography, new types of optics unique to third generation synchrotron sources, in particular, diamond optics, end-station instrumentation with special emphasis on detectors and data acquisition systems. A number of examples will follow to illustrate the impact which insertion devices have had on structural biology. It will also describe a unique, unconventional application of insertion devices: extremely fast, time-resolved protein crystallography.

2

Choice of insertion devices

In making a choice between undulator and wiggler sources, one must consider many practical factors of optics and requirements for protein crystallography experiments. In general, a wiggler source is preferred to a lower energy synchrotron source since undulators cannot provide

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high enough flux at the wavelength required for normal protein crystallography experiments. If a wiggler source is used, its parameters must be optimised for the energy range of 6–16 keV. An X-ray energy range below 6 keV will introduce serious drawbacks such as severe absorption of X-ray photons throughout the beamline and the need for extremely accurate absorption correction of intensity data. There have been extensive discussions on the merit of higher energy X-ray photons. At higher energies, the need for absorption correction is negligible and radiation damage is expected to be less severe. If space is available, a wiggler source can be shared by multiple stations by splitting the beam with wedges, an excellent example being the beamline 5.0 on ALS. At higher energy, 6 GeV and above, however, undulator sources are superior in providing high brilliance, low emittance X-ray beams. An added advantage of undulators is their low heat load, which makes downstream optics less demanding. The lower beam divergence, of the order of tens of microradians, is extremely useful for large unit cell projects. Undulator sources also eliminate the need for collimating mirrors. As discussed below, beam splitting on undulator sources is a relatively new development. Diamond optics provides a suitable solution for multiple stations even though energy changes are somewhat limited. At the ESRF, each straight section of the storage ring is 5 m which can normally fit three 1.6 m long insertion devices. This way each insertion device beamline can have a combination of undulators and/or wigglers suitable for different types of experiments. On High Brilliance Beamline ID2, for example, there are two undulators with two different periods: 26 mm and 46 mm. The former provides a single peak undulator spectrum around 12 keV while the latter gives a complete tunability by changing its gap. On ID14 a similar scheme was adopted with some modification, that is, 23 mm and 42 mm periods, in order to optimise the beam intensities at around 13.4 keV (Figure 9.1). Both undulators have a minimum gap of 16 mm and a length of 1.65 m. The first undulator with a periodicity of 42 mm is tunable over a wide wavelength range and will be used to provide X-rays for experiments outside the standard energy range of 11.5–13.5 keV. The second undulator with a periodicity of 23 mm is a single line undulator with very little tunability but optimised for giving the highest brilliance at around 13.5 keV which is above most of the absorption edges relevant in protein crystallography (Figure 9.2). As this undulator induces only a small additional heat load on optical elements, it is operated in tandem with the 42 mm undulator. The beamline will soon install the third undulator specifically optimised for MAD experiments on Se-Met sample at around 0.979 Å. In almost all cases, horizontal undulators, either single or tandem, are used for protein crystallography beamlines. Only exception so far is the beamline, BL45XU at SPring8 where two tandem vertical undulators are installed in order to accommodate the horizontal scattering geometry of the trichromator set-up [5,6]. Using (400) reflections from Bragg diamond with (100) surface, their scattering angle is almost 90◦ at commonly used wavelengths and therefore a conventional horizontal undulator will be unsuitable since the intensity of scattered X-ray photons will decrease as cos 2θ in the horizontal plane. So far, this is a unique use of vertical undulators. In all other cases, horizontal undulators are used. The direction of polarisation should also match the spindle axis for the oscillation of the protein crystal and the offset (or 2θ motion) of the detector to cover high resolution. Thus, a horizontal spindle axis is preferred for horizontally polarised X-ray beams. In this case, the detector should be translated or swung up (or down) for high resolution data collection (see Section 5). Occasionally a vertical spindle axis is chosen, for example, as on ID13 of the ESRF in order to achieve an extremely small sphere of confusion of the diffractometer, down to a few microns, by avoiding the mechanical sagging of the spindle (C. Riekel, personal communication).

Photon flux [1/(s ×100 mA SR current × 0.1% bandwidth)]

Undulator spectra of the ID14 undulators at the minimum gap of 16 mm

1.4 × 1014 1.2 × 1014

Fixed wavelength, single peak U23

1.0 × 1014 Tuneable U42 0.8 × 10

13

0.6 × 1013 0.4 × 1013 0.2 × 1013 0 5.0 × 103

1.0 × 104

1.5 ×104

2.0 ×104

2.5 ×104

3.0 ×104

Energy (eV)

Figure 9.1 Undulator spectra of the two undulators, U23 and U42, of ID14 at the minimum gap of 16 mm calculated with XOP after transmission through 1 mm of beryllium windows of the beamline and carbon filters of the front-end. Photon flux is calculated through 3 × 3 mm slits at 37 m from the source (figure modified from Figure 3 of [9]).

EH4 Xe U Br Pb Se Hg Au Pt Zn Cu Sm Fe

0.3587 Å 0.7223 Å 0.9202 Å 0.9511 Å 0.9795 Å 1.0093 Å 1.0402 Å 1.0722 Å 1.2837 Å 1.3808 Å 1.6625 Å 1.7433 Å

34.57 keV 17.16 keV 13.48 keV 13.08 keV 12.66 keV 12.29 keV 11.92 keV 11.57 keV 9.66 keV 18.98 keV 17.46 keV 17.11 keV

EH1 EH2 EH3

Figure 9.2 The wavelength ranges of the four experimental stations of ID14, the ESRF and absorption edges of the elements frequently used in protein crystallography.

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More recently, a horizontal spindle axis with about 2 µm sphere of confusion has been developed as part of the microdiffractometer for the same beamline by F. Cipriani et al. Another novel undulator application is one straight section having two slightly misaligned undulators for two independent stations with complete freedom to choose wavelength. An example of such an arrangement has been operational on ID12A/B of the ESRF [7] which provides X-ray photons to two experimental stations operating simultaneously and independently. On this beamline, there are two sets of undulators: Helios I and Helios II. The former is composed of two identical undulators which are separated by 320 µrad from each other to provide a circularly polarised X-ray beam. The second undulator Helios II provides on-axis X-ray beams to another station. Another more recent example of a multiundulator, multiple station beamline is an insertion device beamline, 4ID, at the APS whose two undulators are slightly misaligned with respect to each other by 270 µrad in order to provide X-ray beams to two independent stations [8]. This could be considered as an alternative way to build multiple stations on a straight section of a ring with a precaution that the choice of undulators and dipole(s) is to be made carefully so as not to produce cross talk between the two undulators. Also, a specially designed front-end and a careful arrangement of X-ray optics will be necessary to accommodate the two X-ray beams close to each other.

3 3.1

Optics and choice of wavelengths Wavelength tunability

An essential part of protein structure solution is phase determination which is commonly carried out using the heavy atom derivatives or the MAD techniques. In the former case, it is possible to enhance the heavy atom signals by choosing the X-ray energy just above the absorption edges of the heavy atom label. This necessitates the tunability of the X-ray beam, ideally, for each of the different heavy atom derivatives. In practice, however, wavelength change is not needed so often since the anomalous signal does not change so drastically above their absorption edges. In contrast, for MAD experiments, one needs to be able to scan the absorption edge in order to determine the X-ray wavelengths of choice, with extreme accuracy. The absorption scan must be performed on a real crystal rather than a concentrated solution of a model compound since edge positions shift in energy according to the chemical environment of the heavy atoms in the protein. In some cases, it is even important to control the oxidation state of the heavy atom in the crystal. For example, fully oxidised selenium gives a substantially higher white line than reduced or semi-reduced selenium. More and more structures have been solved using the MAD technique. If there is enough capacity in a synchrotron facility to dedicate a number of straight sections for protein crystallography applications, every effort must be made to provide complete MAD capability to each of the protein crystallography beamlines. If, on the other hand, some of the protein crystallography beamlines cannot be made completely tunable as in the case of the Troika concept, fixed wavelength beamlines should be optimised to provide X-ray energy above most of the important absorption edges, for example, 13.5 keV to take advantage of anomalous signals from heavy atoms.

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3.2

Diamond optics

For the Troika concept, as applied to an undulator source at a third generation synchrotron storage ring, diamond optics is essential to deliver high-brilliance X-ray beams to multiple experimental side stations (Figure 9.3) [3,9]. The absorption of X-ray photons by a thin diamond crystal is very low for X-ray photons with a wavelength of 1 Å or shorter. The excellent thermal properties of diamond single crystals also solve the problem of the high power density of the undulator beam. A diamond (111) reflection which gives scattering angles, 2θ , between 25◦ and 43◦ was chosen to provide the highest possible flux for ID14 at the ESRF. A number of reflectivity measurements on crystals in the symmetric Bragg mode and in the asymmetric Laue geometry were made prior to the construction of the beamline. The asymmetric Laue crystals cut perpendicular to the (100) axis behave almost like perfect crystals and have an integrated reflectivity comparable to that of the symmetric Bragg (111) crystals. In principle, (111) diamonds in Bragg mode will provide a twofold higher flux than (100) diamonds in asymmetric Laue mode. For Bragg diamonds, however, the crystals have to be thinner than 100 µm in order to avoid significant attenuation of the main beam for the downstream stations. This is because the path of the polychromatic beam in the (111) Bragg diamond is about four times as long as that of asymmetric Laue (100) crystals at around 13 keV. Unfortunately, it is still very difficult to obtain large size (111) oriented diamond slabs with the needed perfection. In addition, cleaving and polishing large (111) oriented diamonds to a thickness of less than 100 µm is a formidable challenge. The best (111) oriented diamond that has been made so far is a large diamond, 6 by 10 mm, which was polished down to 65 µm by J. P. F. Sellschop (see below). N. Fujimori of Sumitomo Electronics Ltd, Japan has suggested the possibility of laser ablation of (111) diamond crystals to thin down part of large diamond crystals (6 by 10 mm)

e– beam E3

E2



E1 

 E



E

E

E

EH4 2 3

2 2

2 1

EH3 EH2

Toroidal miror Double crystal monochromator

EH1

Ge220 crystal C111 crystal Multilayer

Figure 9.3 Principle of beam multiplication on the Quadriga beamline, ID14, the ESRF. (Figure prepared by H. Belrhali.)

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to a thickness of ∼ 50 µm. There are other collaborative projects which aim to improve the surface quality of the crystals. A number of yellow, type Ib (100) synthetic diamond crystals with sizes of up to 7 by 7 mm2 and a thickness of 0.1 mm are available on the market. Currently such type Ib (100) diamonds are used for EH1, EH2 and EH3 of ID14 for routine user operation. In order to minimise the number of energy gaps due to the simultaneous excitation of reflections at different energies, a computer program was developed by D. Bourgeois to investigate the optimum orientations of the diamond monochromator crystals for a chosen Bragg angle around the [111] axis. The predicted energy gaps have been verified experimentally on the Optics Beam Line, BM5 of the ESRF. It was found that the [010] axis of [111] Bragg crystals should be inclined by 30◦ from the vertical direction to minimise the number of energy gaps in the energy range of 7–14 keV (Bourgeois et al. unpublished data). The three complete diamond monochromators, with vacuum vessels and associated mechanics for EH1, EH2 and EH3 of ID14 have been designed and constructed in-house [10]. Water cooling schemes for the Bragg diamond crystals will need special consideration due to their small size and thickness. The holding mechanism of the diamond in the double crystal monochromator will inevitably be different from that of standard silicon crystals which are much larger and thus provide more space for thermal contacts. The (100) crystals are held on four sides by Ni-plated Cu holders with indium/gallium eutectic for good thermal contact. The (111) diamonds can be held only at the top and the bottom of the crystals since the other sides need to be kept open for X-ray beam passage, which somewhat limits the removal of heat load. Thus the large (111) oriented diamond prepared by J. P. F. Sellschop suffers from high heat load on the current mounting frame of ID14/EH2 at higher current in the storage ring. At a lower current in 16-buch mode, however, it exhibits the expected flux, about twice as that of a Laue (100) oriented diamond on the same station. In the Troika concept (see Figure 9.3), the monochromatic beam after the first diamond crystal is diffracted again by a second crystal in order to restore the original direction of the X-ray beam in a non-dispersive mode. The second Bragg crystals do not need to be very thin since there is no need of transparent crystals there. In this case, it might make more sense to use Ge (220) which has a much wider bandwidth than the diamond (111) reflection. For standard, that is, non-MAD, experiments, -E/E = 1 × 10−4 is an overkill for protein crystallographic data collection and is to be avoided since the narrow band pass certainly means a less stable beam.

4

End-station instrumentation

A key parameter for high performance of a beamline is the overall duty cycle including automation of sample handling, crystal visualisation and alignment on a diffractometer, reliable and fast area detector, data collection and data analysis software. A number of synchrotron beamlines which have been operating for some time are now concentrating on these ‘down-stream’ projects which will have an enormous influence on the overall throughput. This is particularly relevant in designing a beamline for structural genomics and targetspecific therapeutics. In these applications, a large number of data sets need to be collected and processed within a short period of time. Automatic sample exchange apparatuses have been developed by a number of groups to optimise the overall efficiency of data collection. This will require some co-ordination between synchrotron facilities so as to have a finite number of standard methods of mounting frozen crystals.

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Experiences at various protein crystallography beamlines suggest that kappa diffractometers are not essential for cryo-cooled samples. Very often users bring goniometer heads with long arcs that are used for mounting frozen crystals and saving them after exposure. These arcs can be removed during data collection. The argument on the need of kappa diffractometers are based on the data quality when crystals are properly oriented to measure Bijvoet pairs. Our experience so far indicates that it is not absolutely necessary to do so to collect good MAD data sets, at least on a bending magnet beamline. On insertion device beamlines, however, more thorough investigation is needed to evaluate the optimum strategy for collecting good MAD data sets since radiation damage is a more serious issue with extremely high flux [11–13].

5

Area detectors

Two dimensional area detectors are an essential part of protein crystallography beamlines and are therefore discussed in a separate section. Imaging plate detectors have been widely used on synchrotron as well as laboratory sources. More recently CCD based area detectors have become widely used. This is particularly relevant for third generation synchrotron sources where the high flux X-ray beam allows extremely short exposure, as short as a second or even shorter per degree of oscillation. The advancement of detector technology is so fast, it is often necessary to defer decisions on detectors during the construction of a beamline or frequent upgrades of detectors are necessary. The essential parameters for detectors are pixel size, number of pixels, point spread function (FWHM and FW at 1%), sensitivity, noise level, dynamic range, read-out times, etc. It is also rather important to ensure that direction of offsetting the detector should follow the polarisation of the X-ray beam. For example, a vertical offset of the detector matches a horizontally polarised undulator with a horizontal spindle axis. The reason is twofold. First, intensities of diffracted X-ray photons at a large scattering angle in the horizontal direction are attenuated more severely than in the vertical direction due to the horizontal polarisation of the incoming X-ray beam. Second, diffraction spots in the cusp region created along the spindle axis of the crystal tend to result in poor statistics due to their high partiality. Thus, if one moves the detector upwards, it can collect fully recorded reflections at high resolution without additional decrease in intensity due to the beam polarisation. The first tiled CCD detector was developed by E. Westbrook for the Structural Biology Center at the APS [14]. This detector had a 3 by 3 array of CCD detectors tiled together to provide a large active area of 210 by 210 mm. Various commercial companies have extended the concept of ‘tiling’ CCD detectors and they now provide a number of such detectors. The low divergence of the X-ray beams on ID14, of the order of tens of microradians, enables separation of diffraction spots corresponding to d-spacing of a few thousands of angstroms. The choice of detectors is, therefore, of critical importance to take full advantage of such beam characteristics. Two types of detectors, fibre optics coupled CCD and image plate (IP) detectors are used on ID14 experimental stations. The point-spread function of the detectors must be comparable to the physical dimensions of diffraction spots from crystals of 50 to 200 µm in size. The overall dimension, thus the number of pixels, needs to be sufficiently large in order to collect a large number of diffraction orders. For example, the 2k by 2k CCD detector on ID14/EH4 can record 500–550 diffraction orders with a duty cycle of 10 frames per minute.

Macromolecular crystallography

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Another solution for capturing as many diffraction orders as possible is to use IPs since larger active areas can be achieved more readily. A Weissenberg camera with an active area of 800 by 800 mm has been developed for one of the experimental stations, EH3. Two IPs, 40 by 80 cm, are placed on a flat wall by a robot from an IP storage/erasure device standing next to the robot. The IPs are read out using an off-line IP drum scanner with 100 µm pixel size. Figure 9.4 shows enlarged diffraction spots from blue tongue virus core particles (D. Stuart et al., personal communication) recorded on such IP on ID14/EH3. The spot separation corresponds to 570 Å (20 pixels) and the FWHM is 1.8 pixels. Based on this, one can expect to be able to collect a 3 Å resolution data set from a crystal with 2000 Å cell edges without translating the IP detector in the plane perpendicular to the X-ray beam. While CCD detectors satisfy many of the requirements of protein crystallography beamlines, they have some limitations: a rather low dynamic range, 14–15.5 bits, not so ideal spatial resolution, and severe cross talk between pixels especially when they are overloaded. There is also a strong correlation between the read-out speed and the dynamic range of CCD detectors. In order to circumvent these problems, many groups are currently developing pixel array detectors based on solid state detectors. Even though most of the intrinsic physics of pixel array detectors seems to have been solved, there remains the formidable engineering

570 Å (20 pixels)

Figure 9.4 Part of a diffraction pattern from blue tongue virus core particle 1 (Stuart et al., personal communication), space group P21 21 2 with 755×796×825 Å3 , collected on ID14/EH3 using a large image plate. Crystal to IP distance was 1250 mm and the wavelength was 0.918 Å. Oscillation range was 0.1◦ and exposure time 100 s. The beam size was 100 µm. The pixel size of the scanner was 100 µm, which gives an average spot size of 181 µm (FWHM).

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task of tiling small patches to cover a wide solid angle required for protein crystallography experiments. In addition, since pixel array detectors are photon-counting devices, they will have dead-time induced non-linearity problems at high count rate, which might become critical when used for kinetics experiments on fourth generation synchrotron sources in the future. Thus, there might be a need for solid state area detectors based on other types of techniques.

6

Data acquisition and data analysis

Data acquisition is an important issue to be addressed in conjunction with the choice of detector systems. A comprehensive software package, ProDC, has been developed at the ESRF for beamline control, data acquisition and interfacing with data analysis software in order to fulfil the requirements of the beamlines (Figure 9.5). In particular, ProDC is interfaced to

Figure 9.5 ProDC graphical user interface developed by D. Spruce as a common interface for protein crystallography beamlines at the ESRF. The interface provides uniform experimental environment for the stations with different hardware set-ups: diffractometer, beam slits, detectors.

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commonly used processing programs such as DENZO, SCALEPAC, MOSFLM and other CCP4 programs with common graphical representation of results. The package has been installed on all the ESRF macromolecular crystallography beamlines for various detectors and experimental set-ups. In the future, additional features will be added to the package. On-line data analysis, will run automatically as the images are collected and data acquisitioin is kept track of so that users know the quality of the data, completeness, etc. Optimum data collection strategies based on cell dimensions, crystal orientation and space groups can be suggested. These strategy solutions include orienting the crystal for optimising anomalous signals. If one or more data sets have already been collected from other crystals, the strategy program will suggest the best strategy to fill the missing region of the previous data sets. For instance, a data collection strategy program developed by R. Ravelli (personal communication) can import cell parameters and an orientation matrix from an output file of either MOSFLM or DENZO and calculates the best data collection strategy for simultaneous measurements of Friedel pairs, high completeness or uniform redundancy.

7

Ancillary facilities

It is extremely useful if sample preparation laboratories and computer rooms are provided in the vicinity of the already existing protein crystallography beamlines. As an example, the arrangement of the beamlines, ID13, ID14 and BM14 at the ESRF (Table 9.1), along with various ancillary facilities is shown in Figure 9.6. For more complicated sample preparation, there is a fully equipped biochemistry laboratory for users of the ESRF beamlines located in the EMBL on the same site. In many other synchrotron facilities, similar set-ups are available in the vicinity of the beamlines. In some cases, capacity for handling air-sensitive samples is also important for sample manipulation in anaerobic conditions.

Table 9.1 ESRF beamlines concerned with structural biology Beamline

Time dedicated to structural biology

Operation since

Specific applications

Detectors

ID9

Half

Sep. 1995

CCD, IP

ID13 ID14 A/B EH1 EH2 EH3 EH4

One quater

Sep. 1995

Time resolved Laue/monochromatic protein crystallography Micro crystals Monochromatic protein crystallography

Full Full Full Full

Mid-1999 Early 1999 Dec. 1997 Jul. 1998

BM14 ID29

Full Full

Sep. 1996 Sep. 2000

∗

CRG beamlines are not included in the list.

Large proteins and viruses Multiwavelength anomalous diffraction (MAD) MAD Mad on insertion device

CCD, IP CCD CCD IP, CCD CCD CCD, IP CCD

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Sample preparation labs BM14

Terminal room

OH2 EH4 EH3

OH1 EH2

ID14

EH1 ID13 Terminal room

Figure 9.6 Arrangement of the protein crystallography beamlines, BM14 and ID14, and the micro focus beamline ID13 with the ancillary support facilities of the ESRF.

8

New operation modes of synchrotron radiation protein crystallography

Extremely fast data collection on third generation synchrotron radiation facilities has changed the concept of efficient use of synchrotron beam time. A centralised beam time allocation is much more efficient in assuring fast and flexible access to the synchrotron sources by the community. In the ESRF, it is co-ordinated by Beam Line Operation Managers (BLOMs). Two-thirds of user beam time are used by BAGs (Block Allocation Groups) while the rest is used by individual groups who do not belong to BAGs. Each BAG can decide which projects to bring. A few days of ‘test crystal’ beam time is also set aside for non-BAG users. One proposal round is six months and in each period the Beam Time Proposal Committee evaluates hundreds of proposals for PX beamlines. As a whole, 20–25% of the ESRF beam time is used for protein crystallography applications. In the NSLS, Brookhaven, a remote control is already in operation via network (http://www.xl2c.nsls.bnl.gov/xl2c/remote_tools.html). In this scheme, users can check the progress of experiments in their home institute. In the near future, a number of synchrotron facilities are expected to provide this mode of data collection. There are also projects to develop operation schemes in which users can ship pre-frozen samples to a synchrotron source where they are mounted semi-automatically to a beamline.

Macromolecular crystallography

9

305

Recent results from insertion device beamlines

It is not possible to list all the recent results of protein crystallography using insertion devices. Instead, some examples of structures from the ESRF will be given to illustrate how challenging structural biology projects benefit from the advanced properties of insertion devices, optics and end-station instrumentation.

9.1

Tropinone reductase II

In plants, there are a number of enzymes involved in the synthesis of tropane alkaloids including those which are medically important such as cocaine and hyoscyamine (atropine). In Korean morning glory, tropinone reductases (TR) catalyse key branching reactions in the synthesis of such tropane alkaloids. There are two TRs that have the opposite stereospecificity on the 3-carbonyl group reduction of tropinone: TR-I forms tropine with the 3α-hydroxyl group, while TR-II forms pseudotropine with the 3β-hydroxyl group. In spite of this different specificity for tropinone reduction, two enzymes have the same stereospecificity for NADPH and share 64% identical residues in their amino acid sequences. Thus, the crystal structures of both TR-I and TR-II, were expected to provide details of the structural basis of their enzymatic catalysis, especially details of how these enzymes control their strict stereospecificities. The crystal structures of both TRs have been determined independently at 2.4 Å (for TR-I) and 2.3 Å (for TR-II) resolution using the MIR method. The overall structures of the two enzymes were strikingly similar. Further determination of the crystal structure of TR-II complexed with NADP+ and pseudotropine at 1.9 Å resolution revealed the detailed structure of the active site. In fact, this stable product complex was the key to observe the bound substrate, pseudotropine, and NADP, which was one of the first successful experiments on ID 14 during its commissioning period. The crystal of the tropinone reductase II ternary complex had a hexagonal lattice with a caxis of 338.3 Å. The spots along the c-axis were well separated on the IP. The diffraction spots were clearly observed beyond 1.2 Å (Figure 9.7), whereas on a rotating anode, spots barely reached 2.5 Å and could not be processed due to the large cell dimensions. The structure was solved at 1.9 Å resolution with molecular replacement [15]. Fo-Fc electron density omit map of the active site is shown in Figure 9.8. The NADP(H) binding mode and the positions of catalytic residues were similar between TR-I and TR-II. The substrate binding site of TR-II showed high complementarity to the bound substrate (pseudotropine). In addition, electrostatic interaction between the substrate and the binding site seems to fix the binding position and orientation of the substrate. On the other hand, the corresponding area in TR-I which is presumed to be the binding site, provides different van der Waal’s surfaces and electrostatic natures from that in TR-II. These differences contribute to differentiating the substrate binding modes of TR-I from that of TR-II; TR-I binds the substrate in ‘upside-down’ orientation compared to TR-II. These results suggest that the active site structures of both enzymes control their substrate binding modes and determine their reaction stereospecificities.

9.2

α-actinin double repeats

Dijinovic-Carugo et al. [16] have determined the crystal structure of the two central repeats in the α-actinin rod at 2.2 Å resolution using the large format IP camera on EH3/ID14. The unit

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S. Wakatsuki

1

2

1

2

3

4

3

4

1.2 Å

Figure 9.7 Diffraction spots of reaction intermediates of tropinone reductase II measured on a large IP detector. Some parts of the diffraction image, insets 1 and 2, show a complicated diffraction peak profile due to the inhomogeneous distribution of reaction status across the crystal. This degree of detail, while keeping the possibility of collecting a full data set from the large unit cell (P61 22, a = b = 88.6 Å, c = 338.3 Å) to the resolution limit of 1.2 Å, can only be observed on this type of large area detector.

cell dimension along the c-axis was rather large, 390 Å, and it was difficult to resolve 350 reflections on the CCD detector of the station along the edge for the high-resolution data collection. Thus the use of the large format IP was essential to carry out the experiment. The MAD phasing was carried out using the MAD data set collected on BM14 of the ESRF, while the high resolution native data set was collected on the large IP whose low resolution bins were collected on the MAR 133 mm CCD detector on the same station. The structure (Figure 9.9) shows that the repeats are connected by a helical linker and form a symmetric, antiparallel dimer in which the repeats are aligned rather than staggered. Using this structure, which reveals the structural principle that governs the architecture of α-actinin, the authors have proposed a model of the entire α-actinin rod. The electrostatic properties explain how the two α-actinin subunits assemble in an antiparallel fashion, placing the actin-binding sites at both ends of the rod. This molecular architecture results in a protein that is able to form crosslinks between actin filaments. 9.3

Acetylcholinesterase complexed with an inhibitor

Acetylcholine is a neurotransmitter which opens a cation channel, acetylcholine receptor, on the postsynaptic membrane of a synapse. Thus the receptor mediates the transmission of nerve

Macromolecular crystallography

307

(a)

(b)

L210 L213

L210 L213

E156

E156

V191

V191

V197

Y159

I192

S146 K163

T194

R19

V197

Y159

I192

S146 K163

T194

R19

R41

D66

R41

D66

Figure 9.8 (a) Fo-Fc electron density omit map of the active site of TR-II showing NADP and pseudotropine and (b) a stereoview of the active site model. (Reproduced with permission from [12].)

signals across synapses. The acetylcholine receptors close rapidly, within milliseconds, since acetylcholine is hydrolised to acetate and choline by acetylcholinesterase (AChE). Crystals of mouse AChE, an α/β hydrolase, in complex with the peptidic inhibitor fasciculin (Fas2mAChE complex) belong to space group P65 22 with the cell dimensions a = b = 73.8 Å and c = 548.6 Å. Using the large IP detector with an industrial robot IP changer of the ESRF beamline ID14-EH3, Y. Bourne et al. of CNRS Marseille collected complete data with resolution up to 2.5 Å with a total oscillation of 36◦ and 3◦ oscillation range, with the long c axis of the crystal being roughly aligned to the spindle axis. The crystal-to-detector distance was 590 mm. Five complete data sets were collected: a native Fas2-mAChE complex, and four

(a)

TITIN αA

F-actin

αA

(b) F-actin

αA Tensin

αA Vinculin Talin

αA β α

Zyxin β α

Plasma membrane Extra-cellular matrix

(c)

Figure 9.9 Role of α-actinin and overall structure of the R2R3 dimer and the dimer interface. (a) The muscle Z disk, α-actinin (αA) crosslinks antiparallel actin filaments from adjacent sarcomeres. Titin which acts as a molecular ruler for the sarcomere interacts with two different parts of α-actinin. (b) A simplified representation of a focal contact showing the potential roles played by α-actinin in linking the actin cytoskeleton to such membrane associated structures. (c) Domain structure of α-actinin showing aligned and staggered models for the dimer. ABD: actin binding domain; R1,R2,R3,R4: repeats; C: C-terminal calmodulin like domain. The colour scheme for the repeats is maintained throughout the figure. (I) An aligned arrangement where repeats 1 and 2 are paired with repeats 4 and 3, respectively from the opposing monomer. (II) and (III) Alternative staggered arrangements where either repeat 1 or 4 is not paired with any repeat from the opposing monomer. (d) Ribbon diagram of R2R3 dimer. R2 domain is coloured blue, R3 domain is green, and the linker is red. Structural elements involved in dimer interface are coloured yellow. (e) Dimer interface. The protein backbone is shown as a ribbon; residues involved in dimer interface are shown as red balls for domain R2, and yellow balls for domain R3, centered on Cα atom of the residues. R2 domain is coloured blue, R3 domain of the opposing subunit is green. (Reproduced with permission from Cell – copyright 1999.) (See Colour Plate II.)

Macromolecular crystallography (d)

(e)

309

loop 1⬘–2⬘ helix 1

Top helix 2

Centre

loop 2⬘–3⬘ Bottom

helix 1⬘

Figure 9.9 (Continued.)

ternary complexes of Fas2-mAChE and second inhibitors obtained either by co-crystallisation of the three partners or by soaking of the Fas2-mAChE crystals with the second inhibitor. One of the data sets gave the following statistics RSYM = 7.6 (36) in the highest shell Completeness = 84.5% (53) Net I /σ (I ) = 8.9 (2.0) Multiplicity = 5.4

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S. Wakatsuki

Figure 9.10 Acetylcholinesterase complexed with a peptidic inhibitor fasciculin (Fas2-mAChE) by Y. Bourne et al. CNRS Marseilles, France. (See Colour Plate I.)

This structure is refined to R-factor = 22.7% and R-free = 27.4% in the 20–2.5 Å resolution range. Figure 9.10 shows part of the electron density map of AChE complexed with an inhibitor (Y. Bourne, personal communication).

9.4

Structure of the nucleosome core particle

All the eukaryotic chromosomes consist of regularly repeating complexes of protein and DNA, called nucleosomes. One of the fundamental problems in biology has been the question of how nucleosomes can pack DNAs in such a tight structure and yet make some of the DNAs quickly available for transcription in response to various cellular signals. Each nucleosome particle consists of a protein complex with two sets of four different histones, which, together with a fifth histone, H1, wrap approximately 200 base pairs of DNA around the particle. More than a decade after having solved the 7 Å structure of the nucleosome in 1984, Tim Richmond and his colleagues of ETH, Zurich managed to elucidate the structure at 2.8 Å resolution using the microfocus beamline ID13 of the ESRF. In order to obtain well diffracting crystals, they spent enormous amount of effort to perfect biochemical preparation of the complexes. They used the DNA/protein complex crystals for which all components were made in bacteria and assembled after purification. In the end, the crystals containing DNA of a defined sequence and histones lacking post-translational modifications diffracted to around 2.0 Å (more recent result on ID9) and an even higher resolution structure is expected to become possible on ID14/EH4 in the near future. In the current structure an entire 146 base pair DNA molecule has been identified together with over 80% of the eight histone subunits. The DNA is wrapped around the histone octamer in 1.65 turns of a flat, left-handed superhelix, but its path is distorted by bends at several

Macromolecular crystallography

311

positions. The structure provides a clear template for the interpretation of genetic and biochemical data and will help in understanding how the nucleosomes assemble into the higher order chromatin structures [17]. The work has paved the way for further analyses of histone acetylation.

9.5

Nuclear transport receptor importin-β bound to the IBB domain of importin-α

In eukaryotic cells there is a separation between the cell nucleus, where genetic information is stored, and the surrounding medium called cytosol where the translation of the information stored in the nucleus results in the synthesis of proteins. Hence, there is a need for a wellregulated mechanism of transporting molecules between the two compartments of the cell. In many cases, proteins known as nuclear transport receptors perform this task. The receptor consists of two proteins. Importin-α, which recognises a molecule to be transported into the nucleus, is tightly bound at the centre of the snail-shaped, all α-helical importin-β. The rather closed conformation of importin-β when binding importin-α has led to suggestions that once the complex has entered the nucleus, importin-β unwinds to form a more open conformation. This would allow importin α and the molecule to be imported to the nucleus to move away from importin β. This structure has thus shed important light on the complicated processes of transporting proteins between the cell nucleus and the cytosol. High resolution data were collected on ID02 and ID14/EH3 and MAD data set on BM14 of the ESRF. The structure of the complex [18] showed how importin-β recognises its adapter importin-α for nuclear import (Figure 9.11). Importin-β consists of 19 tandemly repeated HEAT motifs and wraps around the IBB domain of importin-α. The structure of one such receptor has been solved using MAD phasing at BM14. The refinement of the structure was carried using data collected at ESRF beamlines ID14-EH3 and ID02.

9.6

Bovine heart cytochrome bc1 complex (ubiquinol-cytochrome c reductase)

Proteins associated with cell membranes, either partially or fully integrated into the membrane, provide a special challenge to macromolecular crystallography. Not only are these important proteins difficult to crystallise, but the crystals are mostly small and tend to give weak diffraction patterns with medium to high mosaicity, which makes the use of synchrotron radiation mandatory. The mitochondrial cytochrome bc1 complex is a respiratory multi-enzyme complex which may also function as a signal peptidase. Previous studies have characterised the transmembrane section and matrix side of the complex, but Iwata and his colleagues solved the complete structure of the complex which gives detailed information on the intermembrane side [21]. The functional domains containing the redox centres of cytochrome c1 , the ‘Rieske’ [2Fe2S] protein and the entire subunit 8 have been clearly identified, providing the basis for understanding the respiratory function of the complex. In particular, by comparing structures of different crystal forms, the authors were able to discern the details of the reaction mechanism through different positions of the ‘Rieske’ protein relative to the rest of the complex.

(a)

(b)

N

2

1

16

3

15

17

4

18 C

5

19

6 7

14 13

8

12 9 11

10

Figure 9.11 Structure of importin-β bound to the IBB domain of importin-α. (a) view along the superhelical axis. A and B helices are shown in red and yellow respectively while connecting residues are shown in grey. (b) Side view where importin-β is coloured progressively from yellow (N-term) to red (C-term). The structure has been rotated by 90◦ about a vertical axis with respect to (a). This kind of superhelix structure is seen in other molecules, for example, clathrin heavy chain stem [19] phosphates PR65/A [20]. (Reproduced with permission from Nature – copyright 1999.) (See Colour Plate III.)

Macromolecular crystallography

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Intermembrane space Subunit 8 Cytochrome c1 Heme c1 ISP FeS Transmembrane Subunit 10 Subunit 11 Subunit 7 Cytochrome b Heme bL Heme bH

Matrix Subunit 6 Subunit 9 Core 1 Core 2

Figure 9.12 The complete structure of cytochrome bc1 complex with 11 subunits from the bovine heart mitochondria inner membrane. (Reproduced with permission from Science – copyright 1998.) (See Colour Plate IV.)

X-ray data used for this structure determination were collected on beamlines ID2, EH3 and EH4 of ID14 and some key features of the complex are shown in Figure 9.12. In particular, on EH4/ID14, they could collect a complete data set from the P65 form with the longest cell dimension of 720 Å up to 2.5 Å resolution within one hour using ADSC Quantum 4, a 2 by 2 CCD detector of the beam line. With this set-up, 580 reflections along the edge of the detector were collected, and the peak-to-peak distance was about 4 pixels, showing the extremely low divergence of the beam and the sharpness of the point-spread function.

9.7

Microcrystals: structures of the photocycle intermediates of bacteriorhodopsin

At the ESRF a microfocus beamline, ID13, permits good quality data to be collected on small crystals only a few tens of micrometres in size [22,23]. The experimental arrangement on

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S. Wakatsuki

ID13 is based on a 30 µm focused undulator beam, but beams down to a few microns in size are feasible. A major problem concerns sample alignment and much effort has been invested in this area. In addition, radiation damage makes the use of cryogenic techniques mandatory and fast data collection is essential through the use of CCD-based detectors. One of the early experiments conducted on this beamline was bacteriorhodopsin. The protein is found in the purple membrane of Halobacterium salinarum and acts as a lightdriven proton pump across the cell membrane to convert light into chemical energy. At ID13, diffraction patterns of three-dimensional bacteriorhodopsin crystals of about 30 µm in size were, for the first time, measured to high resolution giving data to 2.0 Å. The structure at 2.5 Å [24] was subsequently extended to 1.9 Å after numerous attempts of obtaining non-twinned crystals [25]. The seven transmembrane helices of bacteriorhodopsin encompass a proton translocation pathway containing the chromophore, a retinal molecule covalently bound to Lys216 via a protonated Schiff base, and a series of proton donors and acceptors. Photoisomerisation of the all-trans retinal to the 13-cis configuration initiates the vectorial translocation of a proton from the Schiff base, the primary proton donor, to the extracellular side, followed by reprotonation of the Schiff base from the cytoplasm. Using EH3/ID14, Edman et al. [26] obtained high resolution X-ray structure of an early intermediate in the photocycle of bacteriorhodopsin, formed directly after photoexcitation. A key water molecule is dislocated enabling the primary proton acceptor, Asp85, to move. Movement of main chain Lys216 locally disrupts the hydrogen bonding network of helix G, facilitating structural changes in later stages of the photocycle. Independently, the groups of Luecke and Lanyi also elucidated the structure of the late photointermediate of bacteriorhodopsin, MN at 2 Å resolution [27]. They used the beamline 5.0.2 of the Advance Light Source and the microfocus beamline ID13 of the ESRF for high resolution data collection from thin crystals of about 60 by 60 by 15 µm which were also crystallised by the cubic lipid phase technique.

9.8

Time-resolved macromolecular crystallography

Laue diffraction, using the white X-ray beam of bright synchrotron radiation sources, had opened up new possibilities in studies on dynamics in protein crystals. Although the technique has a longer history than the rotation method using monochromatic X-ray radiation, its revival had to wait until powerful insertion device beamlines became available. With the recent introduction of a cryo-trapping of reaction intermediates, the Laue technique will have its main advantages in extremely fast kinetics at ambient temperatures. Beamline ID9 at the ESRF has been specifically designed for time-resolved macromolecular crystallography experiments so that reaction mechanisms can be studied in situ. Proteins normally undergo various degrees of structural changes while carrying out their biological and catalytic functions. Although the initial and final structures of these reactions are known for some proteins, the pathways connecting these limiting structures are largely unknown and until recently, largely unexplored. The experimental set-up to follow structural changes of macromolecules as they occur in the crystal has been developed at the ESRF on beamline ID9 [28]. K. Moffat and his colleagues (University of Chicago) and the group led by M. Wulff at the ESRF have pioneered the ultra-fast kinetic X-ray crystallography technique to study the ligand binding haem proteins, myoglobin, and haemoglobin and a photosensor called photoactive yellow protein (PYP).

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Myoglobin is a protein found in muscle cells, which stores oxygen for conversion into energy as required. Haemoglobin, a protein found in red blood cells, transports oxygen from the lungs to various tissues of the body. The collaborative team led by K. Moffat and M. Wulff have conducted time-resolved experiments in which they triggered the dissociation of carbon monoxide from CO-bound myoglobin (MbCO) crystal using a 10 ns optical pulse, and measured Laue diffraction patterns with 150 ps polychromatic X-ray pulses. CO was chosen as an alternative for O2 since it is easier to photo-dissociate from MbCO and rebinds to the protein more slowly. Structures were determined at various times after the excitation by varying the time between photolysis and the X-ray pulses. The intensity of the X-ray pulse on ID9 was sufficient to record a Laue pattern averaged from tens of single pulses, making it possible to accumulate a complete set of Laue patterns at different times after excitation with a single MbCO crystal (Figure 9.13). In principle, the sequence of time-resolved structures produces a short ‘movie’ of a protein in action. Simplified snapshot models derived from the diffraction patterns are shown in Figure 9.14. The first model shows the initial state before the optical laser pulse where the CO is still bound to the iron of the haem moiety. The second model, calculated from the data collected 4 ns after photo-dissociation, shows an intermediate state in which the CO is displaced by about 4 Å from the iron. It stays in this configuration for around 350 ns. The function of this ‘docking’ site is to prevent CO from returning to the chemically attractive haem iron. The last picture, taken 1 µs after the optical pulse, shows that the CO has left the binding pocket completely. In this time regime, the CO diffuses about in the outer part of the protein for a fraction of a millisecond before eventually recombining with the iron through random collisions with the fluctuating haem and thereby closing the cycle. The time evolution of these features suggests how the protein responds structurally to ligand dissociation and rebinding under ambient temperature conditions [29]. The time resolution of the technique has been improved recently by nearly a factor of 200 making the fastest time resolution of 50 ps, that is, the shortest duration of the X-ray pulses produced at the ESRF. This is combined with an extremely fast excitation laser pulse of about 85 fs from a femtosecond laser synchronised to the X-ray pulses. Preliminary work has also been undertaken on haemoglobin in which the protein is modified in order to prevent an allosteric transition whose quaternary structural change would destroy the crystal packing. This kind of time-resolved technique could be extended to other photo-biological systems such as bacteriorhodopsin and PYP (see below) and the photo synthetic reaction centre provided that suitable high quality crystals can be obtained. PYP of bacterium Ectothiorhodospira halophila is thought to be a photosensor that allows the bacterium to detect and swim away from dangerously high levels of blue light. The chromophore in PYP is a 4-hydroxycinnamic acid molecule bound to the cystine residue Cys69 via a thioester linkage. When this chromophore absorbs a photon, it isomerises from trans to cis form which triggers a series of structural rearrangements within the protein. The structural change of proteins in its late intermediate state is believed to be transmitted as a signal to the cascade of intracellular events, even though there is no experimental evidence identifying a signal transducer at present. As a net result of the cascade, the bacteria change the direction of motion to avoid the danger of the strong, near ultraviolet light. Even though many of the biological and structural questions related to the photocycle of PYP have been addressed crystallographically using the freeze trapping techniques, there are many more questions that remain to be answered. For example, it is not yet known how the protein prepares the chromophore for the initial photoreaction to achieve the rather high quantum yield of 64% for the trans to cis isomerisation. The question of how the chromophore is able to

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Figure 9.13 Time resolved Laue diffraction patterns from MbCO. The inset on the top-left shows part of the region marked on the overall diffraction image of the dark state. On the right, the changes of the diffraction spots in the same region are shown for the six different times: 4, 15, 60, 150, 520 nsec and 1.2 µsec after the laser flash. (Srajer et al. personal communication.) (See Colour Plate V.)

undergo photo-isomerisation in the tightly packed protein interior, especially under the constraints of crystal contacts, also needs further investigation. These questions cannot be readily answered by the results from the cryo-trapped intermediates whose local environments might be altered due to the low temperature. Thus it is necessary to perform experiments at ambient temperatures, which requires the time-resolution currently available only at the ESRF. In collaboration with ESRF scientists, the groups of K. Moffat and E. Getzoff have determined X-ray structures of photo-excited PYP with microsecond [30] and nanosecond [31] time resolution, respectively. Preliminary results from high quality Laue patterns obtained with

Figure 9.14 Structural models of the transition from MbCO to Mb obtained from 150 ps Laue data. (Courtesy of M. Wulff.) (See Colour Plate VI.)

Figure 9.15 Laue diffraction pattern recorded using a single-pulse focused from a wiggler source on ID09. A high-speed chopper and the single bunch mode of operation are used to isolate the pulse which has a length of 150 ps. The bandwidth of the polychromatic pulse is 6–38 keV and the pulse contains 5 × 1010 photons. There are about 800 diffraction spots in the image with intensities ranging from 5 to 50 000 photons per spot. The image was recorded with an image intensifier CCD camera (1242 × 1152 pixels). In the experiment the CO molecule was dissociated from the Fe binding-site by a l0 ns laser flash and position of the CO molecule and the protein response were recorded for time delays between l0 ns and 1 ms. The data was analysed to 1.9 Å resolution. (Courtesy of M. Wulff.) (See Colour Plate VII.)

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a single electron bunch and sub-nanosecond time resolution are encouraging (Figure 9.15). It appears, however, that for this type of kinetic studies one needs both approaches, fast time-resolved experiments at ambient temperature and cryo-trapped intermediates, in order to arrive at unambiguous structured results and functional analyses.

10

Future outlook

In the next few years, structural genomics efforts will have dramatic impacts on the synchrotron protein crystallography beamlines by making data collection and analysis far more streamlined. As a result, the field of macromolecular crystallography will move in two directions: high throughput protein crystallography [4] and extremely challenging and complex crystallographic project exemplified by, for example, the ribosome projects [32–34]. In both cases, insertion devices, particularly undulator sources, will be a critical component of such endeavours. In-vacuum undulators on smaller third generation synchrotron sources such as the Swiss Light Source will prove extremely useful in providing high-brilliance X-ray photons for protein crystallography experiments.

Acknowledgements The author acknowledges the help and support of all the members of the Joint Structural Biology Group of the ESRF and the EMBL Grenoble Outstation. Thanks go to D. Bourgeois for his suggestions on the manuscript.

References [1] Shapiro, L., A. M. Fannon, P. D. Kwong, A. Thompson, M. S. Lehmann, G. Grubel, J. F. Legrand, J. Als-Nielsen, D. R. Colman and W. A. Hendrickson, ‘Structural basis of cell–cell adhesion by cadherins’, Nature 374, 327–37 (1995). [2] Nielsen, J.-A., A. K. Freund, G. Grubel, J. Linderholm, M. Nielsen, M. Sanchez del Rio and J. P. F. Sellshop, ‘Multiple station beamline at an undulator X-ray source’, Nucl. Instrum. Meth. B94, 306–18 (1994). [3] Wakatsuki, S., H. Belrhali, E. P. Mitchell, W. P. Burmeister, S. M. McSweeney, R. Kahn, D. Bourgeois, M. Yao, T. Tomizaki and P. Theveneau, ‘ID14 Quadriga, a beamline for protein crystallography at the ESRF’, The Proceedings of SRI97, J. Synchrotron Rad. 5(3), 215–21 (1998). [4] Kim, S. H. ‘Shining a light on structural genomics’, Nature Structural Biology 5, 643–5 (1998). [5] Yamamoto, M., T. Kumasaka, T. Fujisawa and T. Ueki, ‘Trichromatic concept at SPring-8 RIKEN beamline’, The Proceedings of SRI97, J. Synchroton Rad. 5(3), 222–5 (1998). [6] Ueki, T. and M. Yamamoto, ‘The start of a new generation: the present status of the SPring-8 synchrotron and its use in structural biology’, Structure 7, R183–7 (1999). [7] Goulon, J., N. B. Brookes, C. Gauthier, J. B. Goedkoop, C. Goulon-Ginet, M. Hagelstein and A. Rogalev, ‘Instrumentation developments for ESRF beamlines’, Physica B 199, 208–9 (1995). [8] Randall, K. J., G. Srajer, D. Mills and E. Gluskin, ‘SRI CAT Sector 41D – new capabilities for polarization dependent science and high heat load testing’, SRI CAT Newsletter 4(2), 2–5 (1998). [9] Burmeister, W. P., D. Bourgeois, R. Kahn, H. Belrhali, E. P. Mitchell, S. M. McSweeney and S. Wakatsuki, ‘Diamonds, a multilayer and a sagitally focusing crystal as optical elements on the ID14/Quadriga-3 beamline at ESRF’, SPIE Proceedings 3448, 188–96 (1998).

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[10] Mattenet, M., T. Schneider and G. Grubel, ‘The multicrystal monochromators of the TROIKA beamline at ESRF’, J. Synchrotron Rad. 5, 651–3 (1998). [11] Burmeister, W. P., Acta Crystallogr. D Biol. Crystallogr. 56, 328–41 (2000). [12] Ravelli, R. B. and S. M. McSweeney, Structure Fold Des. 8, 315–28 (2000). [13] Weik, M., R. B. Ravelli, G. Kryger, S. McSweeney, M. L. Raves, M. Harel, P. Gros, I. Silman, J. Kroon and J. L. Sussman, Proc. Natl. Acad. Sci. U.S.A. 97, 623–8 (2000). [14] Westbrook, E. M. and I. Naday, ‘Charge-coupled device-based area detectors’, Methods in Enzymol. 276, 244–68 (1997). [15] Yamashita, A., T. Nakatsu, T. Tomizaki, S. Wakatsuki and Kato, ‘Structure of tropinone reductase II complexed with NADP+ and pseudo-tropine at 1.9 Å: implication for stereospecific substrate binding and catalysis’, Biochemistry 38, 7630–7 (1999). [16] Djinovic-Carugo, K., P. Young, M. Gautel and M. Saraste, ‘Structure of the actinin rod: molecular basis for cross-linking of actin filaments’, Cell 98, 537–46 (1999). [17] Luger, K., A. W. Mäder, R. K. Richmond, D. F. Sargent and T. J. Richmond, ‘Crystal structure of the nucleosome core particle at 2.8 Å resolution’, Nature 389, 251–60 (1997). [18] Cingolani, G., C. Petosa, K. Weis and C. W. Muller, ‘Structure of importin-β bound to the IBB domain of importin-α’, Nature 399, 221–9 (1999). [19] Ybe, J. A., F. M. Brodsky, K. Hofmann, K. Lin, S.-H. Liu, L. Chen, T. N. Earnest, R. J. Fletterick and P. K. Hwang, ‘Clathrin self-assembly is mediated by a tandemly repeated superhelix’, Nature 399, 371–5 (1999). [20] Groves, M. R., N. Hanlon, P. Turowski, B. A. Hemmings and D. Barford, ‘The structure of the protein phosphatase 2A PR65/A subunit reveals the conformation of its 15 tandemly repeated HEAT motifs’, Cell 96, 99–110 (1999). [21] Iwata, S., J. W. Lee, K. Okada, J. K. Lee, M. Iwata, B. Rasmussen, T. A. Link, S. Ramaswamy and B. K. Jap. ‘Complete structure of the 11-subunit bovine mitochondrial cytochrome bc1 complex’, Science 281, 64–71 (1998). [22] Cusack, S., H. Belrhali, A. Bram, M. Burghammer, A. Perrakis and Christian Riekel, ‘Small is beautiful: protein micro-crystallography’, Nature Structural Biology 5, 634–7 (1998). [23] Perrakis, A., F. Cipriani, J. C. Castagna, L. Claustre, M. Burghammer, C. Riekel and S. Cusack, ‘Protein microcrystals and the design of a microdiffractometer: current experience and plans at EMBL and ESRF/ID13’, Acta Crystallogr. D Biol. Crystallography 55, 1765–70 (1999). [24] Pebay-Peyroula, E., G. Rummel, J. P. Rosenbusch and E. M. Landau, ‘X-ray structure of bacteriorhodopsin at 2.5 angstroms from microcrystals grown in lipidic cubic phases’, Science 277, 1676–81 (1997). [25] Belrhali, H., P. Nollert, A. Royant, C. Menzel, J. P. Rosenbusch, E. M. Landau and E. PebayPeyroula, ‘Protein, lipid, and water organization in bacteriorhodopsin crystals: a molecular view of the purple membrane at 1.9 Å resolution’, Structure with Folding and Design 7, 909–17 (1999). [26] Edman, K., P. Nollert, A. Royant, H. Belrhali, E. Pebay-Peyroula, J. Hajdu, R. Neutze and E. M. Landau, ‘High-resolution X-ray structure of an early intermediate in the bacteriorhodopsin photocycle’, Nature 401, 822–6 (1999). [27] Luecke, H., B. Schobert, H.-T. Richter, J.-P. Cartailler and J. K. Lanyi, ‘Structural changes in bacteriorhodopsin during ion transport at 2 angstrom resolution’, Science 286, 255–62 (1999). [28] Bourgeois, D., T. Ursby, M. Wulff, C. Pradervand, A. Legrand, W. Schildkamp, S. Labouré, V. Srajer, T. Y. Teng, M. Roth and K. Moffat, J. Synchrotron Rad. 3, 65–74 (1996). [29] Srajer, V., T. Teng, T. Ursby, C. Pradervand, Z. Ren, S. Adachi, W. Schildkamp, D. Bourgeois, M. Wulff and K. Moffat, ‘Photolysis of the carbon monoxide complex of myoglobin: nanosecond time-resolved crystallography’, Science 274, 1726–9 (1996). [30] Genick, U., G. Borgstahl, K. Ng, Z. Ren, C. Pradervand, P. Burke, V. Srajer, T. Teng, W. Schildkamp, D. McRee, K. Moffat and E. Getzoff, ‘Structure of a protein photocycle intermediate by millisecond time-resolved crystallography’, Science 275, 1471–5 (1997).

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[31] Perman, B., V. Srajer, Z. Ren, T.-Y. Teng, C. Pradervand, T. Ursby, F. Schotte, M. Wulff, R. Kort, K. Hellingwerf and K. Moffat, ‘Energy transduction on the nanosecond time scale: early structural events in a zanthopsin photocycle’, Science 279, 1946–50 (1998). [32] Cate, J. H., M. M. Yusupov, G. Zh. Yusupova, T. N. Earnest and H. F. Noller, ‘X-ray crystal structures of 70S ribosome functional complexes’, Science 285, 2095–2104 (1999). [33] Clemons, W. M. Jr., J. L. May, B. T. Wimberly, J. P. McCutcheon, M. S. Capel and V. Ramakrishnan, ‘Structure of a bacterial 30S ribosomal subunit at 5.5 Å resolution’, Nature 400, 833–40 (1999). [34] Ban, N., P. Nissen, J. Hansen, M. Capel, P. B. Moore and T. A. Steitz, ‘Placement of protein and RNA structures into a 5 Å-resolution map of the 50 S ribosomal subunit’, Nature 400, 841–7 (1999).

10 Medical applications – intravenous coronary angiography as an example W.-R. Dix

1

Introduction

At most of the synchrotron radiation centers worldwide a number of different medical applications of synchrotron radiation were begun or are planned. First, there is the large field of in vitro applications such as radiation cell biology, structural biology and designer drug design, fluorescence analysis of trace elements, and X-ray microscopy. The high flux and brightness, tunable energies, time structure and polarization of synchrotron radiation are the basis for the research in this fast growing field. Beginning in the late 1970s synchrotron radiation was also applied to diverse in vivo research and clinical problems, the second field of medical applications. There are several advantages of synchrotron radiation compared to conventional X-ray tubes. Several orders of magnitude more flux and a smooth, continuous spectrum are available. This high intensity and tunability allow monochromatic beams at virtually any energy of interest for medical methods. With monochromatic radiation, the beam hardening problem is eliminated since the energy spectrum does not change with passage through tissue – only the intensity decreases. Furthermore, the most suitable energy can be selected for a given procedure. In therapy, this results in more effective dose delivery. In imaging procedures for diagnostics greater image quality is available at the same radiation dose or at least the same quality with less dose. Besides this image and therapy optimization and the dose effectiveness further advantages of synchrotron radiation are the possibility of element separation and contrast enhancement by K-edge subtraction. In computed tomography (CT) the quasiparallel beams allow simple reconstruction algorithms. There are some disadvantages of synchrotron radiation, too. The planar beam results in constraints for two-dimensional (2D) imaging. But the main problems are the limited access to synchrotron radiation and the high facility costs. Dedicated beamlines for medical applications exist at several centers, for example, the National Synchrotron Light Source (NSLS) in Brookhaven, USA, the European Synchrotron Radiation Facility (ESRF) in Grenoble, France, the Sinchrotrone Trieste (ELETTRA) in Trieste, Italy, the Hamburger Synchrotronstrahlungslabor (HASYLAB) in Hamburg, Germany (only part time), the Super Photon ring-8 GeV (SPring8) in Harima Science City, Japan, and the Shanghai Synchrotron Radiation Facility (SSRF) in Shanghai, China (planned). In the field of therapy two methods are under development: microbeam radiation therapy and photon activation therapy. The first one is in an experimental stage with small animals and is performed at the NSLS [1] and at the ESRF [2]. The lesions are irradiated in a stereotactic fashion using synchrotron radiation at 50–150 keV collimated into microscopically thin

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parallel arrays of planar beams. Although endothelium is destroyed by absorbed doses in the direct paths of the microbeams it is later regenerated from similar cells in the minimally irradiated contiguous segments between the microbeams. Therefore, tissue necrosis is avoided except in the crossfired zone. The photon activation therapy is in a very early phase with investigation of cells and was first tested at the NSLS [3]. It is now developed at HASYLAB [4] too. The idea is to incorporate halogen atoms like iodine into the critical site (DNA) of malignant cells. After photoionization with monochromatic photons at an energy above the K-edge, significant damage of the cells due to Auger electrons is expected. More methods than in therapy are under development for image formation in diagnostics. They all use monochromatic X-rays and work with projection images as well as with reconstructed images from CT. Some of the systems are in bending magnet beamlines but most of the systems and the advanced ones are in wiggler beamlines. The CT methods are again manyfold and different physical effects are used for image formation: transmission monochromatic X-ray CT, dual photon absorptiometry CT, K-edge subtraction CT, phase contrast CT, diffraction enhanced imaging CT, fluorescent X-ray CT, Compton scattering X-ray CT, Thomson scattering X-ray CT and combinations of those different methods. Some applications are based on absorption contrast, similar to methods using conventional X-ray tubes but with monochromatic beams. The most advanced one is the intravenous coronary angiography program at the Photon Factory (PF) in Tsukuba, Japan, where the first four patients were investigated in 1996 [5]. Monochromatic X-rays above the iodine K-edge at 33 keV are used to image coronary arteries in projection sequences of very rapid, 2D exposures. The programs in Japan not only aim at intravenous injection of contrast agent but also at aortic [5] or superselective injection [6], respectively. In an experimental phase, with animals, are two CT applications with absorption contrast at the NSLS [7] and the PF [5] to examine the head. The monochromatic X-rays have an energy of 43 and 33 keV, respectively. For investigation of the neck, NSLS and PF plan to use monochromatic X-rays above the K-edge of iodine at 33 keV. In the case of the chest and abdomen, higher energies of X-rays are needed because of the high absorption of 33 keV photons. Therefore, NSLS proposed to work above the K-edge of gadolinium at 50 keV. For mammography, the first in vitro tests were performed to obtain absorption contrast images using 16–25 keV X-rays at NSLS and the Daresbury Laboratory (CLRC) in Warrington, England [8]. In image formation with absorption contrast the imaginary part, β, of the refractive index n = 1 − δ − iβ is the source of information within the images. On the other hand the real part δ allows one to calculate the phase shift caused by the sample. Phase contrast imaging shows a very high sensitivity to light elements which is 100–1000 times higher than conventional absorption. Therefore, phase contrast imaging could be an interesting tool for the identification of tumors. Phase contrast CT is developed at HASYLAB [9] and the PF [10]. A CT mammography system is planned at PF and good results from phase contrast projection images with a reasonable X-ray dose are expected, too. A new radiographic imaging modality called Diffraction Enhanced Imaging (DEI) has been developed at the NSLS [11]. An X-ray analyzer crystal as a scatter rejection optic produces images of thick absorbing objects that are almost completely free of scatter. This system is sensitive to refractive index effects within the object in addition to X-ray absorption and small-angle scattering by the object. Therefore, refractive index effects can be separated from absorption effects with a simple algorithm. In mammography phantoms this allows

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an enhancement of the apparent absorption up to a factor of 17 compared to conventional synchrotron radiation radiograms. The method is developed at different centers such as NSLS, ESRF, ELETTRA, CLRC and HASYLAB. The above mentioned examples are methods with single monochromatic X-rays. But synchrotron radiation also allows multiple-energy methods with monochromatic X-rays like dual-photon absorptiometry (DPA) or K-edge subtraction (KES) imaging. For DPA two images of the object with different energies (e.g. 50 and 80 keV) are used for the separation of low Z- and intermediate Z-elements images. CT methods with DPA for the investigation of the head and the brain are developed at the ESRF and the NSLS. The long term goal at the ESRF is to image neurologically significant elements and to identify concentration changes of endogene components. Also cerebral ischemia is of interest. At the NSLS, in situ imaging for patient orientation and positioning for subsequent radiotherapy treatment is an example of application of DPA-CT. While DPA-CT applications are in a very early phase of development, CT applications with KES are more advanced. For KES two images are taken with monochromatic X-rays, one at an energy below the K-edge and the second above the K-edge of the element of interest, for example, iodine. After logarithmic subtraction of these two images the contrast of that element is largely enhanced. At the ESRF this CT method is used at the K-edge of iodine and of gadolinium which are contained in contrast materials. First animal experiments were performed and the goal is to investigate tumors in the brain [2,12]. Examples are the measurement of blood volume and tissular perfusion for angiogenesis development of tumors or understanding the blood brain barrier rupture and leakage phenomena. At the NSLS the method is aiming at investigations of the head and neck especially the evolution of carotid artery plaques by the use of iodine. KES applications in projection geometry with line scan systems were the first medical applications of synchrotron radiation and, therefore, are the most well-known ones. In 1979, Rubenstein and Hofstadter proposed the use of this method for intravenous coronary angiography (ICA) [13]. Today the ICA programs certainly are the most advanced ones at the different synchrotron radiation centers. ICA is developed at SSRL/NSLS, HASYLAB, ESRF and at the storage ring VEPP of the Institute of Nuclear Physics in Novosibirsk, Russia [14]. While these systems are similar in principle and only differ in the X-ray optics and the type of detectors, at the PF a different system is developed. This iodine-filter system [15] involves a 2D camera and image intensifier as a detector, like all systems in Japan. In 1995, Rubenstein et al. proposed to use KES for bronchography [16]. In this application the patients inhale a gas mixture containing stable xenon. This allows the imaging of the respiratory air passage by KES at the K-edge of xenon. This method could provide the opportunity to image anatomic structure and pathologic processes that cannot be visualized by conventional X-ray based imaging methods. For example, early detection of lung cancer is an important application because the method allows the detection of tumors significantly smaller than with conventional methods. The method is developed at VEPP [17], ESRF and NSLS. Initial studies on three human volunteers have been carried out at the NSLS [18]. More patients than with the KES bronchography method and the intravenous coronary angiography program at the PF [5] have been examined with the ICA applications at SSRL/NSLS and HASYLAB. The early human studies were done at SSRL in 1986 [19]. After investigation of seven patients the system was relocated to NSLS where 21 additional patients have been imaged [20]. In 1990, human studies started at HASYLAB. Up to now 379 patients have been investigated in Hamburg, Germany [21]. Obviously the NIKOS system in Hamburg is the most advanced one in the pioneering medical application of synchrotron

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radiation. Therefore, this system and the underlying method will be described in more detail in the following sections.

2

Medical background of ICA

In most industrialized countries diseases of the heart and the blood circulation are the leading causes of death. For example, in Germany 48% of the cases of death are due to these diseases. Per year 260 000 myocardial infarctions occur in Germany, 100 000 of them are followed by death. In most of these cases the reason is a sudden occlusion at a pre-existent stenoses in the coronary arteries. The imaging modality routinely used for visualization of these stenoses is selective coronary angiography. In recent years, the number of these procedures has considerably increased [22]. In 1997, 485 440 investigations with this method were performed in Germany [23], about 30% of them were follow-up after bypass surgery or interventions like angioplasty, rotablation etc. For selective coronary angiography a catheter is introduced via the arterial system into the origin of the coronary artery of interest. Due to the arterial catheterization there are complications associated with this invasive approach. The rate of complications is of the order of 1.5% (0.5% severe ones) and the mortality is 0.1% [24]. A further reduction of risk appears difficult to achieve because of the invasive nature of the procedure. Therefore, efforts have been made to image coronary arteries by non-invasive or minimally invasive techniques. In the field of synchrotron radiation applications, KES in projection geometry, also known as dichromography, is the imaging method used for these investigations. At present ICA with dichromography appears to be the best minimally invasive imaging modality for coronary arteries as compared to other approaches, for example, electron beam tomography (EBT) [25] or magnetic resonance imaging (MRI) [26]. For ICA the contrast material is introduced not into the arteries, but into the veins, which reduces the risk remarkably.

3

The method

Image formation with KES is a special form of digital subtraction angiography (DSA) which allows the enhancement of very low contrasts. The digital subtraction is based on the logarithmic subtraction of two images at different energies. For ICA the two images with monochromatic X-rays just below (E1 ) and above (E2 ) the absorption K-edge (EK ) of the iodine containing contrast agent at ∼33.17 keV are obtained simultaneously. The change of the mass absorption coefficient between E1 and E2 for iodine (-µ/ρ)I = 28.9 cm2 /g is positive for ln(E2 /E1 ) and its absolute value is about 10 000 times higher than that from soft tissue (-µ/ρ)S = −0.0033 cm2 /g. This allows the visualization of low iodine contrast in small vessels down to a mass density of 1 mg/cm2 , corresponding to a coronary artery of 1 mm in diameter and a dilution of the contrast agent by a factor of 40. After intravenous injection the contrast agent is diluted by a factor 40–50 before it enters the coronary arteries because it has to pass the right heart, the pulmonary system, the left heart and the aorta. The difference in absorption in the two energy images of such a small artery is in the range of that for 10 cm of soft tissue but with different sign. These numbers are obtained for an energy separation of the two monochromatic X-rays of δE ≤ 300 eV. This energy separation demands a bandwidth of maximal ≤250 eV for each monochromatic X-ray beam. Due to the fact that in the dichromography method the two images for subtraction are taken simultaneously, it is possible to image fast moving structures. This is not the case for

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conventional DSA, where the first image is taken before contrast agent injection and the second after injection. This proved to be unsuitable for imaging of fast moving small vessels like coronary arteries because the images do not match. For intravenous injection a complication arises from the fact that not only the coronary arteries but also at least the left heart chambers and the aorta are filled simultaneously with the contrast material. This leads to a superposition of the small arteries on large iodinated structures. To overcome this problem the projection angles must be selected carefully. In this way most parts of the coronary arteries can be imaged without superposition. Only for the main stem and part of the circumflex coronary artery no suitable projections have been found. Here, special image processing algorithms must be applied to get images for the diagnosis. The idea for dichromography was first published by Jacobson in 1953 [27]. But all efforts to use the method by filtering the monochromatic X-ray beams out of the spectrum of conventional X-ray units failed. Monochromatic X-rays of sufficient intensity are only provided by wiggler beamlines at synchrotron radiation facilities. Therefore, the work on dichromography was only continued when synchrotron radiation laboratories became operational.

4

The NIKOS system

In 1981, the work on ICA started at HASYLAB. During the first years the setup was developed. Since 1986 the safety and imaging potential of the method were tested in animal studies and in 1990 the first patients were investigated. The system is a line scan system as all systems for dichromography except the one in Japan. In these systems the data for the images are taken line by line and afterwards the images are composed in the computer. In the NIKOS system (Nicht-invasive Koronarangiographie mit Synchrotronstrahlung) at HASYLAB one line is taken every 0.8 ms and the image is completed within 250 ms. Therefore, the system do not allow one to take sequences of images as fast as in conventional selective coronary angiography with X-ray tubes. But sequences are not possible for dichromography anyhow because of the relatively high skin entry radiation dose per image (see below). There are two advantages with line scan systems: 1 2

The fraction of background due to scattered radiation is much less than with a 2D-imaging system. They optimally match the geometry of synchrotron radiation sources.

Version IV of the NIKOS system was installed in 1996 and is described in this chapter (for more details see [28]). It consists of six main components: a wiggler beamline, a monochromator, a safety system, a scanning device, a detector and a computer system (Figure 10.1). The system was installed at the wiggler beamline W2 of the storage ring DORIS at DESY which runs at 4.5 GeV. It allows the investigation of patients with high intensity (up to 3 × 1011 photons/mm2 /s) quasimonochromatic X-ray beams (below 250 eV bandwidth) at 33 keV as required by the dichromography method with iodine containing contrast agent. 4.1

The beamline

In the beamline a 20-pole wiggler [29] with a length of 2.4 m is installed. The variable vacuum chamber [30] allows a minimal vertical magnetic gap of 30 mm with a maximum field strength of 1.26 T corresponding to a critical energy of Ec = 17 keV.

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Detector Scanning device

E1 E2

Computer system

Safety system E2 Two beam monochromator

E1 White synchrotron radiation

Wiggler HARWI

e– Storage ring DORIS

Figure 10.1 Schematic view of the NIKOS system.

Behind the wiggler a slit system reduces the white synchrotron beam horizontally to ±1.7 mrad which matches the 13 cm wide detector at a distance of 37.3 m from the wiggler. The vertical size of the beam in front of the monochromator varies from 1.2 to 2.5 mm. This size is adjusted by means of the slit system in order to always get the same skin entry radiation dose on the patient independent of the number of scans per injection and the beam current in DORIS. During the studies with patients the storage ring was operated with beam currents between 41 mA and 130 mA.

4.2

The monochromator

The monochromator [31] produces two beams of energies E1 and E2 and a bandwidth of ≤ 250 eV. For each of the two beams it contains a single Si (111) crystal, installed in Laue geometry. The Bragg angle for 33.17 keV photons is Z = 3.42◦ . Therefore, behind the monochromator the two beams rise about 6.84◦ away relative to the incident white synchrotron radiation beam. From Bragg’s law it follows that the higher harmonics are also reflected under the same angle but due to the selected crystal reflection in the NIKOS system 66.34 keV photons are reflected very weakly. The monochromatic beams only contain 0.3–0.6% of 99.51 keV photons depending on the vertical opening angle of the slit system. The crystals are bent to a radius of about 10 m in order to focus the beams inside the patient who is seated at a distance of 3.5 m from the monochromator. The two beams cross just in front of the patient. For the imaging conditions of NIKOS (vertical source size, distance to the source etc.), a vertical beam size of 2.5 mm in front of the monochromator corresponds to a beam size of 0.5 mm at the patient and 1.0 mm at the detector 2.2 m behind the patient. For that adjustment, the bandwidth of the two monochromatic beams is -E = 163 eV and the energy separation δE = 285 eV (center-to-center). At 54 mA current in DORIS a flux of 2.7 · 1011 photons/mm2 /s in the monochromatic beams is measured in front of the patients.

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For higher currents in the storage ring the vertical size of the beam and, therefore, the bandwidth is decreased correspondingly in order to ensure a constant photon flux. This flux corresponds to an upper limit of the skin entry radiation dose per scan of 50 mSv, and 220 mSv for the complete procedure which is about 50 % of the mean of the skin entry radiation dose to the patient during conventional selective coronary angiography. From the skin entry radiation dose the effective dose can be calculated which is the integral of the dose along the path through the patient’s body. For the different tissues along this path tissue weighting factors are used which correspond to the sensitivity of the tissue to radiation. The result of this calculation is about 0.5 mSv per scan for males. For females it is 1.0 mSv because the weighting factor for breast tissue is very high. 4.3

The safety system

The critical safety issue of the system relates to ionizing radiation. For a standard scan a skin entry radiation dose rate of 64 Sv/s was estimated. Therefore, an immediately reacting safety system is compulsory. The central part of the NIKOS safety system consists of three independent shutters which can switch off the monochromatic beams within less than 10 ms [32]. Because a possible change of the scanning speed remains the main potential reason for hazard during the exposure, the speed is permanently measured by two scanning encoders. Two of the fast shutters are connected to these encoders while the third one is closed and opened under computer control. Ionization chambers are used routinely for each scan to measure the expected radiation dose to the patients. The system can only be activated after approval by the physician based on the predicted dose. The dose is controlled with thermoluminescence dosimeters on the chest of the patients. 4.4

The scanning device

The scanning device is driven by a hydraulic system. It allows for vertical movements of the patient over a distance of 20 cm at a constant speed of 50 cm/s. Accordingly, one scan of the 12.5 × 12.5 cm2 large image is finished within 250 ms. This allows the physicians to select a certain phase of the heart cycle for the image (systole or diastole). Therefore, the scanning device is ECG-triggered. Acceleration and deceleration are performed over a distance of 10 cm each. During the movement of the scanning device the readout of the detector is controlled by a precise optical scale. Every 0.4 mm a trigger signal initiates the readout of one line at each energy. This determines the vertical pixel size in the images. The scanning device is equipped with a seat for the patient. The seat can be rotated around a horizontal and a lateral axis to choose different angular projections. Furthermore, vertical and horizontal adjustment relative to the monochromatic X-ray beams is possible. 4.5

The detector

As a detector [33,34] a specially designed fast low-noise, two-line ionization chamber is installed which simultaneously records the X-rays of the two monochromatic beams. The chambers share a common drift cathode. In each chamber a Frischgrid made of thin wires is installed between the cathode and the anodes. The distance between the cathode and the

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Frischgrid is 3 mm, thus defining the conversion volume. The voltage of the grid is selected such that electrons can pass while ions are shielded. Thus the Frischgrid allows a fast readout of the chamber without crosstalk from line to line. The horizontal pixel size of the system is given by the anode strips on a printed circuit inside the detector with a width of 0.3 mm and a pitch of 0.4 mm. The 336 copper anode strips per line with a length of 56 mm in the direction of the monochromatic beams are connected to the frontend electronics. The chamber is filled with a mixture of 90% Kr gas and 10% CO2 gas at 13 bar pressure. This results in a quantum efficiency of 92% for 33 keV photons. The ionization current of each strip is integrated and digitized with 20 bit resolution. Four different integration cycles (L) can be selected in order to optimally adjust the high dynamic range to the absorption in the patients which is related to their thickness. The dynamic range was measured to be 328 000 : 1 (L = 4). The high dynamic range is necessary because the diluted contrast agent must be visible in small arteries which are superposed on large iodinated structures. The electronics allows for a readout time for the two lines (672 pixels) of 170–226 µs. For the scans of the patients a readout time of 800 µs is used. Therefore, the 320 lines per scan are read within 250 ms.

4.6

The computer system

The data of the detector are transmitted to the computer via an optical fiber link. An Alphastation 400 4/233 is used for system control and data acquisition via a VME system, for image processing and presentation and data storage. The user interface is of very large importance. It enables the physicians to perform the sophisticated image processing work on their own, in a simple way.

5

Studies of patients

Since 1990 studies with patients have been conducted in six phases: Phase 1 proved the safety and feasibility of the imaging protocol. With 11 patients it was demonstrated that imaging of coronary arteries is possible and that radiation exposure is comparable to conventional coronary angiography. In phase 2 technical amendments were made according to the initial experience. The aim of this phase was to optimize projection angles for the imaging of the different regions of the coronary arteries without superposition [35]. Phase 3 was designed to improve image quality with the aid of further technical developments. In this phase 46 patients were investigated and the results appeared promising for a large scale study. In phases 4 and 5 another 60 patients were investigated in order to generate and evaluate the parameters for a large scale patient study. After investigation of these 136 patients in the pre- and pilot-studies, a large scale study, from June 1997 to 1998 was started, which included 243 patients. The study is designed to validate ICA in comparison with conventional selective coronary angiography. Therefore, every patient enrolled into the study has to have a selective coronary angiography within six months before or after intravenous coronary angiography. From these patients, 128 had previously had a balloon angioplasty or rotablation of a narrowed coronary artery, with a consecutive implantation of a stent to keep the vessel open.

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Unfortunately these stents tend to restenose in up to 20%. Fifty-two patients had only a balloon angioplasty with a restenosis rate higher than 20%; and, finally, 59 patients had prior coronary artery bypass surgery. In accordance with the study protocol, the indication for ICA is a follow-up after the interventions mentioned above. The protocol of the study was approved by the ethics committee at the Hamburg Medical Board. To enter the study, the patients have to give written informed consent. No complication occured related to the investigation. For the investigation, the contrast agent (370 mg iodine/ml) is injected into the brachial vein or in 7% of the cases into the superior vena cava. In all cases the injection device is basically a plastic tube of 10–25 cm in length and 1.8–2.1 mm in diameter (introducer) positioned in the patient’s arm vein. For the injection into the superior vena cava a tube extension is used (pig-tail catheter) to minimize contrast agent dilution. The latter technique is used for patients with an unusually low heart rate or poor pump function of the heart muscle for instance. After positioning the patient in an upright position in the scan chair the transit time of the contrast agent from the brachial vein to the heart is measured. For this purpose indocyanine green is injected into the brachial vein and the arrival time in the ear is determined densitometrically. This arrival time is practically identical with the transit time which varies from 6 to 25 s. After entering the coronary arteries the duration of their optimal opacification lasts only 1–2 s. Therefore, the transit time must be known exactly before the investigation is started. After injection of 5–10 ml of contrast agent a scout image is taken with a reduced skin entry radiation dose of about 10 mSv. This image allows to optimize the patient’s position with respect to the area of interest and the individual transit time. Thereafter, the diagnostic scans are taken using a bolus of 30– 40 ml of contrast agent with an injection rate of 15–18 ml/s. During the upward and downward movement of the scanning device the fast beam shutters are opened over 320 lines each. The images are taken ECG-triggered to obtain end-systolic or diastolic images. If an end-systolic image is taken, there is less superposition (the left ventricle is contracted) compared to end-diastolic images (the left ventricle is filled with blood). On the other hand, the heart moves slower in diastole and this could sometimes enhance the contrast. Routinely, two diagnostic runs are taken with nearly perpendicular projections. Each run consists of two (50 mSv per scan) to four (25 mSv per scan) scans at a time distance of about 2 s. In order to reduce background iodine contamination, intervals of 30 min are required between imaging runs. For each scan two energy images are obtained simultaneously, one for the X-ray energy below the K-edge of iodine and the second above the K-edge. These images are corrected for the offset from the detector, for fixed pattern noise due to individual efficiencies of the detector elements, fluctuations in beam intensity and irregularities of the scanning device. These corrections must be performed very carefully because the error must be less than 0.1%. The reason is that for a coronary artery of 1 mm diameter the difference between the two energy images is 3% of the signal only. Because the signal-to-noise ratio (SNR) in the subtracted image should be at least 5 and because the noise due to electronics etc. must be negligible compared to the statistical noise due to the detected photons, the condition given above must be fulfilled. Because the mass density cI for iodine must be determined the two energy images must be subtracted logarithmically [21]. In the existing software the physician can decide whether the resulting image is presented on a linear or a logarithmic scale. The decision depends on the region of interest but in most of the cases the linear presentation is preferred. Figure 10.2 shows an example of a subtraction image with linear scale.

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(b)

(c)

Figure 10.2 Intravenous angiogram of a 69-year-old female (66 kg) in LAO 30◦ projection: (a) energy image above the K-edge of iodine; (b) energy image below the K-edge; (c) subtraction image with linear scale. Visible are the ascending and descending aorta, pulmonary veins and the right coronary artery (arrows).

This simple and fast image processing algorithm is sufficient for the diagnosis in most of the cases. During the physician’s session for diagnostics the grey levels within the images are largely stretched or squeezed. Only if the superposition problem must be overcome, that is, mainly if the target vessel is the circumflex coronary artery (Cfx), subsequent sophisticated image processing algorithms must be started. Two techniques routinely used by physicians are the creation of “iodine images” or “unsharp masking images,” respectively [21].

6

Results

The results of 379 patients investigated so far demonstrate the feasibility and safety of ICA with synchrotron radiation and high diagnostic accuracy. Experience over the years showed that at least two scans are necessary, after injection of the contrast agent, to be able to

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distinguish between coronary arteries and pulmonary veins. This is possible with two scans because, due to the blood circulation, the contrast in the pulmonary veins is decreased at the same moment when it is increased in the coronary arteries. If the Cfx and the left anterior descending coronary artery (LAD) must be imaged, it is even better in most cases to perform 3 scans with correpondingly less skin entry radiation dose per scan. The first scan is used for diagnosis in the LAD and the second and third one for the Cfx. Right coronary arteries (RCA) were seen with acceptable quality in left anterior oblique (LAO) projections and the LAD in right anterior oblique (RAO) projections. Additional caudal or cranial (CC) angulation based on the scout scan improved the images of the individual patient. Vein bypass grafts and the internal thoracic artery (IMA) were reliably imaged with high diagnostic accuracy. Most problems were caused by the main stem and the Cfx due to superposition on the aorta or the left ventricle. However, the postprocedural image processing helped considerably to improve image quality. The progress in ICA has led to a possible alternative to the presently used selective coronary angiography. Due to the non-invasive nature of this technology the related risk to the patient can be considerably reduced. Dichromography does not aim to replace invasive coronary angiography but could find its place for follow-up of coronary interventions, like percutaneous transluminal coronary angioplasty or coronary artery bypass graft surgery. The acceptance of the method by the patients is very high. Further developments aim at an improvement of the image quality. The use of ICA in hospitals requires the development of compact, cost effective synchrotron radiation sources. At HASYLAB a design study for a 1.6 GeV storage ring is in progress. In Japan and USA similar studies have started.

7

Conclusion

The described NIKOS system at HASYLAB is currently the most advanced facility available for human studies with synchrotron radiation. There is a large amount of additional medical applications of synchrotron radiation. They are different in the underlying physical effects as well as in the goal of work. Most applications are in an initial state, only some of them are in the phase of animal or human studies. Work on ICA showed that the development of medical systems at synchrotron radiation sources is a long term one. It needs about 5 years for the tests of phantoms etc. In that phase, very often, bending magnet beamlines can be used. But in the next phase of 5 years for animal studies a wiggler beamline should become available. After the very first human studies it again takes about 5 years before the method is available for routine investigation of patients. This duration of 15 years is not reduced much if a method is transferred to another synchrotron radiation center. The reasons are the demands on the environment and safety, which is remarkably higher than for physics beamlines. Nevertheless, worldwide a number of exciting methods with synchrotron radiation are under development. A lot more work needs to be done to bring them to the level of human studies and later on to clinical applications. For the installation of the new methods in the clinical world the development of compact sources will be required. Furthermore, there is a lot of competition from advances in conventional imaging and the synchrotron radiation based methods must provide significant advantages over the conventional methods in order

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to be accepted by physicians. Today, on the other hand, methods like ICA described in this chapter cannot be realized without synchrotron radiation. So the future of medical applications of synchrotron radiation is not guaranteed. But there is no doubt that these applications will establish a “golden standard” which must be targeted by the conventional modalities.

Acknowledgments I would like to thank all colleagues of the DESY workshops, electronics group and machine group involved in the NIKOS project. The work on human studies has been conducted by a team of physicists, engineers and physicians, including: T. Dill, C. W. Hamm, M. Jung, W. Kupper, M. Lohmann, B. Reime and R. Ventura. The fruitful discussions with W. Thomlinson (ESRF) are gratefully acknowledged. The work on NIKOS is supported by the BMBF (Bundesministerium für Bildung, Forschung und Technologie) under the contract number 05350 GKA 9.

References [1] Laissue, J. A., G. Geiser, P. O. Spanne, F. A. Dilmanian, J.-O. Gebbers, M. Geiser, X.-Y. Wu, M. S. Makar, P. L. Micca, M. M. Nawrocky, D. D. Joel and D. N. Slatkin, “Neuropathology of ablation of rat gliosarcomas and contiguous brain tissues using a microplanar beam of synchrotronwiggler-generated X-rays,” Int. J. Cancer 78, 654 (1998). [2] Elleaume, H., A. M. Chavret, P. Berkvens, G. Berruyer, T. Brochard, Y. Dabin, M. C. Dominguez, A. Draperi, S. Fiedler, G. Goujon, G. Le Duc, M. Mattenet, C. Nemoz, M. Perez, M. Renier, C. Schulze, P. Spanne, P. Suortti, W. Thomlinson, F. Esteve, B. Bertrand and J. F. Le Bas, “Instrumentation of the ESRF medical imaging facility,” Nucl. Instrum. Meth. A428, 513 (1999). [3] Thomlinson, W., “Medical applications of synchrotron radiation at the National Synchrotron Light Source,” in Synchrotron Radiation in the Biosciences, 674, Chance, B. et al. (eds) (Clarendon Press, 1994). [4] Lawaczeck, R., “Bimodale Tumortherapien – Photonen- und Neutronen-Akti-vierung: MößbauerKernabsorptions-Therapie (MKT) und 157 Gd Neutronen-Einfang-Therapie (NET),” Klin. NeuroRad. 5, 39 (1995). [5] Takeda, T., Y. Itai, K. Hyodo, M. Ando, T. Akatsuka and C. Uyama, “Medical applications with synchrotron radiation in Japan,” J. Synchrotron Rad. 5, 326 (1998). [6] Mori, H., K. Hyodo, E. Tanaka, M. Uddin-Mohammed, A. Yamakawa, Y. Shinozaki, H. Nakazawa, Y. Tanaka, T. Sekka, Y. Iwata, S. Handa, K. Umetani, H. Ueki, T. Yokoyama, K. Tanioka, M. Kubota, H. Hosaka, N. Ishikawa and M. Ando, “Small-vessel radiography in situ with monochromatic synchrotron radiation,” Radiology 201, 173 (1996). [7] Dilmanian, F. A., X. Y. Wu, E. C. Parsons, B. Ren, J. Kress, T. M. Button, L. D. Chapman, J. A. Coderre, F. Giron, D. Greenberg, D. J. Krus, Z. Liang, S. Marcovici, M. J. Petersen, C. T. Roque, M. Shleifer, D. N. Slatkin, W. C. Thomlinson, K. Yamamoto and Z. Zhong, “Singleand dual-energy CT with monochromatic synchrotron X-rays,” Phys. Med. Biol. 42, 371 (1997). [8] Lewis, R. A., A. P. Hufton, C. J. Hall, W. I. Helby, E. Towns-Andrews, S. Slawson and C. R. M. Boggis, “Improvements in image quality and radiation dose in breast imaging,” Synchrotron Radiation News 12 (1), 7 (1999). [9] Beckmann, F., K. Heise, B. Kölsch, U. Bonse, M. F. Rajewsky, M. Bartscher and T. Biermann, “Three-dimensional imaging of nerve tissue by X-ray phase-contrast microtomography,” Biophy. J. 76, 98 (1999). [10] Momose, A., T. Takeda, Y. Itai, A. Yoneyama and K. Hirano, “Phase-contrast tomographic imaging using an X-ray interferometer,” J. Synchrotron Rad. 5, 309 (1998).

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[11] Thomlinson, W., D. Chapman, Z. Zhong, R. E. Johnston and D. Sayers, “Diffraction enhanced x-ray imaging,” in Medical Applications of Synchrotron Radiation, 72, Ando, M. and C. Uyama (eds) (Springer-Verlag, 1998). [12] Chavret, A. M., J. F. Le Bas, H. Elleaume, C. Schulze, P. Suortti and P. Spanne, “Medical applications of synchrotron radiation at the ESRF,” in Proceedings of the International School of Physics Enrico Fermi, Course CXXVIII, 355, Burattini, E. and A. Balerna (eds) (IOS Press, 1996). [13] Rubenstein, E., E. B. Hughes, L. E. Campbell, R. Hofstadter, R. L. Kirk, T. J. Krodicki, J. P. Stone, S. Wilson, H. D. Zeman, W. R. Brody, A. Macovski and A. C. Thompson, “Synchrotron radiation and its application to digital subtraction angiography,” SPIE 314, 42 (1981). [14] Barsukov, V. P., V. G. Cheskidov, E. N. Dementyev, I. P. Dolbyna, K. A. Kolesnikov, Y. M. Kolokolnikov, G. N. Kulipanov, S. G. Kurylo, A. S. Medvelko, N. A. Mezentsev, A. A. Nesterov, V. F. Pindyurin and M. A. Sheromov. “Digital subtraction angiography with synchrotron radiation in Russia,” in Synchrotron Radiation in the Biosciences, 646, Chance, B., J. Deisenhofer, S. Ebashi, D. T. Goddhead, J. R. Helliwell, H. E. Huxley, T. Iizuka, J. Kirz, T. Mitsui, E. Rubenstein, N. Sakabe, T. Sasaki, G. Schmahl, H. B. Stuhrmann, K. Wutrich and G. Zaccai (eds) (Oxford University Press, New York, 1994). [15] Umetami, K., T. Takeda, H. Ueki, K. Ueda, Y. Itai, I. Anno, T. Nakajima and M. Akisada, “Iodine-filter imaging system for subtraction angiography and its improvement by fluorescentscreen Havpicow detector,” in Medical Applications of Synchrotron Radiation, 99, Ando, M. and C. Uyama (eds) (Springer-Verlag, 1998). [16] Rubenstein, E., J. C. Giacomini, H. J. Gordon, J. A. Rubenstein and G. Brown, “Xenon k-edge dichromographic bronchography: synchrotron radiation based medical imaging,” Nucl. Instrum. Meth. A364, 360 (1995). [17] Kondratyev, V. I., G. N. Kulipanov, M. V. Kuzin, N. A. Mezentsev, S. I. Nesterov and V. F. Pinduyrin, “First tests on subtraction bronchography study at the angiography station of the VEPP-3 storage ring,” in Medical Applications of Synchrotron Radiation, 29, Ando, M. and C. Uyama (eds) (Springer-Verlag, 1998). [18] Giacomini, J., H. Gordon, R. O’Neil, A. Van Kessel, B. Cason, D. Chapman, W. Lavender, N. Gmür, R. Menk, W. Thomlinson, Z. Zhong and E. Rubenstein, “Bronchial imaging using xenon K-edge dichromography,” Nucl. Instrum. Meth. A406, 473 (1998). [19] Rubenstein, E., R. Hofstadter, H. D. Zeman, A. C. Thompson, J. N. Otis, G. S. Brown, J. Giacomini, H. J. Gordon, R. S. Kernoff, D. C. Harison and W. Thomlinson, “Transvenous coronary angiography in humans using synchrotron radiation,” Proc. Natl. Acad. Sci. USA 83, 9724 (1986). [20] Rubenstein, E., G. S. Brown, D. Chapman, R. F. Garrett, J. C. Giacomini, N. Gmür, H. J. Gordon, W. M. Lavender, J. Morrison, W. Thomlinson, A. C. Thompson and H. Zeman, “Synchrotron radiation coronary angiography in humans,” in Synchrotron Radiation in the Biosciences, 639, Chance, B. et al. (eds) (Clarendon Press, 1994). [21] Dill, T., W. -R. Dix, C. W. Hamm, M. Jung, W. Kupper, M. Lohmann, B. Reime and R. Ventura, “Intravenous coronary angiography with synchrotron radiation,” Eur. J. Phys. 19, 499 (1998). [22] Gleichmann, U., H. Mannebach and P. Lichtlen, “Bericht über Struktur und Leistungszahlen der Herzkatheterlabors in der Bundesrepublik Deutschland,” Z. Kardiol. 84, 953 (1995). [23] Bruckenberg, E., “Herzbericht, Krankenhausausschuß der Arbeitsgemeinschaft der obersten Landesgesundheitsbehörden der Länder (1997). [24] Johnson, L. W. and R. Krone, “Cardiac catheterization 1991: a report of the registry of the society of Cardiac Angiography and Interventions: I. Results and complications,” Cathet. Cardiovasc. Diagn. 28, 219 (1993). [25] Achenbach, S., W. Moshage and D. Ropers, “Value of electron beam computed tomography for the non-invasive detection of high grade coronary artery stenoses and occlusions,” NEJM 27, 1964 (1998).

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[26] Pennell, D. J., H. G. Bogren, J. Keegan, D. N. Firmin and S. R. Underwood, “Assesment of coronary artery stenoses by magnetic resonance imaging,” Heart 75, 127 (1996). [27] Jacobson, B. “Dichromatic absorption radiography. Dichromography,” Acta Radiologica 39, 437 (1953). [28] Dix, W.-R. “Intravenous coronary angiography with synchrotron radiation,” Prog. Biophys. Molec. Biol. 63, 159 (1995). [29] Graeff, W., L. Bittner, W. Brefeld, U. Hahn, G. Heintze, J. Heuer, J. Kouptsidis, J. Pflüger, M. Schwartz, E. W. Weiner and T. Wroblewski, “HARWI – a hard X-ray wiggler beam at DORIS,” Rev. Sci. Instrum. 60, 59 (1989). [30] Pupp, W. and H. Hartmann (eds), Vakuumtechnik: Grundlagen und Anwendungen, 353 (HansaVerlag, 1991). [31] Illing, G., J. Heuer, B. Reime, M. Lohmann, R. H. Menk, L.Schildwächter, W.-R. Dix and W. Graeff, “Double beam bent Laue monochromator for coronary angiography,” Rev. Sci. Instrum. 66 (2), 1379 (1995). [32] Makin, I. “Entwicklung und Test eines Strahlschnellverschlusses für Angiographie mit Synchrotronstrahlung,” Diploma thesis, Fachhochschule Hamburg (1989). [33] Menk, R. H., W. Thomlinson, N. Gmür, Z. Zhong, D. Chapman, F. Arfelli, W.-R. Dix, W. Graeff, M. Lohmann, G. Illing, L. Schildwächter, B. Reime, W. Kupper, C. Hamm, J. C. Giacomini, H. J. Gordon, E. Rubenstein, J. Dervan, H. J. Besch and A. H. Walenta, “The concept of spatial frequency depending DQE and its application to a comparison of two detectors used in transvenous coronary angiography,” Nucl. Instrum. Meth. A398, 351 (1997). [34] Lohmann, M., H. J. Besch, W.-R. Dix, O. Dünger, M. Jung, R. H. Menk, B. Reime and L. Schildwächter, “A high sensitive two-line detector with large dynamic range for intravenous coronary angiography,” Nucl. Instrum. Meth. A419, 276 (1998). [35] Kupper, W., W.-R. Dix, W. Graeff, P. Steiner, K. Engelke, C. C. Glüer and W. Bleifeld, “Projection angles for intravenous coronary angiography,” Ital. Phys. Soc. 10, 165 (1988).

11 Polarization modulation spectroscopy by polarizing undulator Hideo Onuki, Toru Yamada and Kazutoshi Yagi-Watanabe

1

Introduction

Many kinds of polarizing undulators and wigglers have been developed since Onuki and his group succeeded in generating polarized radiation of any ellipticity [1,2]. They can provide not only polarized radiation over a wide range of wavelength, but can modulate the polarization of the radiation between two orthogonal polarization states. A general description of the polarizing undulator and wiggler is given in Chapter 6 of Part 1. Even though polarization is a spectroscopic variable of radiation, as are intensity and wavelength, it is obvious that the use of arbitrarily variable polarized light will have an important impact on all aspects of structure determination on species and of electronic structure determination on solid samples ranging in size from atoms, to surfaces, to biological cells. In particular, the advent of polarizing undulators has made possible circular dichroism (CD) spectroscopy and magnetic circular dichroism (MCD) spectroscopy in the X-ray through the vacuum ultraviolet (VUV) and extreme ultraviolet (XUV) energy ranges. Since the CD or MCD signal is generally thought to be very small, these measurements require phase-sensitive detection (synchronized rectification) with the modulation frequency of the polarization. For this reason, polarization modulation spectroscopy using the polarizing undulator is understood to be a powerful tool for the CD and MCD measurements in the VUV, XUV and X-ray regions. In this chapter, CD and MCD measurements, and two-dimensional CD imaging are presented as typical examples of polarization modulation spectroscopy, which uses a polarizing undulator as a light source.

2

Circular dichroism measurement and two-dimensional CD imaging

Thus far, in ordinary CD spectroscopy, transparent type modulators, such as Pochels cells or a photoelastic modulator as a phase shifter were used together with conventional light sources such as xenon, mercury–xenon or mercury lamp to generate circularly polarized radiation. Since the first measurement of UV-CD by Johnson [3], a great deal of effort has been made to extend the range of measurement wavelength and to improve measurement sensitivity. Synchrotron radiation (SR) provides a highly bright light in the broad range of wavelengths, from the visible region to the X-ray region. Sutherland et al. have developed a VUV-CD measurement system to measure CD spectra of biomolecules by using a photoelastic modulator, and linearly polarized synchrotron radiation which is generated in the bending magnet of the electron storage ring [4].

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Recently, two groups independently made CD measurements using a polarizing undulator. One group is the ESRF group, which developed an X-ray linear and circular dichroism beamline employing two helical undulators (Helios-I and Helios-II) [5]. The other group is the Onuki group. In this chapter, we will describe developments in CD measurement and the CD imaging, and the results of the Onuki group [6–8]. A polarizing undulator installed in the compact electron storage ring NIJI-II in the ETL was used for CD measurement and CD imaging. NIJI-II is shown in Figure 11.1. The magnetic lattice in the NIJI-II is composed of four dipole magnets, four focusing quadrupole magnets, two defocusing quadrupole magnets, and two sextupole magnets. The polarizing undulator is located in the straight section between two dipole magnets, B2 and B3 , as shown in Figure 11.1. The undulator installed in the NIJI-II is a crossed and overlapped type (Onuki type) polarizing undulator [1,2]. The polarizing undulator consists of two pairs of permanent magnet arrays whose sinusoidal magnetic fields are orthogonal to each other. The state of the polarization of the undulator radiation can be switched between right- and left-handed circular polarization by changing the phase between the two magnetic fields. This switching is accomplished by moving one pair of magnet arrays reciprocally along the axis of the undulator with a motor-driven crank. This type of polarizing undulator is capable of producing brilliant, quasi-monochromatic, wavelength-tunable and polarized radiation of any ellipticity. The principal parameters of the polarizing undulator are summarized in Table 6.2 of Chapter 6, which contains details of the polarizing undulator. Figure 11.2 shows the power spectrum and the degree of circular polarization of the first harmonic radiation from the undulator. The full width at half-maximum (FWHM) of the power spectrum was theoretically expected to be about 7%. It was measured to be about 10% in the region between the visible and UV region because of the large size of the electron beam. The degree of circular polarization of radiation in the peak wavelength was almost Septum magnet

Electrons QF

RF cavity

QF B1

B4

Sr

QF⬘ Sx

QF⬘ Mirror chamber B3

Convex mirror

Kicker magnet

QF

B2

Polarizing undulator

Sx Q F

3700 mm

Schwarzschildtype mirror

Quartz window

6600 mm

Figure 11.1 Schematic view of the compact electron storage ring NIJI-II. B1 –B4 : Bending dipole magnets. QF , QF : Focusing and defocusing quadrupole magnets. SX : Sextupole magnets.

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Brightness (arb. units)

10 0.8 8 0.7

6 4

0.6 2 260

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300

320 340 360 Wavelength (nm)

380

Degree of circular polarization

0.9

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Figure 11.2 Spectrum of the first harmonic radiation from the polarizing undulator installed in the NIJI-II with E = 229 MeV, gap distance = 80 mm, and retardation phase α = π/2 (solid curve). The closed circles indicate the degree of circular polarization of the radiation. (Reproduced with permission from [7].)

100% within the measurement uncertainty (a few percent). In the microscope system, the sense of the rotation of polarization of undulator radiation was modulated at a frequency of 1.7 Hz to alternately generate the left- and right-handed circularly polarized radiations. The peak wavelength λn of the nth harmonic radiation from the polarizing undulator is written as follows: λn = λu /2nγ 2 (1 + K 2 + γ 2 θ 2 )

(1)

where λu is the period length of the magnetic fields, γ = E [MeV]/0.51, and K the field parameter = 0.093B0 [kG]×λu [cm]. E is the electron beam energy and B0 the peak value of the magnetic field. θ is the angle between the directions of observation and the undulator axis. The higher harmonic radiations are very weak on the undulator axis in the helical magnetic field configuration that generates the circularly polarized radiation. The peak wavelength of the first harmonic radiation can be changed by changing E or B0 , where B0 can be controlled by the gap distance of the undulator’s magnet arrays. Figure 11.3 shows relationships between the peak wavelength of the first harmonic radiation and the gap distance of magnet arrays at given electron energies. In the experiment, the electron beam energy was fixed at 218 MeV, and the scanning of the peak wavelength was performed by changing the gap distance from 80 to 110 mm. 2.1

Optics of microscopic imaging system

Figure 11.4 is a schematic diagram of the CD microscope using the polarizing undulator as a light source. The present microscopy is of the scanning microbeam type. A convex mirror (focal length (f.l.) = 94.1 mm) in the mirror chamber is positioned at 3700 mm from the center of the undulator. The distance between the convex mirror and Schwarzschild-type mirror (f.l. = 13.5 mm, N. A. = 0.38) is 6600 mm. This mirror system focuses the cross section of SR light source (equal to the electron beam cross section) in the undulator and forms

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Figure 11.3 Relationship between the gap distance of the undulator’s magnet arrays and the peak wavelength of the undulator radiation.

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Figure 11.4 Schematic diagram of the system for microscopic imaging of CD. Circularly polarized radiation from the undulator is reflected by a convex mirror and is focused on the sample in a 3D piezo-driving stage by a Schwarzschild-type mirror.

a microbeam on the sample surface; its reduction rate is theoretically about 5 × 10−5 . A knife edge was set at the focusing point to measure the microbeam profile of the undulator radiation. The transmitted undulator radiation outside the knife edge was measured while it was moved in the direction perpendicular to the optical axis and was differentiated with respect to the displacement of the knife edge. Figure 11.5 shows profiles of the microbeam focused by the convex mirror and the Schwarzschild-type mirror, which are axially symmetric. The sizes of

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–4

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Figure 11.5 Profiles of the microbeam of the undulator radiation focused with a convex mirror and a Schwarzschild-type mirror. The open and closed circles indicate the profiles at peak wavelength = 400 nm and at peak wavelength = 200 nm respectively.

the microbeams were 0.96 µm around the peak wavelength of 400 nm and 0.66 µm around 200 nm. The result in Figure 11.5, also indicates that the size of the microbeam depends on the wavelength of the undulator radiation. It is certain from this fact that the actual microbeam size is determined by the light diffraction and the reduced beam size of light source. It is easily understood from Eqn (1) that the peak wavelength of undulator radiation increases as the observation angle θ (in γ 2 θ 2 ) increases. The aperture of the primary mirror in the Schwarzschild-type mirror is 12 mm (the radius, 6 mm) whose value corresponds to θ = 0.05 mrad. The γ θ of the undulator radiation becomes 2.1 × 10−2 , where γ = E/0.51, and E = 218 [MeV]. In this case, the γ θ (2.1 × 10−2 ) is small enough to keep the peak wavelength and the FWHM of the spectrum of the undulator radiation almost constant and not to degrade the degree of circular polarization inside the aperture. The sample stage was scanned in three dimensions (3D) by piezoelectric translators. To monitor the intensity of the incident radiation, a photomultiplier was placed behind the quartz window. The radiation transmitted through the sample was detected with another photomultiplier. 2.2

Circular dichroism measurement

The present system has about 1% difference in intensity between the left- and right-handed circularly polarized radiations from the undulator. This degree of difference cannot be ignored in measuring CD signals smaller than 1%. To eliminate this influence to the measurement of CD spectra, an electrical servo system was introduced [7]. Instead of the Schwarzschild-type mirror and the 3D sample stage, shown in Figure 11.4, a diaphragm (diameter, 5 mm) and a quartz sample cell (path length, 1 cm) were placed behind the quartz window to measure the CD spectrum of d-10-camphorsulfonic acid, which is a standard samples for measuring a CD spectrum. Figure 11.6 shows the measured CD spectra of d-10-camphorsulfonic acid dissolved in water. The sample is well known because it is a standard sample for CD measurement,

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Figure 11.6 CD spectra of d-10-camphorsulfonic acid in water. The closed circles indicate the measured spectrum with a concentration of 6.13 mM. The dashed curve, which was published elsewhere [9], is shown as a reference CD spectrum. The solid curve indicates a calculated spectrum based on the standard spectrum (dashed curve) taking into account the spectral distribution of the incident radiation (by the convolution method).

and the CD spectrum (dashed curve) is taken from a paper published elsewhere [9]. The peak wavelength in the spectrum is at 290.5 nm and the FWHM is about 35 nm. Compared to the dashed curve, the spectrum measured by the present system (closed circles) shows a broadening in the FWHM and a slight blue-shift in the peak wavelength. The broadening in the FWHM is probably due to about 10% FWHM of the spectrum of the undulator radiation, and the blue-shift is caused by the asymmetric spectral shape of the undulator radiation [7]. Figure 11.7 shows the dilution effect of the CD of d-10-camphorsulfonic acid solution at the peak wavelength (285 nm). A linear region exists in the range of concentration from 0.4 to 10 mM. The standard deviation in the measurement uncertainty was about 1×10−4 . The molar ellipticity of d-10-camphorsulfonic acid in water is known to be 7260 [deg cm2 /decimole] [9]. Thus, our system has a sensitivity of about 0.1 [deg] for the CD measurement. The CD spectra of d-10-camphorsulfonic acid in water were also measured at the focusing point on the 3D stage by the system in Figure 11.4. In this case, all optical components, especially the Schwarzschild-type mirror, were well aligned. The observed CD spectrum (open circles) is shown in Figure 11.8 and agrees relatively well with that observed without the Schwarzschild mirror (closed circles). This agreement is due to the axial symmetry of the optical mirror system used (Figure 11.4). 2.3

Microscopic imaging of circular dichroism [8]

Dried films of d-10-camphorsulfonic acid solution were employed to study microscopic imaging of circular dichroism. The solution was dropped on a copper grid made for electron microscopy (Mesh 1000, pitch = 25 µm, bar = 7 µm, Nissin EM, Japan) and was dried in the air at room temperature. Needle-shaped crystals of several thickness of films were made

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Circular dichroism at 285 nm

10–2

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100 101 Concentration (mM)

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Figure 11.7 Changes of CD intensity at a wavelength of 285 nm by diluting a solution of d-10camphorsulfonic acid. A quartz sample cell with a path length of 1 cm was used.

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Figure 11.8 CD spectra of d-10-camphorsulfonic acid in water measured with (open circles) and without (closed circles) use of the Schwarzschild-type mirror.

on the copper grid. The sample arrangement is schematically drawn in Figure 11.10 (a) based on the picture taken by an ordinary microscope. Figure 11.9 shows the CD spectra of the filmlike regions of the sample. In the thickest case, the thickness of the d-10-camphorsulfonic acid was about 5 µm. The spectra show their single peaks around 275 nm, which are shorter wavelengths than those of sample solution. Also, each CD spectrum with different thickness in different regions shows various spectral baselines. The offset of the baseline of the CD spectrum is caused by a slight tilt of the sample surface from the plane perpendicular to

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Figure 11.9 Spectral shapes and base lines of dried films of d-10-camphorsulfonic acid with different thickness. (•), (+), and (
) indicate thick (about 5 µm), medium, and thin samples, respectively.

the incident radiation. Where there is misalignment, the light scattering from the sample surface increases the baseline offset. The thick film composed of large microcrystals has a large degree of surface roughness, which increases the light scattering. To reduce these influences on the CD imaging, a subtraction process between the two images obtained at different wavelengths was done. Each image was obtained by scanning its sample surface within 45 × 45 µm range of the scan stage. The width of one pixel width was 1.0 µm at the sample stage. The data acquisition time per pixel was 6.0 s. Images of CD of dried d-10-camphorsulfonic acid films were taken at two wavelengths, 275 and 230 nm. The 275 nm value is the peak wavelength of the CD spectrum whereas 230 nm is far from the peak position. The final CD image was obtained by subtracting the image at 230 nm from that at 275 nm, as shown in Figure 11.10(b). Two pieces of needle shaped crystals of d-10-camphorsulfonic acid which have widths of several to ten micrometers were visualized in the lower-left part of the subtraction image. The shape of the longer piece has some discontinuities with a width of several micrometers, which coincides with the width of the copper grid, 7 µm. This was caused by interruption of the incident radiation by the grid, where the CD signal of the sample vanished. A CD sensitivity of 0.01% (0.1 [deg]) was achieved. The sensitivity of the CD by using the system is sufficient to image objects having relatively high optical activity. However, application to other ordinary biological materials will require a higher level of sensitivity, which could be achieved by increasing the modulation frequency of the polarization. Compared to the transparent type of polarization modulators, the present type of CD spectroscopy that uses a polarizing undulator offers the advantage that a wavelength suitable to the CD spectral measurement can be chosen or, in our case, imaging between the visible region and XUV region. Generally speaking, we can even extend the wavelength even to the X-ray region by increasing the electron energy in the electron storage ring.

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(b)

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Figure 11.10 (a) Schematic view of sample arrangement on a copper grid (dashed areas indicate the sample). (b) CD image of dried d-10-camphorsulfonic acid films on a copper grid by subtracting the image at 230 nm from the image at 275 nm (white areas indicate strong CD signals of the sample). (Reproduced with permission from [8].)

3

Magnetic circular dichroism measurement

Magnetic circular dichroism (MCD) spectroscopy is a powerful technique for obtaining information about the electronic structures of magnetic materials. Iwamoto and Onaka have extended MCD measurement to the VUV region using the linearly polarized synchrotron radiation (SR) from the bending magnet, and a stress modulator made of fluorite [10]. However, measurement of the MCD spectrum in the wavelength region shorter than 155 nm is difficult with this method. Magnetic-field modulation, which is applied to samples, has recently been used to observe the MCD spectrum instead of the polarization modulation [11–14]. However, strictly speaking, polarization modulation spectroscopy must be applied to MCD measurements to achieve quantitatively reliable MCD data. 3.1

System for MCD measurement

Yagi et al. developed a method of polarization modulation spectroscopy in the XUV region for MCD measurement using a polarizing undulator, and they observed the MCD spectrum of non-magnetic Y3+ core transition in a ferrimagnetic YIG lattice [15,16]. Figure 11.11 shows the schematic diagram of the beamline that they designed and constructed in the storage ring TERAS in the Electrotechnical Laboratory. The crossed and overlapped undulator, which is an Onuki-type undulator with four periods of magnetic fields, was used as the polarizing undulator [2,3]. The main parameters of the undulator are indicated together with a 15-period polarizing undulator in Chapter 6, Section 2.1.2. The SR from the polarizing undulator is reflected and its image is focused at the entrance slit of the monochromator by a toroidal mirror

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Figure 11.11 Schematic diagram of polarization modulation spectroscopy for the MCD using a polarizing undulator. (Reproduced with permission from [16].)

which is located 9.5 m away from the undulator. A set of four-blade slits is placed between the polarizing undulator and the toroidal mirror to eliminate stray radiation. The grazing incidence toroidal grating monochromator has a Pt-coated toroidal holographic grating, which is used with a deviation angle of 142◦ (Jobin Yvon LHT 30). The wavelength range covers 8–120 eV by two different gratings with blazed wavelengths at 15 and 30 nm. The gratings have 550 grooves per mm. The dispersion is 2 nm per mm. The degree of circular polarization Pc of the radiation from the polarizing undulator depends on the phase difference between the two crossed magnetic fields. It can be varied in the range of −0.9 < Pc < 0.9 and modulated with frequencies up to 3 Hz [17]. The MCD signal is defined as the relative difference between the reflectances of two circularly polarized photons, -R/R = (R + − R − )/((R + + R − )/2), where R − and R + are the reflectances of photons of parallel and anti-parallel spins against the magnetization of the sample, respectively. The right- and left-handed circularly polarized lights were irradiated alternately and normally onto a sample surface. The modulation frequency was 3 Hz. The reflected light was detected with a photomultiplier. A fractional difference in reflectivity, -R/R, was amplified by a lock-in amplifier. The sample was attached directly to the Nd–Co–B permanent magnet block that produced a field of 0.4 T at the position of the sample. 3.2

Simultaneous scannings of the magnet gap in the undulator and the wavelength of the monochromator

The radiation from the polarizing undulator is quasi-monochromatic. The wavelength of the nth harmonic is given by Eqn (1) in Section 2. The photon energy is changed by varying the electron beam energy E and by changing the gap of its magnet arrays. Figure 11.12 shows the photon intensity spectra from the monochromator by varying the magnet gap from 64 to 80 mm. In this measurement, the magnetic field was a right-handed helix and the electron beam energy was 630 MeV. The grating of the monochromator was used with a blazed wavelength at 30 nm. The peak photon energy of the undulator radiation shifts to the high-energy side as the magnet gap increases. The intensity at the first harmonic peak is slightly reduced when the magnet gap is changed from 64 to 80 mm. A second harmonic is also observed in these spectra. The second harmonic radiation is caused by the finite

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emittance of the stored electron beam and the large aperture of the four-blade slits. However, the calculated degree of circular polarization exhibited high value (>90%) in the region of the second harmonic radiation [16]. This indicates that the second harmonic radiation is useful for polarization modulation spectroscopy such as MCD studies. Figure 11.13 illustrates the relationship between the undulator magnet gap and the output photon energy from the monochromator. It shows a plot of peak photon energy of both 64

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Figure 11.12 Photon intensity spectra from the monochromator made by scanning the magnet gap from 64 to 80 mm. (Reproduced with permission from [16].) 90 750 MeV 2nd harmonic

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Figure 11.13 Experimentally determined relationships between the undulator magnet gap and the output photon energy from the monochrometor. (Reproduced with permission from [16].)

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the first and second harmonic components in the electron beam energies E = 630 and 750 MeV. To measure the MCD in the wide photon energies, the undulator magnet gap and the monochromator wavelength were simultaneously scanned by computer control. Because the on-axis magnetic field produced by the undulator is about 0.15 T at most, tune shifts and dynamic aperture reduction were negligible. Any effect on storage ring performance due to the undulator, for example, due to movement of the electron beam, could not be detected. Figure 11.14 shows the MCD spectrum of Y3 Fe5 O12 (YIG) which is obtained by the simultaneous scanning of the polarizing undulator magnet gap and monochromator in the region 22– 40 eV. In this energy region, the wavelength bandwidth of the monochromator was set at 1 nm. The spectrum in the energy region between 22 and 35 eV (filled circles) was measured by scanning the magnet gap from 60 to 80 mm with E = 630 MeV; the spectrum between 30 and 40 eV (open circles) was measured by scanning the magnet gap 9

∆R /R (×10–3)

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Figure 11.14 MCD spectrum of Y3 Fe5 O12 (YIG) obtained by the simultaneous scanning of the polarizing undulator magnet gap and the monochromator wavelength in the region between 22 and 40 eV. (Reproduced with permission from [16].)

∆R /R (×10–2)

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Figure 11.15 MCD spectrum in the energy region between 22 and 60 eV. The spectrum in the region between 50 and 60 eV was measured with the second harmonic of the undulator radiation by scanning the magnet gap from 64 to 70 mm with E = 630 MeV. (Reproduced with permission from [16].)

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from 60 to 74 mm with E = 750 MeV. These MCD features are due to the inner core transitions of Y3+ from the 4p6 1 S0 to 4p5 (4d + 5s). Strong MCD features were observed, even though the ground state of the Y3+ ion has J = 0. An extension of measurement to a higher photon energy region was achieved with the second harmonic of the undulator radiation. Figure 11.15 shows the MCD spectrum in the energy region between 22 and 60 eV. For measurement of the spectrum in the region between 50 and 60 eV, the second harmonic of the undulator radiation was used and the magnet gap was scanned from 64 to 70 mm with E = 630 MeV. The MCD signals were clearly observed at the M2,3 edges in the Fe ion. The MCD measurement mentioned above demonstrated that even a four-period short undulator could produce extremely bright radiation and a high degree of polarization. The first and second harmonic radiations can be changed in the VUV region by scanning the gap of the undulator magnet and the monochromator wavelength. Polarization modulation spectroscopy for the MCD was established and the MCD of YIG between 22 and 60 eV was measured successfully. Detection of (R + − R − )/((R + + R − )/2) as small as 0.1% was demonstrated.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

Onuki, H., Nucl. Instrum. Meth. A246, 94 (1986). Onuki, H., N. Saito and T. Saito, Appl. Phys. Lett. 52, 173 (1988). Johnson, W. C., Jr., Rev. Sci. Instrum. 42, 1283 (1971). Sutherland, J. C., E. J. Desmond and P. Z. Takacs, Nucl. Instrum. Meth. 172, 195 (1980). Yamada, T., M. Yuri, H. Onuki and S. Ishizaka, Rev. Sci. Instrum. 66, 1493 (1995). Goulon, J., A. Rogalev, C. Gauthier, C. Goulon-Ginet, S. Paste, R. Signorato, C. Neumann, L. Varga and C. Malgrange, J. Synchrotron Rad. 5, 232 (1998). Yamada, T., H. Onuki, M. Yuri and S. Ishizaka, J. Electron Spectros. Rela. Phenom. 80, 501 (1996). Yamada, T., H. Onuki, M. Yuri and S. Ishizaka, Jpn. J. Appl. Phys. 39, 310 (2000). Cassim, J. Y. and J. T. Yang, Biochem. 8, 1947 (1969). Iwamoto, H. and R. Onaka, J. Phys. Soc. Jpn. 52, 3992 (1983). Chen, C. T., F. Sette, Y. Ma and S. Modesti, Phys. Rev. B 42, 7262 (1990). Koide, T., T. Shidara, H. Fukutani, K. Yamaguchi, A. Fujimori and S. Kimura, Phys. Rev. B 44, 4697 (1991). Schütz, G., W. Wagner, W. Wilhelm and P. Kienle, Phys. Rev. Lett. 58, 737 (1987). Schütz, G., M. Knülle, R. Wienke, W. Wilhelm, W. Wagner, P. Kienle and R. Frahm, Z. Phys. B: Condens. Matter 73, 67 (1998). Yagi, K., M. Yuri and H. Onuki, J. Phys. Soc. Jpn. 63, 3941 (1994). Yagi, K., M. Yuri and H. Onuki, Rev. Sci. Instrum. 66, 1592 (1995). Yagi, K., H. Onuki, S. Sugiyama and T. Yamazaki, Rev. Sci. Instrum. 63, 396 (1992).

12 Solid state physics Tsuneaki Miyahara

This chapter deals with the basic physics included in soft X-ray spectroscopy, putting some emphasis on future opportunities using high brightness undulator radiation. We would rather not show a particular example confined to a particular material but try to show some typical examples indicating the general advantages of undulator radiation.

1 1.1

Basic physics underlying absorption, photoemission and fluorescence spectroscopy in the soft X-ray region Sudden approximation and non-stationary states

If we know the exact ground state and excited states of a material, we can in principle calculate the photoabsorption spectrum using the Fermi golden rule. In the time domain picture, the probability amplitude of occupation of the ground state Cg (t) gradually decreases from unity to zero, while that of the excited states Ce (t) gradually increases from zero to unity with the probability conservation |Cg (t)|2 + |Ce (t)|2 = 1. Though the change of these probabilities is very slow with the so-called Rabi frequency, superposition of the ground and the excited state generally causes a non-stationary motion with the frequency of the incident light. This non-stationary motion is largest when contribution of the superposition is maximum, namely, both the above amplitudes are (1/2)1/2 . This is an important aspect of any optical excitation. Unfortunately, however, it is generally very difficult to describe exactly the excited states and even the ground state of a many-electron system. Therefore, the “sudden approximation” is often employed, where the system is divided into two parts. The first part A in Figure 12.1 is an electron which accepts the most part of the incident photon energy and is excited into some state while the second part B is the residual system. In the sudden approximation the excitation of A is assumed to happen very quickly within the period of the non-stationary motion corresponding to the inverse of the frequency of the incident light, because of a large photon energy in the soft X-ray region. In fact there arise various non-stationary motions following the above quick motion, depending on various interactions among electrons and lattices. It should be noted that once the system A is excited the state remains for long time because the Rabi frequency is very small. Therefore, even after this single electron excitation the total system A + B continues to move in a complicated way. The important thing is that the time constant of this motion is much slower than that of A and this motion corresponds to a non-stationary state that could be described as motions of a many-electron wave packet,

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which may generally include spin motions. In quantum mechanics this non-stationary motion is described by some superposition of many different energy eigenstates. In general any non-stationary motion is described as superposition of different energy eigenstates. For example, a single photon state with energy
ω/(2π) does not give any observable oscillation as an electromagnetic wave packet motion. However, superposition of a zero photon state and a one photon state results in an observable oscillation with frequency ω. The reason why the motion of A + B is slow can be explained as follows. In general any motion of a wave packet can be constructed with superposition of states with different energies, because, for example, superposition of the time evolution factor exp(iω0 t) and exp(i(ω0 + ω)t) gives the wave packet motion of cos(ωt) when we take the square of the absolute value of the wave function. The situation is illustrated in Figure 12.1. Here
ω of A is very large almost corresponding to the incident photon energy, while those of A + B in the excited state are small and correspond to various interaction energies among electrons and lattices, which are at most of the order of several electron volts. In the time domain picture this means that the wave packet motion of system A is much faster than the wave packet motion of system A + B after the excitation. Generally the motion of a wave packet is not periodic and may even have a component with infinitely large time constant. The exception is a simple harmonic oscillator, where the motion of the wave packet repeats periodically. The wave packet of a harmonic oscillator can be described by superposition of different quantum numbers n. One typical case of a harmonic oscillator is the Poissonian superposition of n’s, giving a well-known “coherent” motion with frequency ω. This motion, illustrated in Figure 12.2(a), approximately gives a many-line spectrum with a Gaussian envelope in the energy domain. A more general squeezed-coherent state is also possible as shown in Figure 12.2(b), where the width of the wave packet is modulated with frequency 2ω. A harmonic oscillator with just a single quantum number is a stationary eigenstate, giving a constant value of the square of the wave function. This gives a delta-function-like line spectrum. When the oscillator has some damping factor, it generally gives a Lorentzian spectrum in the energy domain. It is the process of damping that makes the state non-stationary.

B Non-stationary state with slow wave packet motion

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h /(2) A

Figure 12.1 Sudden approximation: The wave packet motion of system A is much faster than the wave packet motion of system B.

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(b) Harmonic squeezed-coherent motion Harmonic non-stationary motion

Gaussian wave packet

Figure 12.2 (a) Coherent wave packet motion of a harmonic oscillator: The Gaussian wave packet is the same as the modulus square of the wave function of the zero-point motion of the oscillator. (b) Non-classical squeezed coherent state: The width of the Gaussian wave packet oscillates with frequency twice as large as the frequency of the oscillator. This is an example of a non-stationary state which does not have a classical counterpart.

Now the possible condition of breakdown of the sudden approximation is clear. If there are no electrons which can absorb the most part of the incident photon energy and a large number of electrons are excited simultaneously with each electron absorbing only a small part of the incident energy, the sudden approximation breaks down. Fortunately, however, such a case is very rare because in general a single electron excitation has a large transition probability, though it is still much smaller than unity in the absolute sense. The simultaneous many electron excitation usually has a very small transition probability because this corresponds to the product of probabilities smaller than unity. Therefore, the sudden approximation is a good approximation in most types of spectroscopy in the soft X-ray region. In addition, we mention the future possibility of breakdown of the sudden approximation. If the incident light is very intense and the corresponding Rabi frequency becomes comparable to the non-stationary motion of the system B or A + B, then the approximation breaks down. We might have to worry about this situation when the Bose degeneracy of the undulator radiation exceeds 1000.

1.2

Spectral profiles

In all of the three types of spectroscopy we should remember the law of energy conservation. For example, in absorption spectroscopy, the intensity of the absorption at a particular photon energy is proportional to the square of the weight of a particular state that is included in the wave packet motion of A + B and satisfies the total energy conservation. In photoemission spectroscopy, for a given kinetic energy of a photoelectron (system A), the intensity (counting

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h

Figure 12.3 A simplest spectrum that causes a non-stationary state: The frequency of the motion is hν. There still remains a stationary motion corresponding to the difference in the intensities of the two components.

rates) corresponds to the square of the weight of a particular state that is included in the non-stationary motion of system B and satisfies the total energy conservation. Figure 12.3 shows the simplest model case with two conspicuous absorption structures. Here, we have spectral intensities I1 and I2 with the energy separation hν. This figure suggests that after the sudden excitation the system has non-stationary motion with the frequency hν and that the system also includes a stationary state which is proportional to the difference between I1 and I2 . Of course there may be cases where there are more than two different (orthogonal) states which satisfy the energy conservation. In this case, two contributions to the spectrum should be added incoherently, which means that we should take the sum of many squares of weights. This situation is sometimes expressed in the notion that spectral contributions to different final states should be added incoherently. In some cases, however, we may approximately divide the wave packet motion of system A +B (absorption) or the motion of system B (photoemission) into two parts. This is really an approximation because exactly speaking separation of a single coherent motion is impossible. Actually, however, we sometimes meet the situation where one group of electrons moves very differently from the other group of electrons. These two motions are, for example, localized motion and itinerant motion, or rapid motion and slow motion, or motion of the center of gravity and the motion relative to the center of gravity, and so on. The separation is carried out in such a way that different motions should include different “modes” of motion. Therefore, for example, separation of the electron system from the lattice system is possible if the latter motion is much slower than the former. Separation of the electron system into two groups of electrons is also possible if the corresponding two motions could be regarded as “different” modes. It is this type of separation that provides a physical ground for the sudden approximation. Though at first sight this division seems only for convenience of calculations there is a physical reality corresponding to such division as will be described later, which is often called “interference effect” and particularly “resonant photoemission” in photoemission spectroscopy. It should be noted that even though the motion is divided into several parts the corresponding weights of components of the same state included in the different parts are coherent, meaning that the weights of the common state included in the different parts should be added coherently. In other words, different quantum mechanical amplitudes to give the same final states should be added coherently. Hence comes the terminology “intereference effect” such as “Fano effect”.

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(c)

Figure 12.4 Interference of two different wave packet motions reaching an identical final state: (a) First a core electron is excited into a valence state; (b) then the excited electron fills the core hole, simultaneously exciting another valence electron; (c) a direct excitation of a valence electron could make the same final state. These two processes (b) and (c) give an interference effect.

Figure 12.4 shows the case of resonant photoemission. First a core electron is excited into a localized valence state (Figure 12.4(a)). Next the excited electron fills the hole simultaneously exciting another valence electron into a continuum state (Figure 12.4(b)). The other possibility is that a valence electron is excited directly into the continuum state (Figure 12.4(c)). Either of the two processes give the same final states as the valence electron is directly excited to the continuum state. Though both motions included in these processes are very rapid and cannot be included in the slow motion mentioned before, the weights of a particular state included in the associated slow motions included in the two processes should be added coherently. This is one of the important aspects of soft X-ray spectroscopy and is a great advantage because the coherent sum could either be constructive or destructive. In fact, when we subtract the “off-resonance” spectrum from the “on-resonance” spectrum we get information on the localized electronic states because the matrix element diagramatically shown in Figure 12.5 generally has a large value only at the local resonance condition. The situation in fluorescence spectroscopy is a little more complicated, but could be interpreted in a similar way. Though the sudden approximation appears twice, first in the photoabsorption process and, second, in the emission process, the energy conservation should be strictly remembered and the energy difference between the incident photon and the emitted photon is observed as the “Raman shift”. However, the wave packet motion of electrons and lattices still has many degrees of freedom and generally shows a very complicated behavior. Therefore, we usually need some simplification to calculate the spectrum. First, we notice that the final state has an energy higher than the ground state by the Raman shift. We further assume that the wave packet motion consisting of many energy eigenstates

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Figure 12.5 Diagram of the process (b) of Figure 12.4 and its exchange process: The right-hand side corresponds to the process where the excited electron is re-excited with another valence electron which fills the core hole.

of the system A + B, which include states with different energies and symmetries, could be divided into two parts. The first is the motion after the absorption of a photon but before the emission of a photon, and the second is the motion after the emission of a photon. The time with which the first motion continues apparently depends on the time difference between absorption and emission. The important thing is that we should add coherently contributions by different time durations to the same final states. This process is always possible because any wave packet motion which is terminated suddenly, damped gradually, or has a strange time dependence, can be Fourier decomposed with the energy eigenstates of the excited states. As will be shown later this is the physical reason why the formula of the emission spectra includes the coherent sum of many (or infinitely large number of ) intermediate states. Here, we consider the case where the distribution of the eigenenergies is not uniform. When there exists a state, isolated energetically from other states, that satisfies energy conservation, the weight with which this state is included in the motion could be remarkable if the overlap integral has an appreciable nonvanishing value. This weight conforms to a conspicuous peak in the emission spectrum. This type of peak is often called the discrete excitation causing the Raman shift.

2 2.1

Fluorescence spectroscopy Technical considerations

In all the spectroscopic studies the most important figure is the number of photons impinging on the surface of a sample. Because of the technical limitation of current fluorescence spectroscopy the apparent high brightness of undulator radiation does not always mean it is better than bending magnet radiation, especially in the incident photon energy region lower than 100 eV. The specific technical situation of fluorescence spectroscopy in the soft X-ray region is as follows: (1) The typical energy resolution of the second monochromator is about 0.1 eV, which is mainly limited by the spatial resolution of a two-dimensional photon detector, such as microchannel plate. This means that the resolution of the first (beamline) monochromator should be better than the above value. This requirement is not very severe in

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Sample

Diffraction grating

Incident light

Figure 12.6 The shape of image focused on the sample: The image could be of linear shape which is perpendicular to the direction of a diffraction grating of a monochromator.

the lower photon energy region. (2) Considering the dispersion type of a second grating monochromator, a linear shape of the focused light on the sample is allowed if it is perpendicular to the direction of dispersion and the size of the sample is sufficiently large, as is shown in Figure 12.6. This would give some chance for bending magnet radiation to be utilized for fluorescence spectroscopy in the lower photon energy region only if the vertical emittance of the stored beam is as small as the diffraction limit of the incident photons concerned. However, when we go to a higher photon energy region, undulator radiation has a great advantage over bending magnet radiation. The figure of merit in this photon energy region is roughly proportional to ω2 for the following reasons, where
ω/(2π) is the photon energy:

1

2

3

For a given width of the entrance slit of a beamline monochromator, the vertical divergence angle of the incident beam is proportional to ω−1 when a diffraction limited beam is used. Then the coma aberration of the monochromator is roughly proportional to the square of the above divergence angle. This angle could be reduced by increasing the vertical size of the beam on the entrance slit, but, as a result, sacrificing the intensity. Therefore, the throughput of the beamline monochromator with the given energy resolution required for fluorescence spectroscopy is roughly proportional to ω. The optics of the post focusing system usually has an effect of vertical demagnification of the order of 1/10 to guarantee high resolution of the second monochromator. This is because the total size of the second monochromator is much smaller than the beamline monochromator, which requires the vertical beam size on the sample to be much smaller than the width of the exit slit of the beamline monochromator. To obtain such a small beam size we should reduce the magnified divergence angle again to avoid possible coma aberration. Then, the throughput is regarded to be roughly proportional to ω. If the size of the sample is smaller than the image length of the focused light, then the horizontal divergence angle of the incoming beam should also be small, meaning that undulator radiation is much more advantageous than bending magnet radiation because horizontal focusing by the postfocusing system could also cause aberration which would harmfully couple the horizontal and the vertical defocusing.

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Consequently the total throughput is supposed to be proportional to at least the square of the incident photon energy and in some cases the cube of that. In fact, recently, most activities in this field have been associated with undulator radiation except for some effort to use lower photon energies of bending magnet radiation. Therefore, it is not an exaggeration to say that fluorescence spectroscopy in the soft X-ray region has made outstanding progress stimulated strongly by the advent of high brightness undulator radiation.

2.2

Emission intensity

It might be widely believed that a lighter atom has less chances of radiative decay and more chances of nonradiative decay, such as Auger decay. However, the situation is not so simple as one may expect. In fact, if we are interested in emission related to valence–core transition and we examine the characteristics of the electronic states of an atom or a compound in more detail, then we know that, for example, germanium is a much more difficult element compared to carbon. In a carbon atom the 2p–1s transition is the only possible radiative decay, while in a germanium atom the 3p–3s decay and the 3d–3p decay largely reduce the chances of the 4p(valence)–3s decay and 4s(valence)–3p decay, respectively. In general, when a particular atom has a shallow core level which can fill a deep core level with dipole allowed transition, it would be difficult to observe the emission corresponding to the valence-to-deep core level transition. Furthermore, even though the above shallow-todeep transition is dipole forbidden, a strong Auger decay could also occur, which may also reduce the desired valence-to-deep core level transition. Though, in principle, such a weak emission as described above could be observed if we used an extremely intense synchrotron radiation, we are aware of only a few reports such as 4f–4d fluorescence in some rare earth metals. Practically, undulator radiation with much higher brightness would be necessary to observe it.

2.3 2.3.1

What can be seen through soft X-ray fluorescence spectroscopy Basic considerations

Probably the most important application of this spectroscopy is to observe the density of the occupied electronic states around a particular atom. Though resonant photoemission can give similar information on the occupied states it is usually much more surface sensitive than fluorescence spectroscopy. In order to observe the occupied valence band as accurately as possible, the final state should be just a state with a single hole in the valence band. Some additional excitations, such as valence exciton in an insulator, phonon or magnon excitations, or multi-electron–hole pair excitations in a metal could easily deform the spectral shape from the single-body electronic density of states. A similar deformation could also occur in photoelectron spectroscopy, but in a different fashion. In fact, we should be careful about the various time constants of the additional excitations relative to the lifetime of the core hole created, because, in fluorescence spectroscopy, the energy of the emitted photon is smaller than that of the incident photon by making some additional excitations. Sometimes the energy difference between the incident and the emitted photons is called “Raman shift” as is already mentioned. How the various Raman shifts happen in a fluorescence process is the key question of fluorescence spectroscopy.

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The general expression for fluorescence spectra is given by the Kramers–Heisenberg formula as  2 2π   f |W  |mm|W |g  f |W |mm|W  |g  F = + (1)    h Eg + . − E m Eg − ω − E m  m m f

where . is the incident photon energy, ω is the emitted photon energy, and W and W  are the corresponding dipole operators. Eg and Em are the energies of the ground states and of the intermediate state, respectively. In general the intermediate states, |m, are very complicated and may include photoemission or Auger processes. When this type of nonradiative decay happens the wave packet motion could be largely modified and some of the expected emission will not happen because of energy conservation. If we take Eg = 0 as the energy origin, then Eqn (1) shows the Fourier transform of oscillations with the frequency . − Em , or ω + Em , which is the beat frequency between the incident or emitted optical field and the induced non-stationary motion. In this way the above equation may be easily understood in terms of the correspondence principle between classical and quantum mechanics. Furthermore, it should be noted that, exactly speaking, all the possible radiative and nonradiative decays are to be included in the wave packet motion and that the wave packet motion continues even after the emission. A fixed photon energy of emission corresponds to selection of the energy eigenstates to satisfy the law of energy conservation. Therefore, Eqn (1) is a very general expression to calculate the spectrum, obtained just by quantum mechanical Fourier transform (or Green’s function) of the non-stationary motion induced. However, Eqn (1) is too general for practical purposes. In fact, we usually need some approximation such as the sudden approximation in the description of intermediate states. The summation over the different final states f could be confined only to some important states. The summation should be done incoherently for different states even though the energies are accidentally the same. The second term in the above equation corresponds to the process where the emission occurs before the absorption of a photon. This term is usually much smaller than the first term because the denominator of the second term is much larger than that of the first term. Then, we usually neglect the second term. We further make the following approximation to simplify the complicated wave packet motion, and adopt an approximate expression for fluorescence spectra as follows:  2 2π   f |W  |mm|W |g  F = (2)   δ(Eg + . − Ef − ω)  h Eg + . − Em − iΓm  f

m

where the imaginary part Γm is introduced phenomenologically to show the finite lifetime of the intermediate states. Obviously this lifetime should reflect only the events that prohibit the concerned emission with the photon energy determined by the law of energy conservation. More exactly Γm should be derived through some self-energy corrections caused by various excitations associated with the wave packet motion that prohibit the emission concerned through filling the core hole. For example, Auger decays are included in these excitations because they fill the core hole to prohibit the emission. Therefore the introduction of Γm is interpreted to be the following approximation: As already mentioned, the calculation of the emission spectrum needs the Fourier decomposition of the wave packet motion in the time duration between absorption of a photon and

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emission of a photon. Then, in order to obtain the exact spectrum we should add coherently contributions by different values of time duration. However, in general it is impossible to count all the possible events that prohibit the emission concerned and to estimate the probabilities when they occur. Therefore, in the present approximation, this coherent superposition is “carried out” phenomenologically with a weight which becomes exponentially smaller with longer time duration. It should be further noted that the above two equations should be modified if the core electron is excited into some valence state, because the excited electron can participate in the relaxation. In this case, some particular intermediate states play an important role and the denominator of Eqn (1) could be very small and there may be wave packet motion even after the emission occurs. This “after motion” could also be treated phenomenologically. This case is called “resonance fluorescence”. The modified formula is expressed as  2   f |W  |mm|W |g  Γf /π F =    Eg + . − Em − iΓm  (Eg + . − Ef − ω)2 + Γf2 m f

(3)

where Γf is phenomenologically introduced to take into account the lifetime of a possible shallow core hole left behind after the emission. Here, we have distinguished the lifetime of “quasi-particle” in the valence band from that of shallow core hole. Strictly speaking, we cannot always separate Γf from Γm , because some process that prohibits the emission concerned could also affect the lifetime of a shallow core hole left behind after the emission. However, we have made an approximation to assume that the system could be separated into two parts, each having different modes of the wave packet motion. To simplify complicated intermediate states we present the following typical cases: 1

2

Figure 12.7 shows the case where the incident photon excites a core electron into a high continuum state without any further relaxation, and then a valence electron fills the core hole before redistribution of the valence electrons by the core hole potential. In fact, this is a relatively rare case because usually redistribution of the valence electrons occurs very quickly. However, if all the valence electrons around a particular atom have the same symmetry and spatial extension, the attractive force induced by the core hole acting on each valence electron can be regarded as the same, which just causes the common shift of the valence-to-core emission spectrum. Therefore, in this optimum case the obtained spectrum correctly reflects the partial density of state of the valence electrons. Figure 12.8 shows the case where the incident photon excites a core electron into a high continuum state with some relaxation. The relaxation energies are reflected in the spectrum in the energy domain. An interesting example is a spike structure around the photon energy corresponding to the case where an electron just below the Fermi level of a metal fills the core hole. This is called “Fermi edge singularity” similar to absorption spectroscopy because in the excited (or intermediate) state many electrons are excited from levels just below the Fermi level to those just above the Fermi level. In the time domain picture the Fermi edge singularity reflects the wave packet motion with which the core hole potential is gradually screened by conduction electrons. The time dependence of the screening is not exponential but can be described by a power law as t −γ , causing the spike structure in the energy domain.

hΩ/(2) h /(2)

Figure 12.7 Fluorescence without relaxation: First a core electron is excited into a continuum nonresonant state. Then a valence electron fills the core hole without any redistribution of the other electrons.

hΩ/(2) h /(2)

Figure 12.8 Fluorescence with relaxation: First a core electron is excited and then some valence electrons redistribute slowly compared to the speed of the excitation. Next a valence electron fills the core hole, which is again a fast process. The Fermi edge singularity observed in fluorescence spectra of a simple metal is one of the examples of such relaxation.

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T. Miyahara The Fermi edge singularity is a good example to show how the slow relaxations could modify the emission spectrum to a large extent when the lifetime of a core hole is sufficiently long.

2.3.2

Advantages of fluorescence spectroscopy over absorption or photoemission spectroscopy

It is needless to say that all the three spectroscopies have a common advantage of “site selectivity” in the soft X-ray region. However, fluorescence spectroscopy has several definite advantages derived from Eqns (1)–(3) over the other two techniques. There are many cases where absorption spectrum or photoemission spectrum has satellite structures caused by some induced non-stationary motions. Usually, in this case, we try to guess the origin of a particular satellite through comparison of the experimental data and the theoretical calculation. On the other hand, fluorescence spectroscopy fully utilizes tunability of synchrotron radiation in changing the incident photon energy. As seen from Eqn (2) we can cut off some particular intermediate states by reducing the incident photon energy. This “cut-off” effect greatly simplifies the interpretation of the origin of a satellite, because a satellite usually corresponds to a particular electronic configuration. Resonance fluorescence could also be used to confine a particular intermediate state with narrow energy width. Furthermore, as is the case in photoemission spectroscopy, the final state is confined to some particular state among many possible configurations through the law of energy conservation. Whether polarization of the incident photon is conserved or not is an important check-point of the detailed confinement, which cannot be done in photoemission spectroscopy and obviously is an advantage of fluorescence spectroscopy. Based on the above advantages fluorescence spectroscopy gives us information on the electronic states in the ground and excited states of a material with less ambiguity compared to the other two techniques. In fact, there are many cases where fluorescence spectroscopy clarifies the electronic state that has been controversial in interpretation of the data by photoemission or absorption spectroscopy. Similar to absorption spectroscopy, fluorescence spectroscopy is not contaminated by charge-up effect and is much less surface sensitive than photoemission spectroscopy. However, we believe that the greatest advantage of fluorescence spectroscopy is to get information of creation or destruction of coherence in a material, though it has rarely been utilized except in measurements of polarization dependence. In order to fully utilize this advantage we probably need synchrotron radiation with characteristics of the coherent (Glauber) state. As already mentioned in Section 1.1, even a suitable superposition of zero and single photon states could make an almost coherent state. Unfortunately, in this situation, it is very difficult to induce coherent motion in a material and see how the coherence is created or destroyed. Therefore, to utilize the advantage, we probably need synchrotron radiation with peak(instantaneous) Bose degeneracy larger than 1000. Some possible applications will be discussed in Section 3. 2.4 2.4.1

Problems on coherence Basic considerations

How coherence is created or destroyed is the key question of fluorescence spectroscopy or even could be one of the central questions of solid state physics in investigating the progress

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of materials on the earth. Generally speaking the Kramers–Heisenberg formula (1) includes all the mechanisms to create or destroy the coherence in fluorescence when we could take into account all the possible processes to induce the emission of a photon. Practically, however, it is impossible to do so, and even worse, the formula is too general to give us a physical insight into creation and destruction of coherence. Therefore, we consider some typical processes to create or destroy the coherence. Here, we point out one important guideline, which claims that coherence or incoherence depends on how we observe a given system and that destruction of coherence could occur with finite resolutions (energy, time, momentum etc.) of the observation. We start from the simplest case, where the target material is in the definite energy eigenstate. When a photon (ω, k) is absorbed by the material and a photon (ω , k  ) is emitted, there remains the coherence between the incident photon and the emitted photon regardless of mechanisms to produce the corresponding Raman shift, if the energy and k resolutions are infinitely good. Hereafter, we regard k as a momentum though it is actually a wave number. It is obvious that there is no mechanism to destroy the coherence. This is because the initial state of the material is a single quantum mechanical state and the final state is uniquely determined through the laws of energy and momentum conservation. One may consider the question of what the coherence would be when the kinetic energy or the momentum of the photoelectron participating in the Raman shift has some ambiguity due to poor resolution or when the resolution to detect the emitted photon is not good. The answer is that it depends on how good a resolution the coherence is measured with. In fact, the above coherence can be checked in principle by an interference effect. To illustrate the actual observation we consider the following ideal experiment. There are two identical targets that are separated by a distance d and are irradiated by a photon. In this experiment the angular distribution (k-dependence) of the emitted photon shows some interference pattern determined by the wavelength of the emitted photon and the distance d between the two targets as shown in Figure 12.9. The two neighboring intensity peaks are separated by the angle λ/d, where 2π/d is the momentum transfer between the

Emitted light ( ⬘,k⬘)  = 2 c / ⬘

d

Incident light ( ,k)

Figure 12.9 Inelastic scattering by two identical targets: Diffraction effect due to the phase relation between the two scattered waves could occur because there is no process to destroy the coherence if two targets are in an exactly identical quantum mechanical pure state. This is possible if two identical atoms are prepared.

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incident and the emitted photon and λ is the wavelength of the emitted photon. Therefore, the momentum resolution to observe the emitted photon should be better than 2π/d to observe the above interference pattern. The energy or wavelength resolution required to observe the interference pattern is not very large. Classically, this is just the diffraction effect through interference of waves produced at two separated origins. This corresponds to an ideal case where even inelastic scattering keeps the coherence. The classical interpretation of the above coherence is as follows. First the incident electromagnetic wave induces two non-stationary motions in the two targets. Simultaneously, there arise two identical non-stationary motions, the frequencies of which correspond to the Raman shift. Then, finally, the non-stationary motion corresponding to the difference frequency is induced and radiates the electromagnetic wave. The important point is that the phases of every non-stationary motion are common in the two identical targets, and, as a result, we observe the coherence. Another check of the coherence is observation of quantum beats between two different radiations with different energies. Practically, we can observe the quantum beats corresponding to two different discrete Raman shifts in the intensity change of the emitted photon (ω , k  ), if the the energy difference is very small. The situation for a single target is illustrated in Figure 12.10. It should be noted here that the quantum beat can be observed only when the time resolution is better than the period T of the beat. This means that the time duration of the incident photon should be shorter than T , and accordingly the bandwidth of the incident photon should be much larger than the energy difference for the quantum beat to be observed. Under these conditions quantum beats are observed even for many identical targets. This kind of quantum beat has been observed in the time domain spectrum in nuclear Bragg scattering in the X-ray region. However, ordinary situations are not so simple. If the initial state includes some nonstationary motion, the situation could be completely different. For example, we have a single target in a non-stationary state, expressed by superposition of two different states |x and |y with different energies. Even so, we may observe the identical emitted photon (ω , k  ) if the respective Raman shifts cancel the energy difference -E between the two states and if the

Intermediate state

Quantum beat Incident light (not monochromatic) Final state

Initial state

Figure 12.10 A process to cause a quantum beat: For a quantum beat to be observed the time resolution of the detector should be shorter than the period of the beat oscillation, which means that the band width of the incident light should be larger than the energy difference between two intermediate levels. Each energy level should be sharp enough to have an energy width smaller than the above energy difference.

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Intermediate state

Incident light (not monochromatic) Final state ∆E

y x

Initial non-stationary state

Figure 12.11 A process that destroys the quantum beat: When a target has a non-stationary motion as superposition of two different energy eigenstate, we could not observe a quantum beat because for an ensemble of such targets there is no means to control the phase relation in many non-stationary motions of the “identical” targets.

incident photon is not exactly monochromatic and covers the energy difference -E. The situation is shown in Figure 12.11. It is important to note that the phase of this non-stationary motion has no correlation to the phase of the incident electromagnetic wave. In other words, the phase relative to those of the two components of the incident light that causes the two excitations shown in the figure are uncontrollable. Therefore, it is expected that if there are many identical targets we could not observe the quantum beat at all. To repeat the point, the problem here is that the phase relation in the superposition of the two states included in the non-stationary motion together with the phase relation in the superposition of two monochromatic photons included in the incident light is not controllable. Therefore, the phase relation of the emitted photons through |x and |y is not controllable. This may present a real problem when we have more than two targets. To clarify the importance of the above phase we present the following example. Here, we have again two identical targets in the above non-stationary state separated by distance d as shown in Figure 12.12. In this situation there is no interference effect in the angular distribution of the emitted photon (ω ). What happens here is that, though the two independent scattering processes through |x and |y should give some angular distribution, the interference between the two processes destroy the angular distribution when the phases of the above non-stationary motion are uncontrollable. When the ratio of the above superposition is 1 : 1, the above uncontrollable phase completely destroys the angular distribution. Fortunately, however, the ratio is not always 1 : 1 and we may observe partial coherence in this case. In addition when the energy difference between |x and |y is very small, the phase difference in the superposition could be generally small if the non-stationary motions are created by an excitation of some classical coherent motion. This is discussed in the following section. 2.4.2

How the coherence in the non-stationary motion is created

In the previous section we considered the uncontrollable phases between two identical targets in non-stationary motions. In this section we consider the opposite situation where the two stationary motions in the initial state are coherent.

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Emitted light

d y

y

x

x

Two identical targets in a non-stationary state

Incident light

Figure 12.12 Incoherence in fluorescence by two identical targets but with uncontrollable phases of non-stationary motions: When there is no mechanism to correlate the two nonstationary motions in two identical targets, coherence is completely destroyed. However, if there is such a mechanism, as is the case in a solid, where a long wavelength phonon correlates the motions of neighboring atoms, some coherence could be recovered.

First, we consider coherent motions in a solid. For example, a phonon in a solid is a quasi-particle derived from quantization of lattice vibrations. When the wavelength of the phonon is much longer than the separation d of the identical targets, then two non-stationary motions expressed by a superposition of different phonon-number states have almost the same phase. In other words this case corresponds to the situation where the two targets are commonly related to the same mode of lattice vibration. Practically, d corresponds to the distance between two identical atoms and is of the order of nanometers in a solid, while the wavelength of a phonon is of the order of 10 nm. If this phonon is in an acoustic mode with a long wavelength, corresponding to small k, it has a very small energy compared to the local electronic excitations. Therefore, generally, excitations with very small energy quanta included in the initial state do not destroy the coherence, even though they are thermally excited. This condition is usually satisfied also in the thermal excitations of electron–hole pairs around the Fermi level of a metal because such electron–hole pairs usually have small momentum of the center of gravity. However, when the distance d is much larger than the wavelength of the quasi-particle the above coherence is largely destroyed through the mechanism discussed in the next section. This is the case, for example, in a zone plate where fluorescence X-rays are not focused, while the ordinary Thomson scattered X-rays are focused through interference among scattered rays by the different rings of a zone plate. It should be remembered here that the Thomson scattering is an ideal elastic scattering and that the actual Bragg scattering is not exactly elastic. In the case of the ordinary Thomson scattering by a zone plate or of Bragg scattering by a crystal, many phonons with almost zero energy and very small momentum are excited in the scattering. As a result, scattered light has a relatively large k shift as the sum of many

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small momenta but has an almost vanishing energy shift. This is why Thomson or Bragg scattering is often called “elastic” scattering and regarded as coherent scattering. Second, local excitations with relatively large energy quanta are considered. This corresponds to cases of optical phonons, local magnons, or any other local excitations. Here “local” means a short wavelength in the wave picture and “large” energy means the excitation energy is larger than the thermal energy. Then, the quantum occupation number of such particles is mostly 0 or 1. Because the Raman shifts should cancel the single quantum energy of the particle for the two targets to have the same final state, one process contains the Raman shift of a single quantum and the other does not. Therefore, if the energy bandwidth of both the incident and the emitted photons is smaller than that of the single quantum, then we could distinguish the two processes with occupation number 0 and 1 in the initial state and forget about the superposition of the state 0 and 1. This means that we could select one of the two states in the initial state. Accordingly the emitted photon would not lose the coherence. Unfortunately, however, the energy resolution of an emission monochromator is not yet good enough for the above excitation to be resolved, though the energy resolution of an incident photon has been greatly improved recently with use of undulator radiation. Consequently detection of the above coherence is still difficult nowadays but will be possible in the future.

2.4.3

How the coherence is destroyed

There are many cases where a small momentum change between incident and emitted photons occurs as a result of the vector sum of several excitations with large momentum. For example, a small momentum transfer could be made with photoelectrons with large momentum k 1 and a phonon with large momentum k 2 . Under this situation even if the net momentum transfer k 1 − k 2 is very small k 1 or k 2 could be larger than 2π/d. Even in this case, the coherence is not destroyed if the initial state is in a single energy eigenstate, as discussed in Section 2.4.1. The only thing we have to worry about might be that there are so many different mechanisms to give the common emitted photon (ω , k  ), which means the corresponding final states are “different”. However, unless we observe the electron or phonon states, the coherence between two identical targets is not destroyed. This is a result of the basic principle of quantum mechanics. On the other hand, if the system in the initial state is in a non-stationary state with superposition of a zero-phonon and a single-phonon states, then there is no correlation between the phases of non-stationary motions of the two identical targets, because the wavelength of the phonon with momentum k 2 excited in the initial state is shorter than d. Therefore, the coherence is destroyed when the energy bandwidth of the incident photon is larger than the energy of the phonon. The important point here is that a phonon with large momentum could have a small energy, while a photoelectron with large momentum has a large kinetic energy. Then, two identical targets would not be commonly related to the same phonon when 2π/d is smaller than the momentum of the phonon. As a result, even when the emitted photon has a definite momentum, the corresponding mechanism to cause the Raman shift is not coherent between one target and the other due to the uncontrollable phases in the initial non-stationary motions. This means that two “identical” targets could not be regarded as really identical. The above situation is easily realized in a solid at room temperature, where phonons with large momentum could be thermally excited in the initial state.

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2.4.4

Some check points to evaluate the coherence

Here we present some criterions to check whether a particular process is coherent or not. 1 2 3

4

5

6

7

3

The de Broglie oscillation included in an energy eigenstate has nothing to do with coherence or incoherence, because this phase is canceled out in the bra and ket vectors. The difference between two different de Broglie oscillations can be observed and can produce a non-stationary motion. If the above non-stationary motion does not exist in the initial state and it is induced only by an external field, then there is a definite coherence between the non-stationary motion and the external field. If the above non-stationary motion exists in the initial state without the external field, then there is no relation between the phases between the non-stationary motion and the external field. If the external field is strong enough to induce a large non-stationary motion, then the contribution of the phase of the existing non-stationary motion without the external field becomes negligible. If the wavelength of a non-stationary motion in the initial state is larger than the distance among identical targets, then, all the targets have an almost common phase of the nonstationary motion. Even if the above wavelength is much shorter than the distance, the phases could be common if the non-stationary motion is induced by a strong external field such as a sound wave or an additional electromagnetic wave.

Future prospects

The recent development of undulators promises to give us some new opportunities in soft X-ray spectroscopy, especially, fluorescence spectroscopy. Here, we present some possible experiments to utilize the coherence property with large Bose degeneracy discussed in the previous section. 1. Coherent diffraction effect of the emitted photon with large Raman shift. Here “large Raman shift” means, for instance, the Raman shift of about 100 eV observed in the excitation of the Si K shell, corresponding to Si 2p–3p energy separation. The problem here is whether or not a diffraction effect is observed if we have two Si atoms separated by a distance of 5 nm. Obviously the sample should have a definite periodic structure in this experiment. It should be noted that the above effect could be a single-photon effect that could be observed even with weak incident light if there is no mechanism to destroy the coherence in the material. However, because there are so many excitations in the material in the initial state that may destroy the single-photon coherence we would rather not expect to observe the diffraction effect. On the other hand very intense coherent light could induce coherent excitations such as lattice vibrations that have a wave number much smaller than 2π/d. Then the phase of this oscillation could be almost common in both Si atoms and the momentum induced by the wave number 2π/d could be added to the emitted light and cause diffraction. The situation is illustrated in Figure 12.13. 2. Multi-beam fluorescence spectroscopy. If we have two or more incident beams with large Bose degeneracy, then one beam plays the role of inducing a coherent wave packet

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Inelastically diffracted emission (2p–1s emission)

d

Two identical Si targets

Incident light causing 1s–3p excitation

Figure 12.13 Inelastic diffraction by two identical 1s atoms: If two identical 1s atoms are prepared with separation d, then we could observe a diffraction effect through 1s–3p excitation followed by 2p–1s emission. This is an example of general cases shown in Figure 12.9.

motion, while another works as a modulator of the motion. We could observe fluorescence with the beat frequency other than ordinary Raman shifts. An induced Raman effect could also be observed when the photon energy of one of the incident beams is equal to that of the fluorescence. As a special case an interesting thing would happen when the energy of one beam is much lower than the other and is of the order of the band gap of insulators. First, we expect Raman shift of the gap energy, which is just an ordinary soft X-ray fluorescence. Here, we notice that the final state created in the material through this scattering is identical to the state with valence electron excited into the conduction band by the low-energy light. Then we expect an interference effect in the material system, where the phase relation between the identical wave packet motions to cause the Raman shift is very important. As a special case of the above effect, electromagnetically induced transparency (EIT) could be possible in the soft X-ray region, if the Bose degeneracy is large enough for the corresponding Rabi frequency to become larger than the inverse of the core hole lifetime. Similarly coherent trapping or creation of a dark state is possible if we can choose three appropriate levels. All the above interference effects utilize three levels, |a, |b, and |c, as illustrated in Figure 12.14, where radiation coupling exists between |a and |b, and between |a and |c. Then we have the Rabi frequencies .ab and .ac characterizing these radiation couplings, which of course depend on the intensity of the incident light. Therefore, the interference effect occurs at |a either destructively or constructively depending on the phases of the Rabi oscillations. This is the physical origin of EIT or a dark state. 3. Reconstruction of wave packet motion. Up to now all the spectroscopy in the soft X-ray region could just give a spectrum of optical responses. This means that information

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c

b

Figure 12.14 A three level system interacting with strong high fields: When there are three levels, the energy width of which are smaller than the Rabi frequency of each transition, then we could expect various exotic effects such as electromagnetically induced transparency.

of the phases of the motion is completely lost. Even magnetic circular dichroism gives some information on the phase only of a particular eigenstate but not of the wave packet motion. Then, it is impossible to reconstruct the wave packet motion only from the energy spectrum. Though in theories of soft X-ray spectroscopy the wave packet motion is “reconstructed” only through some models to describe the initial state, there is no way to experimentally observe the wave packet motion. For example, if we observe a Gaussian spectrum in the energy domain, it may suggest the existence of a coherent wave packet motion. However, the Gaussian spectrum could be also obtained for a Gaussian density of states. Then we could not experimentally distinguish one from the other without having information about the phases. On the other hand, spectroscopy using intense visible lasers gives information on phases of the wave packet motion through techniques such as “pump and probe” spectroscopy. Similarly, if we have soft X-rays with very short pulse and high Bose degeneracy, then we could observe the real wave packet motion induced by photo-excitation. Of course, we can use visible or UV lasers to probe the relatively slow coherent motion induced just after core excitation by high energy photons. 4. Multi-photon photoemission spectroscopy. First, we notice that the probability of a two-photon process is smaller than a single-photon process by the factor of about α(kr)2 , where α is the fine structure constant 1/137, k is the wave number of the incident light, and r is the spatial extension of the initial state. When the photon energy is 100 eV, and r is 1 nm, then the above factor is 2 × 10−5 . Consequently if we have the Bose degeneracy of 500, we can detect the two-photon spectrum which is about 1% of the intensity of the single-photon spectrum. Of course, because r of a core state is much smaller than 1 nm, while that of a valence state could be larger than 1 nm, the above probability could be larger or smaller depending on r. The important point here is that kr is associated with the transition matrix element, which is not quantitatively measured in single-photon photoemission spectroscopy. However, if we compare the results between single-photon and two-photon spectra, we could obtain information on the spatial extension of the initial state as the second moment. When we detect the three-photon spectrum, we can get the third moment of the spatial extension. The above feature of multi-photon photoemission spectroscopy would have a “revolutionary” effect on solid state physics, because we can qualitatively describe the degree of localization or itinerancy of an electronic state in a solid.

13 X-ray crystal optics Wah-Keat Lee, Patricia Fernandez and Dennis M. Mills

1

Introduction

Although dedicated synchrotron facilities for X-ray production (second-generation facilities, c.1980s) have been available for more than a decade, the advent of the third-generation sources (c.1990s) and the accompanying development of insertion devices have enabled a wide range of new experiments. Third-generation sources (the European Synchrotron Radiation Facility (ESRF) in Grenoble, France; the Advanced Photon Source (APS) in Illinois, USA; and the Super Photon ring (SPring8) in Harima, Japan) are exemplified by their low particle beam emittance and the incorporation of insertion devices in the storage rings. The result is the generation of X-rays with unprecedented brilliance. In many of these facilities, new and innovative X-ray optics are often employed as an integral part of the beamline. Some of these new developments are driven by necessity, such as high-heat-load optics, while others are now made possible due to the high beam brilliance, such as high-energy-resolution optics. A review and discussion of the basic principles of synchrotron radiation and X-ray crystal optics will be presented. Due to space constraints, this chapter will concentrate on X-ray crystal optics that are relevant for X-ray radiation from insertion devices at third-generation synchrotron sources. In particular, the discussion will focus on crystal optics for hard X-rays from undulator sources in the 6–60 keV regime. Reviews on general aspects of synchrotron radiation, and X-ray diffraction and its application to synchrotron radiation can be found in references [1–3], [4,5] and [6,7], respectively. It will become clear in this chapter that the combination of high flux and low beam angular divergence has greatly improved the effectiveness of many X-ray optical components and, thus, the overall success of many synchrotron experiments.

1.1

A brief review of undulator radiation

In this section, some unique aspects of radiation from third-generation sources and insertion devices will be discussed. The goal here will be to provide a basis for subsequent discussions in later sections. The focus will be on the radiation properties of “standard” (not helical or tapered) undulators. Interested readers can learn more about the properties of synchrotron radiation from the comprehensive articles by Kim [1,3] and Krinsky et al. [2]. For this discussion, the following nomenclature is used: the beam propagates in the z-direction, x and y are transverse to the propagation direction, with the y-direction normal (vertical) to the particle orbital (horizontal) plane. Third-generation synchrotrons are exemplified by their small particle beam natural emittances. The particle beam natural emittance, ε0 , is the sum of the particle beam emittances

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in the horizontal (orbital) plane, εx , and the particle beam emittance in the vertical plane, εy . The emittances, εx and εy , define an area in the x–x  and y–y  phase spaces, respectively. Usually, the quoted emittances are for ±σ values of the phase space areas. The ratio εy /εx is known as the coupling. In many cases, experiments are only sensitive to the beam emittance in one plane (e.g., the scattering plane). It can, therefore, be useful to reduce the beam emittance in that plane. For this reason, a small coupling is desirable. Note, however, that since the natural emittance is a constant (for a particular machine), a smaller coupling also implies a greater percentage of beam emittance in the other plane. The actual particle beam sizes, σx,y , and divergences, σx  ,y  , at any point around the ring depend on the properties of the magnetic lattice and are quantified by the “amplitude functions,” βx and βy . When β˙ = 0, which is normally the case at the location of an insertion device,  εx βx  σy = ε y β y

 εx /βx  σy  = εy /βy

σx =

σx  =

(1)

While the natural emittance of the storage ring is a constant (for a given operational mode), the amplitude functions are dependent on the focussing properties of the magnetic lattice and can vary around the ring. Typical particle beam design parameters for the centers of straight sections at thirdgeneration synchrotron storage rings are shown in Table 13.1. Actual operational beam parameters are usually much better than the design parameters as smaller and smaller couplings are achieved. For beamline users and designers, the important parameters are the X-ray beam source size and divergence. The expressions for the X-ray source size and the divergence are given by source size

Qx,y =

source divergence

Q

x  ,y 



2 + σ2 σx,y r  = σx2 ,y  + σr2

(2)

where σx,y , σx  ,y  are the particle beam sizes and angular divergences at the source location in the x- and y-directions, respectively; σr and σr  are the inherent X-ray photon beam size and divergence, respectively, which, for an odd undulator harmonic, are given by [1] 

√ 2λL σr = 4π

σr  =

λ 2L

(3)

Table 13.1 Particle beam design parameters (for the straight sections) for third-generation synchrotron storage rings [8]. Quoted values are for the high-β sections and are based on a 10% coupling

ESRF APS SPring8

σx (µm)

σx  (µrad)

σy (µm)

σy  (µrad)

412 325 350

15 23 15

90 86 78

7 9 7

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where λ is the wavelength of the radiation, and L is the length of the insertion device. Normally, σx,y  σr , while, depending on the wavelength, the natural photon divergence σr  can be significant. Unlike bending magnet and wiggler radiation, the spectral and spatial distribution of undulator radiation is quite complex. An undulator spectrum consists of many sharp peaks, which are referred to as the harmonics of the radiation. The photon energy (in keV) of the nth harmonic at an observation angle θ (relative to the undulator axis) is given by [1] En =

0.95 Ep2 n

(4)

λu (1 + (K 2 /2) + γ 2 θ 2 )

where Ep is the particle beam energy (in GeV), λu is the undulator magnetic period (in cm) and γ is the ratio of the particle beam energy to the particle rest energy. K is the particle beam deflection parameter, given by [1] K = 0.934 λu B0

(5)

where B0 is the peak magnetic field (in Tesla). Figure 13.1 shows the undulator spectrum from the APS undulator A, with K = 2.6. The odd harmonics of the radiation are mainly confined to the central part of the beam, known as the central cone. The beam brilliance and intensity are at their maxima at the center of the central cone. Thus, for most users, the undulator is tuned (by changing the magnetic gap, and hence, the magnetic field strength) such that the desired energy is an odd harmonic (usually the lowest possible) of the undulator output. The spatial distribution of the undulator radiation is particularly complicated in the case of non-odd-harmonic energies. Figure 13.2(a) shows the calculated spatial distribution of the first harmonic energy of the undulator radiation, while Figure 13.2(b) shows the spatial distribution at an energy close to the second harmonic.

1.2 ×1015 Flux (photons/s/0.1% bw)

3rd harmonic 1 ×1015 5th harmonic 8 ×1014 6 ×1014 4 ×1014 2 ×1014

1st harmonic 0

5

10 Energy (keV)

15

20

Figure 13.1 A typical undulator spectrum. The sharp peaks denote the odd undulator harmonics, which spatially, are mainly confined to the central cone of the radiation. The calculation here is for the APS undulator A, K = 2.6, 100 mA ring current, with no slits.

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(a)

3.0 × 1014

2.0 × 1014

1.0 × 1014

0 2 1 ica ld im

Ve rt

0 sio

en

n ( –1 mm )

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ntal dim Horizo

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(b)

4.0 ×1014 3.0 × 1014 2.0 × 1014 1.0 × 1014 0 2 1 ica ld im

Ve rt

0 sio

en

n ( –1 mm )

–2 –3

–2

–1

0

1 sion (m

en ntal dim Horizo

2

3

m)

Figure 13.2 (a) Spatial 3196 eV (first undulator harmonic) photon distribution at 30 m from the source for the APS undulator A with deflection parameter K = 2.6. (b) Spatial 5900 eV (close to second harmonic) photon distribution at 30 m from the souce for the APS undulator A with deflection parameter K = 2.6.

Photons from the synchrotron can be represented in a five-dimensional phase space Qx –Qx  –Qy –Qy  –λ “ellipsoid”. For brevity, this five-dimensional space can also be denoted as x–x  –y–y  –λ. At most synchrotron facilities, the particle beam is controlled (via the coupling) such that the beam size and divergence are much smaller in the vertical direction (y). For ease of representation, Figure 13.3(a) shows the phase space of the radiation projected

X-ray crystal optics (a)

y⬘

(b)

(c)

y⬘

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 High wavelength cutoff

y

y

Σy ⬘

Low wavelength cutoff

Σy ⬘

y⬘

Σy

Σy

∆y = Σy⬘ .D

Σy ⬘

Figure 13.3 (a) Phase space of the undulator radiation projected onto the vertical y–y  plane at the source (see text). (b) Phase space of the undulator radiation projected onto the vertical y–y  plane at a distance D from the source. (c) Phase space of the undulator radiation projected onto the λ–y  plane.

onto the y–y  (same as the Qy –Qy  ) plane at the radiation source, and Figure 13.3(b) shows the same ellipse at a distance D downstream. The source phase space ellipse is upright only at certain positions around the storage ring where the β-function is an extremum. At the APS, this is true at the center of the straight section. For later discussion involving crystals, it is useful to show, in Figure 13.3(c), the projection of the ellipsoid onto the λ–Qy  plane. It is convenient here to introduce the term brilliance, which is defined as the number of photons/s/mm2 /mrad2 /energy bandwidth. Thus, brilliance is directly related to the photon density in the five-dimensional Qx –Qx  –Qy –Qy  –λ phase space of the radiation. A related term, brightness, is defined as the space integrated brilliance, that is, photons/s/mrad2 /energy bandwidth. Figure 13.4 shows the on-axis X-ray beam brilliance tuning curves for the first three odd harmonics of the APS standard 3.3 cm period undulator A (100 mA operation). The plot shows that, in the classical hard X-ray regime (6–20 keV), the beam brilliance is about 1019 photons/s/mrad2 /mm2 /0.1% bw, which is four to five orders of magnitude higher than the corresponding bending magnet radiation. The spatial distribution of the wavelength-integrated power is quite different from that of the harmonic. For an undulator, the power envelope opening angle for K ∼ 1 (undulator regime) is approximately [3]  

2  0.9K 0.6 2 POWER POWER 2 Qx  ≈ + σx  Qy  ≈ + σy2 (6) γ γ At high-energy (6–8 GeV) third-generation synchrotrons, it can easily be shown that the power opening angles are considerably larger than the odd harmonic central cone opening angles. Thus, it is possible to use white beam slits to limit the total power hitting the downstream components without losing any beam intensity from the central cone. Third-generation synchrotrons are exemplified by their small particle beam sizes and angular divergences. With the addition of insertion devices in the synchrotron straight sections,

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1019

n=3

n=1

n=5 1018

1017

1016

0

10

20 30 Energy (keV)

40

50

Figure 13.4 APS undulator A tuning curve (100 mA) using the particle beam parameters shown in Table 13.1.

third-generation synchrotrons are able to deliver X-rays with unprecedented beam brilliance. This aspect of the radiation has greatly contributed to the success and vast improvements of many synchrotron experiments. 1.2

Review of X-ray dynamical diffraction theory

Perfect crystals are still the mainstay of optical components used at synchrotron radiation facilities. In this section, the basic aspects of dynamical diffraction theory [4,5], which are applicable to perfect single crystals will be presented. The goal here is not to give a rigorous derivation of this theory, but to provide an outline of the theory and elaborate on the key results that are relevant for this chapter. The basic problem in dynamical diffraction concerns the interaction of X-rays with the electron charge distribution in a perfect crystal structure. Assuming zero current density and unit magnetic permeability (µ = µ0 ), two of Maxwell’s equations can be written (MKS units) as ∇ ×E=−

∂B ∂t

∇ ×H=

∂D ∂t

(7)

where E is the electric field, B the magnetic induction, H the magnetic field and D the displacement field. Equations (7) can be combined (assuming B = µ0 H) to give ∇ × ∇ × E + µ0

∂ 2D =0 ∂ 2t

(8)

The electric displacement D is related to the electric field, E, by the electric susceptibility χ: D(r) = ε0 (1 + χ)E(r)

(9)

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Here, ε0 and µ0 are the vacuum dielectric constant and magnetic permeability, respectively. √ The index of refraction, n, defined as the square root of the dielectric constant, is 1 + χ. Far from an absorption edge, the electric susceptibility is given classically by χ (r) = −

re λ2 ρ(r) π

(10)

where re is the classical electron radius = 2.82 × 10−15 m, λ is the wavelength of the electromagnetic field and ρ(r) is the electron density. In the X-ray regime (λ ∼ 10−10 m), χ is small (∼10−5 –10−6 ) and negative. Therefore, Eqn (9) can be approximated as D(r) (1 − χ)D(r) ≈ (1 + χ)ε0 ε0

E(r) =

(11)

Inserting Eqn (11) into (8) ∇ × ∇ × (1 − χ)D + ε0 µ0

∂ 2D =0 ∂ 2t

(12)

Only elastically scattered waves that have the same time dependence as the incident wave (eiωt ) will be considered. Since 1 c2

ε0 µ0 =

ω = 2πk c

∇ ·D=0

(13)

Eqn (12) can be rewritten as ∇ 2 D + ∇ × ∇ × χD + 4π 2 k 2 D = 0

(14)

where k = 1/λ is the vacuum wave number. Equation (14) is the wave propagation equation for an electromagnetic wave in a material with a small susceptibility that is a function of spatial coordinates. Next, consider the properties of the crystalline material. A crystal is by definition spatially periodic. Its electron density can, therefore, be expressed as a Fourier sum over its reciprocal lattice vectors, H: 1  FH exp(2π i H · r) V

ρ(r) =

(15)

H

and  FH =

ρ(r) exp(−2π i H · r) dv

(16)

V

where V is the volume of the unit cell. FH is the structure factor, and, assuming that the atoms behave as rigid spheres, it can be written as a sum over the unit cell: FH =

 n

fn exp(−2πi H · rn )

(17)

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where fn are the atomic scattering factors. The atomic scattering factors are usually written as f = f0 + -f  + i-f  , where -f  and -f  are the Honl or dispersion corrections. Parts of the structure factor can be conveniently expressed as follows:  FrH = (f0 + -f  )n exp(−2πi H · rn ) = |FrH |eiϕrH n

FiH

 (-f  )n exp(−2πi H · rn ) = |FiH |eiϕiH =

(18)

n

Similarly, the electric susceptibility can be written as  χ= χH exp(2π i H · r)

(19)

H

where χH is related to the structure factor FH (through Eqn (10)) by χH =

−re λ2 FH πV

(20)

That the susceptibility χ can be expressed as a Fourier sum suggests that solutions to Eqn (14) can take the form of Bloch waves:  DH exp(−2πi KH · r) (21) D(r) = exp(iωt) H

where KH = KO + H

(22)

The vector relation in Eqn (22) is simply a restatement of Bragg’s law: λ = 2 d sin θB

(23)

where the reciprocal lattice vector is related to d, the interplanar spacing, by d=

1 |H|

(24)

Assuming that only one reciprocal lattice vector is excited (the so-called two-beam approximation), neglecting the time dependence, Eqn (21) simplifies to D(r) = DO exp(−2πiKO · r) + DH exp(−2πiKH · r)

(25)

Electromagnetic fields are transverse in nature, and it is convenient to consider their polarization vectors in two separate orthogonal directions. In scattering applications, the two orthogonal polarization directions are denoted as σ and π . For σ -type radiation, the polarization vector is perpendicular to the scattering plane (defined as the plane spanned by KO and H). For π-type radiation, the polarization vector is in the plane of diffraction. Insertion of Eqns (19) and (25) into Eqn (14) yields a set of simultaneous equations: DO = DH =

2 KO 2 − k2 KO 2 KH 2 − k2 KH

[χO DO + CχH DH ]

(26)

[CχH DO + χO DH ]

(27)

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The factor C in Eqns (26) and (27) takes into account the two possible polarization directions: C = 1 for σ -polarization, and C = cos(2θ ) for π-polarization. Keeping in mind that the difference between KO (or KH ) and k is small compared to k (∼10−5 –10−6 ), these two simultaneous equations can be linearized to give 2XO DO − kCχH DH = 0

(28)

2XH DH − kCχH DO = 0

(29)

where  χO  k XO = KO − 1 + 2  χO  X H = KH − 1 + k 2

(30) (31)

In order that Eqns (28) and (29) have non-zero solutions,

X O XH =

k 2 C 2 χ H χH 4

(32)

It must be pointed out that, except for H, C and k, all the variables in the above equations may be complex. The imaginary parts of the wave-vectors lead to absorption in the crystal. For now, consider the real parts of the variables. Under this assumption, Eqn (32) represents the equation of a hyperbola about the two axes, XO and XH . Furthermore, XO and XH , from Eqns (30) and (31), represent the difference between the length of the wave-vectors (real part) inside the crystal and the vacuum wave-vector corrected by the average index of refraction (n = 1 + χO /2). For each polarization, there are thus two hyperbolic curves associated with Eqn (32). The hyperbolas, known as dispersion surfaces, therefore, represent the locus of all possible real wave-vector solutions inside the crystal. The hyperbola that is further away from H is usually referred to as the α-branch dispersion surface (XO , XH > 0), while the one closer to H is usually referred to as the β-branch dispersion (XO , XH < 0). The relative wave amplitudes DO and DH are given (Eqns (28), (29) and (32)) by ξ=

−2πV XO DH = DO re λCFH

(33)

The actual selection of a particular point on the dispersion surfaces (called a tie-point) is achieved through the use of boundary conditions on the entrance surface of the crystal. Application of the boundary conditions requires that the external wave-vectors differ from the internal wave-vectors only in the direction normal to the surface. Once the tie-points are selected, the wave properties inside the crystal are known. Similarly, application of the equivalent boundary conditions at the exit surface gives the external-diffracted and forwarddiffracted waves.

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For brevity, the geometrical details on the determination of the tie-points will be omitted. Following the notation used by Authier [4]: K O = K O sO

KH = KH sH

γO = n · s O γO b= γH

γH = n · s H

η=

-θ = θ − θB

(34)

[b-θ sin(2θ ) − 1/2χO (1 − b)] πV |b|1/2 b|C|[FH FH ]1/2 re λ2

where η is a scaled, dimensionless angular variable, and b, the asymmetry factor, is <0 for the Bragg geometry (incident and reflected beams enter and exit the same crystal surface) and >0 for the Laue geometry (incident and reflected beams enter and exit from different crystal surfaces). For symmetric Bragg (crystal planes parallel to surface) geometry, b = −1, and for symmetric Laue (crystal planes normal to surface) geometry, b = +1. n is the normal to the crystal entrance surface, and θ is the incidence angle with respect to the surface. Given the above definitions, for an incoming beam at an incidence angle θ 
1/2  DH |C| b [FH FH ]1/2 b ξ(θ ) = η ± η2 + (35) =− DO C |b|1/2 FH |b| Solutions with a positive sign in Eqn (35) correspond to tie-points on the α-branch while those with a negative sign correspond to tie-points on the β-branch. Illustrating the above geometrically, as in Figures 13.5(a) and (b), is helpful. Inside the crystal, the “tails” of all the wave-vectors must lie on the dispersion surfaces, while the “tips” of all the wave-vectors must lie at the reciprocal lattice points O or H. The locations of the tie-points depend on the external (vacuum) incoming beam direction and the inward crystal normal. In the Laue case (Figure 13.5(a)), b > 0, while in the Bragg case (Figure 13.5(b)), b < 0. In the Laue case, tie-points can exist everywhere on all dispersion surfaces. In the Bragg geometry, all the dispersion surfaces can be activated for thin crystals (thin compared with the X-ray attenuation depth), while in the thick crystal case, the tie-points only exist for half of each dispersion branch. (The tie-points within the bold dashed regions correspond to physically unrealizable solutions where the field amplitudes approach infinity deep inside the crystal.) Equation (33) shows that the amplitude of DH is proportional to |XO |. Therefore, DH is very strong and dominates at tie-points high up (closer to H) on the α-branch and low down (closer to O) on the β-branch. Conversely, DO is very strong and dominates at tie-points low down on the α-branch and high up on the β-branch. At the waist of the hyperbola, the amplitudes of the waves are equal. In the Bragg case, there is a region in between the hyperbolas for which there are no real solutions to the wave-vector equations (28) and (29). This corresponds to the region of total reflection in which only a surface evanescent wave exists inside the crystal. The angular range of this region of total reflection is known as the Darwin width and, in the symmetric case (b = −1), is given by  √   2|C|  χH χH  2re λ2 |C|  FH FH  ω0 = = (36) sin(2θB )πV sin(2θB )

(a) Crystal surface

KH,

H

KH,

Sphere of radius nk about O

KH,

re he Sp

KH,

ius rad of H ut bo ka

n Laue point O

KO,

Incident k

rad of re he

h nc bra h ,  ranc b , 

Sp

KO,

KO, K O,

,  bra nc ,  h bra nc h

ius

ka

bo

ut

Lorentz point

O

Sphere of radius nk about H (b)

h

bra

nc

h nc

, 

bra , 

re

H

he Sp

re

he

tH

Sp

ou

ab

of

nk

rad

ius

ius

rad

nk

of

ab

re

ou

he

tO

Sp

Crystal surface

of

KH,

iu rad sk

KH,

tH

ou

ab ,  ,  bran bra ch nc h

Sp

he

re

of

rad

ius

ka

bo ut O

Incident k

O

n

KO,

KO,

Figure 13.5 (a) Schematic of the dispersion surfaces for the Bragg geometry. The dashed line regions of the dispersion surfaces are not active for thick crystals (see text). (b) Schematic of the dispersion surfaces for the Laue geometry.

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That the range of total reflection is finite and not a δ-function can be intuitively understood by the fact that the incoming X-rays do not “see” the total or infinite periodicity of the crystal but only interact with a finite number of atomic planes, usually in the 104 –105 range. For a symmetric reflection, the angular acceptance and the reflected beam divergence are identical and are given by Eqn (36). For an asymmetric reflection (where the reflecting planes are not parallel or normal to the crystal surface), the angular acceptance, ωacc and the reflected beam divergence ωrefl , are no longer the same and are given by ω0 ωacc = √ |b|  ωrefl = |b|ω0

(37) (38)

Figure 13.6 shows the symmetric and asymmetric (b = −5.48) reflectivity curves for Si(111) reflection at 8 keV. Figure 13.7 shows the APS undulator A vertical X-ray beam opening angles (FWHM, Qy  and symmetric Darwin widths (Eqn (36)) for the Si(111) reflection as a function of X-ray energy in the 5–20 keV range. As seen from the figure, in this energy range, there is a reasonable match between the two. Later, it will be shown that the narrow opening angles of the undulator radiation at third-generation sources has dramatically improved the performance of many X-ray optical components. This, in turn, has led to the success of many experiments that were not feasible before. While the real parts of Eqns (30)–(32) lead to the construction of the dispersion surfaces, the imaginary parts of the same equations lead to information on absorption. As mentioned above, boundary conditions require that the external wave-vectors differ from internal wavevectors only in the direction normal to the surface. This, together with Eqn (22), implies that the imaginary parts of the internal wave-vectors (KiO and KiH ) are equal and have directions normal to the crystal entrance surface. The absorption coefficient, µ (not to be confused with the magnetic permeability), is usually defined by the ratio of the transmitted beam intensity to

Reflectivity

1 0.8

b = –5.48 acc

0.6

refl

b = –1 0

0.4 0.2 0 –20

–10

0

10 20 30 – B (arc secs)

40

50

Figure 13.6 Symmetric (ω0 ) and asymmetric (b = −5.48) acceptance (ωacc ) and asymmetric (b = −5.48) outgoing (ωrefl ) reflectivities for Si(111) reflection at 8 keV. In the symmetric case, the acceptance and the outgoing divergences are the same, while in the asymmetric case, they can be very different.

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60

Microradians

50 40 0 30 Σy ⬘ 20 10 5

10

15 Energy (keV)

20

25

Figure 13.7 Plot of symmetric Si(111) Darwin widths (ω0 ) and the APS X-ray undulator beam (on odd harmonics) opening angles (FWHM,Qy  as a function of energy. The beam opening angles are based on Table 13.1.

the incident beam intensity. Although KiO and KiH are normal to the surface, the absorption coefficient in the direction of the real part of the wave, µ, can be conveniently defined as I |E|2 = = e−µz I0 |E0 |2

(39)

where z is the distance traversed by KrO (or KrH ) in the crystal. (Since Ki Kr , the direction of the wave-vectors KO (or KH ) is essentially the direction of the real part of the wave-vectors, KrO (or KrH ).) Thus, from Eqns (30) and (33), µ in the direction of KO can be expressed as µ = −4π γO KiO = µ0 − 4πXiO = µ0 −

2re λC I (FH ξ ) V

(40)

where I (a) denotes the imaginary part of a. µ0 is the linear absorption coefficient, given by µ0 =

2re λFiO V

(41)

Many interesting and subtle effects result from absorption, and they are discussed in detail in Batterman and Cole [5]. Only a few results that are important for subsequent X-ray optics discussion will be mentioned. For the Laue case, from Eqns (34), (35) and 40), it can be shown that   √ 1/2ηr (1 − b) + |C| b |FiH /FiO | cos φ  µ = µ0 1/2(1 + b) ∓ (42) ηr2 + 1 where φ = φrH − φiH is the phase difference between FrH and FiH (see Eqn (18)), and the minus sign corresponds to the α-branch and the positive sign to the β-branch. ηr is the real

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part of η. Notice that under the appropriate conditions (C = 1, b = +1, η = 0, FiO = FiH and φrH = φiH = 0) µ = 0 for the α-branch, while µ = 2µ0 for the β-branch. Although µ for the α-branch is zero only for this special case, it is generally true that near the reflection peak (η = 0):  σ   π   π   σ  K  > K  > K  > K  α α β β (43) µσα < µπα < µπβ < µσβ Near the reflection peak, the α-branch waves undergo reduced absorption, while the β-branch waves undergo increased absorption. Far away from the reflection peak (|η|  1), the wave-vectors and absorption coefficients approach their normal values:    χO   σ,π  k Kα,β  → 1 + 2 (44) µσ,π → µ 0 α,β Standard dynamical diffraction theory, as presented above, breaks down at θ ∼ 0◦ or θ ∼ 90◦ . At these angular positions, the approximations (linearization) in Eqns (28)–(31) are no longer valid and Eqns (26) and (27) must be used instead. The Bragg case where θ ∼ 90◦ is of interest here because of its utility in high-energy-resolution monochromators (see Section 15.3). Several studies have been done to extend dynamical diffraction into this regime [9–11]. One result √ from these studies that is of interest here is that, for θ ∼ 90◦ , the angular acceptance ω0 ∼ |χH | instead of ω0 ∼ |χH | as seen in Eqn (36). Since |χH | ∼ 10−6 , the angular acceptances for θ ∼ 90◦ increase dramatically, by as much as 102 –103 times. 1.3

Dumond diagrams

A useful representation of X-ray crystal optics is through the use of Dumond [12] diagrams. A Dumond diagram is a plot of the wavelength–angle relation for a given crystal reflection. From Eqn (23), it is thus, a sine-curve with a certain width to the line that represents the acceptance or outgoing width. Points within the width (or band) indicate good transmission, while points outside the band indicate poor transmission. In most cases, only a small angular region of the Dumond diagram is of interest, and thus the curves can be linearized to be straight lines instead. Figure 13.8 shows an example of a Dumond diagram. It is sometimes useful to consider a three-dimensional version of the Dumond diagram. Depending on the situation, the third axis can be the spatial dimension or the reflectivity of the crystal. In most of the subsequent discussions, it is assumed that the crystals are large enough to accept the entire spatial extent of the beam. Thus, usually, the spatial dimension on a Dumond diagram is not drawn. In multi-crystal systems, the relative orientation of the reflections must be defined. For example, Figure 13.9 shows two different possible configurations of a two-crystal reflection and their respective Dumond representations. Figure 13.9(a) is usually denoted as the (+, −) or nondispersive geometry, while Figure 13.9(b) is usually denoted as the (+, +) or dispersive geometry. In the Dumond representation, the lines in the (+, −) geometry then have slopes of the same sign, while in the (+, +) geometry, the slopes have opposite signs. Clearly, the angular and energy ranges of the transmission overlap region varies between these two geometries.

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Dumond diagrams are especially useful in providing qualitative representation of synchrotron radiation transmission through various beamline components, such as slits and monochromators. Starting with the beam from the synchrotron (see Figure 13.3(c)) and adding the Dumond representations for subsequent optics and slits, the fraction of the original beam that is transmitted downstream of the slits and optics is easily grasped, qualitatively. 

Darwin width,



Figure 13.8 A simple example of a Dumond diagram.

(a) 1



2

Slope =

o tan 1

Slope =

o tan 2

o



Figure 13.9 (a) A two-crystal nondispersive (+, −) setup and its corresponding Dumond representation.

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2 1



Slope =

o tan 1

o

Slope =

o tan 2

Figure 13.9 (b) A two-crystal dispersive (+, +) setup and its corresponding Dumond representation.

2

High-heat-load crystal monochromators

Undulators at third-generation synchrotron sources produce X-ray beams of unprecedented brilliance (photons/s/mm2 /mrad2 /0.1% bandwidth), with total power and power density (power/mm2 ) incident on the first optical component of the order of 1 kW and 100–300 W/mm2 , respectively [13–18]. These intense beams present a challenge for the design of white beam optical components, since the optics must preserve the high quality of the beam while being subject to extreme thermal loads. In a majority of cases, the first optical component in the beamline is either a cooled mirror or a cooled single-crystal monochromator. Beam-induced thermal distortions that will affect the optical performance include the bowing of the optic due to the temperature gradient perpendicular to the surface and the thermal mapping distortion due to the temperature gradient parallel to the surface. To preserve the small size and low divergence of the undulator beam, the thermally induced slope error in a mirror should be less than 1–2 µrad. There is also a third type of thermal effect for crystals, namely the change in lattice constant due to the thermal expansion [17,19,20], although this effect is typically not the dominant distortion. Thermal deformation of the diffraction planes in crystals should be much less than the Darwin width of the diffraction, typically several microradians [21]. As will be seen, the solutions that result in maintaining thermal slope errors of the order of a few microradians are not always those that provide the best cooling but are rather those that consider the thermal management of the

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system (i.e. control of size and direction of temperature gradients, use of thermal/mechanical properties of the optical components, etc.). A closed-form algebraic solution for the thermal stress and strain in a finite cylinder with uniform heat flux, Q (W/mm2 ) at the center of the top surface, uniformly cooled from the bottom surface, has been developed by Subbotin et al. [22]. Although this model does not accurately represent typical X-ray monochromators or mirrors, it is nevertheless useful because it provides some insight into the general problem of thermal-induced strains. From Subbotin et al., the slope error, -θ , on the surface (directly related to the radial strain gradient) is given by α -θ ∝ Q (T )F k

(45)

where α is the thermal expansion coefficient, and k is the thermal conductivity, both of which can be temperature dependent. F is a factor that depends on Poisson’s ratio and the dimensions of the cylinder and the size of the incident beam. Note that Eqn (45) does not depend on the cooling boundary condition, coolant properties or flow rates. The importance of efficient heat removal manifests itself through the temperature dependence of α/k. For common X-ray optic materials, such as silicon, germanium or diamond,1 α/k decreases (nonlinearly) with decreasing temperature in the 150–300 K temperature regime. Improved heat removal results in a lower temperature of the crystal surface and thus improved thermophysical properties, that is, a lower α/k ratio. Therefore, three general methods to minimize the thermal-induced slope errors are (1) reducing the heat flux Q, (2) selecting materials (or operating temperatures) that have favorable thermophysical properties, that is, low α/k ratio, and (3) improving the heat removal efficiency and/or limiting the power absorbed in the optic. 2.1

Cryogenic optics

As discussed above, thermal deformation is proportional to the coefficient of thermal expansion, α, and inversely proportional to the thermal conductivity, k. The choice of a suitable first optic material can therefore be guided by looking at the ratio α/k and choosing the material with the lowest ratio at the desired operating temperature [23]. Two scenarios have emerged as the most favorable: liquid-nitrogen-cooled silicon crystals and water-cooled, single-crystal diamonds. Table 13.2 lists α, k and α/k for silicon and diamond at 300 and 100 K. Based on the figure of merit α/k, one can expect that silicon crystals at cryogenic temperatures will significantly outperform silicon at room temperature [13,24], and that the performance of water-cooled (room temperature) diamond will be comparable to that of cryogenically cooled silicon [25,26]. Liquid-nitrogen cooling of silicon crystals has been successfully implemented at several ESRF and APS beamlines, and recently at SPring8. Following the original proposal by Bilderback [24], several experiments were carried out at second-generation synchrotrons that established the feasibility of using silicon crystals at cryogenic temperatures as high-heatload monochromators [30,31]. This research resulted in the development of indirectly-cooled silicon crystals that are now routinely used on ESRF beamlines. The crystals are side-cooled through contact with a copper block; liquid nitrogen flows through channels in the copper.

1 Germanium, silicon and diamond are available in near perfect crystal form. The high crystalline perfection of the optic material is required to preserve the undulator beam quality, that is, its high brilliance and low divergence.

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W.-K. Lee et al. Table 13.2 Coefficient of linear expansion α and thermal conductivity k for silicon and diamond at 100 and 300 K. The range for diamond reflects the values for type I and type IIa. Type I diamond contains nitrogen as the dominant impurity, while type IIa diamond is virtually free of impurities [27,28]. The figure of merit for thermal performance α/k is also listed. Data for this table were taken from [29] T = 300 K

α(10−6 /K) k(W/cm/K) α/k(10−6 cm/W)

T = 100 K

Silicon

Diamond

Silicon

Diamond

2.6 1.5 1.7

1.0 9–23 0.11–0.043

−0.4 8.8 −0.045

0.05 30–100 0.002–0.0005

Figure 13.10 Cryogenic silicon monochromator developed at the APS. The liquid nitrogen flows through cooling channels in the crystal. The seal between the crystal and the Invar manifold is achieved with indium-coated C-rings, which can be seen on the downstream manifold shown on the right. (Reproduced with permission from [36].)

Thermal contact is achieved by using a thin indium foil between the silicon and the copper. These monochromators typically operate under a thermal load of 100 W and a power density of 20 W/mm2 (normal incidence) [13,14,32,33]. The power incident on crystal monochromators at the APS is higher than at the ESRF (due to the higher particle beam energy, 7 GeV at the APS compared with 6 GeV at the ESRF) and can be in the range of 500 W to 1 kW, depending on the size of the white beam defining slits. At the APS, the normal incidence power density at the monochromator location is typically 150 W/mm2 . The higher thermal loads prompted the development of a crystal design, proposed by Knapp and collaborators [34], which incorporated cooling channels in the silicon (see Figure 13.10). The seal between the cooling manifold and the crystal is achieved by polishing the mating surfaces and using indium-coated C-rings [35]. Experiments conducted at the ESRF and the APS demonstrated the ability of direct cryogenically cooled silicon to withstand large thermal loads with no or minimal distortion [16,23,35–38]. Direct cooling of silicon crystals using liquid nitrogen has become the standard for the majority of the beamlines at the APS. The performance limits of this design have been recently established

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by testing the cryogenically cooled silicon crystal under the heat load delivered by two APS 2.4 m long undulators in tandem. The results show that if the beam is limited to the size of the radiation central cone, the crystal will perform well under the doubled power load [38]. 2.2

Diamond optics

High thermal conductivity and low expansion coefficient make single-crystal diamond an ideal material for high-heat-load monochromators [25,26]. Compared to cryogenic silicon, the use of diamond presents advantages and disadvantages. A water-cooled diamond monochromator is easier to build and operate than a liquid-nitrogen-cooled silicon monochromator, and, since diamond is less absorbing than silicon, the beam transmitted through a diamond monochromator can be used to feed experiment stations downstream (beam multiplexing) [39–41]. On the other hand, for a given (hkl) reflection, the energy bandpass of a diamond monochromator is roughly half that of silicon and results in a lower photon flux (photons/s) at the sample [42]. This reduction in bandpass may be a benefit or a disadvantage depending on the application. The available size and quality (from a diffraction point of view) of diamond crystals may also present a problem: the best commercially available synthetic diamond crystals with a (111) orientation are 8–10 mm by 5 mm in size and show mosaic spreads of the order of 1–3 arcseconds [43–45]. Efforts continue to improve the quality and increase the size of synthetic diamond crystals [46–50]. Berman and collaborators performed the first high-heat-load tests of an indirectly cooled diamond monochromator [51,52]. The diamond showed little distortion (1–2 arcseconds) under an incident power and power density of up to 75 W and 260 W/mm2 , respectively. An edge-cooled diamond crystal was tested at the ESRF focussed undulator beamline, where there was no significant thermal distortion of the crystal for 280 W of incident power and 3.5 kW/mm2 of incident power density [53,54]. At the ESRF, diamond monochromators have been installed as permanent components of the multiple station Troika beamline (see Figure 13.11), where the typical heat load is 80 W and 20 W/mm2 [39,55,56], and will be used at the Quadriga multiple station beamline [40,41]. At the APS, water-cooled diamond monochromators have been successfully tested to power loads of 280 W and 120 W/mm2 using one 2.4 m long undulator and to 600 W and 300 W/mm2 using two 2.4 m long undulators in tandem [43,45,57]. A diamond high-heat-load monochromator is routinely used at the high-energy-resolution beamline (sector 3) at the APS, where it has withstood up to 426 W and 178 W/mm2 with no appreciable thermal distortion [58]. Diamond monochromators have also been proposed for beamlines at SPring8 [17,46,59] and are currently installed in one beamline. 2.3

Geometrical reduction of the absorbed power density

Several schemes have been proposed to mitigate the thermal distortions of high-heat-load monochromators by reducing the power and surface power density absorbed by the first crystal. These approaches include: the use of thin (<1 mm) and ultrathin (10–20 µm) silicon and diamond crystals [17,20,60,61]; self-filtering crystals in extremely asymmetric diffraction and grazing-angle diffraction configurations [62]; rotary crystals with adjustable diffraction asymmetry [63]; and crystals in symmetric Bragg geometry but with the surface at a high inclination angle with respect to the diffraction planes (inclined geometry) [64,65]. The inclined geometry has been the most widely studied of these proposed solutions. Several experiments were performed to determine the high-heat-load performance of cooled, inclined

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Figure 13.11 Diamond monochromator at the Troika beamline (ESRF): (a) diamond crystal in a watercooled copper holder, with Ga/In eutectic to achieve thermal contact; (b) diamond placed in the undulator beam; the fluorescent beam spot is due to nitrogen impurities in the crystal. (Reproduced with permission from [39].)

silicon crystals, with inclination angles as high as 85◦ [66–69]. Substantial effort has also been devoted to understanding the diffraction characteristics of inclined monochromators [70–72]. The staff at SPring8 is pursuing an approach that combines an extension of the inclined diffraction geometry with a pin-post heat exchanger (see Section 2.4 below) for enhanced water cooling. In the inclined diffraction geometry, the surface of the crystal has a large angle with respect to the diffraction planes (typically 70◦ to 85◦ ) and the Bragg reflection is symmetric [64,70]. The extension of this geometry consists of a rotation of the (symmetrical) inclined crystal around the active reciprocal lattice vector, thereby introducing an asymmetry in the reflection. This asymmetry results in a further reduction of the glancing angle between the surface and the incoming beam as compared to the (symmetric) inclined geometry. The small incidence angle results in an extended beam footprint that decreases the surface power density by factors of up to 25 over the standard Bragg geometry. This diffraction geometry is called rotated inclined geometry [18,46]. At SPring8, a pin-post heat exchanger is used (see Figure 13.12), which causes turbulent flow in the coolant at a moderate pressure drop, to improve the heat transfer coefficient over that of standard cooling holes or channels [73,74]. The fabrication of the pin-post heat exchanger is an involved process that requires extensive machining of several layers of silicon and the subsequent bonding of the assembly [15,18,74,75]. Tests carried out on prototypes have indicated that this monochromator will perform adequately under the thermal load of SPring8 undulators (total power of the order of a few kW, and 300 W/mm2 normal incidence power density at 40 m) [18]. However, the tests have also shown that there are residual strains in the crystal due to the fabrication process

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Top plate (Si) with pin-post cell

Return

Supply

Coolant flow

Distributor (Si) Manifold (metal)

Figure 13.12 Schematic of the pin-post heat exchanger used in the water-cooled, rotated-inclinedgeometry silicon crystals at SPring8. (Reproduced with permission from [18].)

and the cooling water pressure. These manufacturing strains must be eliminated to provide the full brilliance of the beam to the experiment; efforts are currently under way to eliminate or reduce these strains.

2.4

Improved heat removal

Efficient heat removal is an essential component of the thermal management of high-heat-load monochromators. Jet cooling has been proposed as a means to improve the thermal transfer coefficient between the optic and the cooling fluid [76–80]. Crystals having internal channels with long ribs use the so-called “fin effect” to conduct heat away from the diffracting surface and increase the cooling area [17,81,82], while microchannels and the pin-post geometry heat exchanger improve the cooling efficiency by encouraging the turbulent flow of the coolant [73–75,83–87]. Another way to enhance the heat transfer coefficient is to use cooling channels fitted with a porous mesh; this approach has been pursued to improve the heat removal capacity in liquid-nitrogen-cooled optics [15,88–90]. The use of more efficient cooling fluids, such as liquid metals, has also been proposed [91,92], with most of the effort devoted to the implementation of liquid-gallium cooling [19,93–97].

2.5

Adaptive optics

Adaptive techniques attempt to compensate for the residual thermal deformations in the cooled optic. This approach includes mechanically bending the cooled crystal [85,98], applying gas pressure on the surface of the crystal [77,79], using a reverse thermal moment to flatten the crystal surface [99], and curving the second crystal in the monochromator to offset the thermal bowing of the first crystal [100]. Self-adaptive cooled crystals, where the cooling geometry and shape of the crystal are carefully chosen to produce reverse thermal moments that negate the thermal deformation induced by the X-ray beam, have also been proposed [101,102] and implemented [103,104].

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High-energy-resolution monochromators

On many beamlines, the first monochromator (e.g. the high-heat-load monochromator) is usually a Si(1 1 1) or Si(2 2 0) or some relatively low order (h, k, l) reflection for maximum throughput. These monochromators usually have an energy bandwidth -E/E ∼ 10−4 . For example, for Si(1 1 1) at 10 keV, the intrinsic energy bandpass is about 2 eV wide. For some experiments, X-ray beams with a high degree of monochromaticity (smaller bandpass) are desirable. In the study of lattice dynamics, for example, it is necessary to detect energy shifts of the scattered X-rays of the order of a few millielectron volts for an incoming beam with energy of about 10–20 keV. This requires a monochromatic beam with -E/E ∼ 10−7 . Highenergy-resolution monochromatic beams are also required for nuclear resonant scattering. There, synchrotron X-ray beams are used to excite a very narrow (∼neV) nuclear resonance. In this case, a high degree of monochromacity in the incoming synchrotron beam is needed to reduce the background noise. These two fields, inelastic scattering and nuclear resonant scattering, have driven the development of high-energy-resolution monochromators. From Bragg’s law (Eqn (23)), it can be shown that -E -λ ωacc -ϑ = = + E λ tan θB tan θB

(46)

The first term on the right comes from the angular acceptance (Darwin width) of the reflection. It is inherent to the particular reflection and wavelength. The second term on the right is due to the incoming beam divergence in the diffraction plane, -ϑ. Thus, in order to have good energy resolution, one would like to have θ ∼ 90◦ . For the X-ray range of interest here, this implies that a high-order reflection would be needed. Equation (46) also suggests that a small angular beam acceptance, ωacc , is desirable. In practice, however, the total throughput of the monochromator, that is, how many photons within the narrow bandwidth can be provided to the sample, has to be considered. If ωacc -ψ, then the bandpass will be dominated by the beam divergence. In that case, a large fraction of the incoming photons will be rejected by the high-energy-resolution monochromator. Therefore, in practice, the need for a narrow bandwidth is often balanced by the need to maximize the throughput. One common practice is to match the angular openings, such that ωacc ≈ -ψ. In cases where there is a pre-monochromator upstream, the common practice is to match the angular acceptance of the high-resolution monochromator to the output angle of the pre-monochromator. As noted in Section 1.2, the expression for ω0 (Eqn (36)) breaks down for θ ≈ 90◦ . At √ θ ≈ 90◦ , ω0 ∼ |χH | instead of ω0 ∼ χH as seen in Eqn (36). For the hard X-ray regime, χH is typically ∼ 10−6 . Therefore, the angular acceptances at θ ≈ 90◦ increases by as much as 102 –103 times. In general, there are two common schemes for achieving high-energy-resolution monochromators. The first is through the use of a single backscattering reflection, where θ ≈ 90◦ . This approach is used in many inelastic scattering experiments. The second approach is through the use of multiple crystal reflections. 3.1

Back-reflection monochromators

In 1982, Graeff and Materlik [105] showed that with a back-scattering geometry (θ ∼ 89.84◦ ), millielectron volt resolution in the hard X-ray regime can be achieved. Using the

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∆ Acceptance stripe for Si(8 8 8) Slope = /tan( ) Si(8 8 8) ( acc/tan( ))

∆

Slits (∆ϑ/tan( )) ∆ϑ (slit)

Figure 13.13 Schematic of a back-reflection monochromator and its corresponding Dumond diagram. Notice that since θ ∼ 90◦ , the acceptance band for the crystal lies almost horizontally, and thus, the energy resolution contribution from the beam divergence (-ϑ) is greatly reduced.

Si(8 8 8) reflection at ∼15.84 keV, they achieved an energy resolution better than 8.3 meV at HASYLAB. However, because they were working with a bending magnet beam and a relatively large particle beam source size and divergence, they had to use a small pinhole before the monochromator to reduce the effect of the beam divergence. This results in a heavy penalty on the monochromatic flux. The back-reflection geometry and its corresponding Dumond diagram is illustrated in Figure 13.13. Notice that as a result of θ ≈ 90◦ , the crystal transmission band is almost horizontal. A vertical cut across the band gives the ωacc /(tan θB ) contribution to the energy resolution as expressed in Eqn (46). An improvement of the Graeff and Materlik approach was suggested by Dorner and Peisl [106] and implemented by Dorner et al. [107]. The concept is to use a spherically bent crystal instead of a flat one to focus the beam and increase the beam acceptance. The radius of curvature is derived from the white beam divergence and size, and the desired geometry and energy resolution of the setup. Great care must be taken in the fabrication of the spherically bent crystal in order to minimize crystal strains. Elastic deformations in the lattice prevent the use of a thin curved disk. (The extinction depth of the X-rays is deep enough that they are sensitive to the lattice deformation.) Therefore, crosshatch strain relief cuts are often made on the diffraction surface [108]. These strain relief cuts unfortunately come at the expense of focus size and usable diffraction surface. Using a spherically bent monochromator and analyzer crystals in their “INELAX” beamline at HASYLAB, Burkel and others have been able to directly measure phonon dispersion curves of single-crystal diamond and beryllium [109,110]. Their setup is shown in Figure 13.14. The energy resolution of that beamline is in the 9–40 meV range, depending on the actual beam size and setup used [111]. With the advent of undulator radiation from third-generation synchrotron sources, the use of a bent crystal monochromator is no longer necessary because the inherent undulator

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I⬘ d I

L⬘

L d⬘

II

X VIII

V

IX VI IV

I

VII

III

Figure 13.14 Setup of the INELAX instrument at HASYLAB. The beam passes the premonochromator (III) and the monochromator (I) before it illuminates the sample (IV). The scattered intensity is focussed by the analyzer (II) into the detector (VI). The primary intensity is controlled by a monitor unit (V). (VII–X) indicate slit systems. Source to (I) is 38 m, L is 2.6 m, l = 8 m, l  = 2.5 m, d = 6 mm, d  = 4 mm. (Reproduced with permission from [122].)

beam divergence is small and is comparable or smaller than the Darwin acceptance angle of the back-reflection monochromator. (Spherically bent back-reflection crystal analysers are still needed in inelastic scattering experiments.) Using a flat Si(13 13 13) back-reflection monochromator, Verbeni et al. [112] at the ESRF have achieved an energy resolution of 450 ± 50 µeV at 25.70 keV. There are two main disadvantages to the use of back-reflection monochromators. The first is the spatial constraints of the geometry. In order for the sample to not be in the white beam path, long beam paths are usually required for practical white-monochromatic beam separation. (A channel-cut version of the back-reflection monochromator, which circumvents this problem, has been reported by Kushnir et al. [113] using the Si(7 7 7) reflection at 13.84 keV.) Second, because the reciprocal lattice vectors (h, k, l) only occur at discrete integer values, back-reflection monochromators with high energy resolution are not possible at all energies. (Although finding a material with the right d-spacing for the desired energy is theoretically possible, in reality, currently, the only commercially available materials that have the necessary degree of crystalline perfection and sizes required for high-energy-resolution X-ray monochromators are silicon and germanium.) Small energy variations can be accomplished by changing the crystal temperature, thereby changing the lattice spacing. However, the range of energy tunability is rather limited with this method. 3.2

Multireflection monochromators

An alternate approach to high-energy-resolution monochromators is to use multiple reflections. For example, Figure 13.15 shows a dispersive (+, −, −, +) geometry and its associated Dumond diagram. Notice that because the first two reflections are identical and in a nondispersive mode, they can be represented by just one stripe on the Dumond diagram. The same applies to the third and fourth reflections. Using this approach, with dual channel-cut Si(10 6 4) reflections, Faigel et al. [114] achieved a 5 meV energy resolution at 14.4 keV. By using channel-cut crystals in this way, the exit beam is collinear with the incoming beam. The disadvantage of this system, however, is that its angular acceptance is only 2 µrad, compared with the angular emittance of the Si(1 1 1) pre-monochromator of ∼20 µrad. Thus, the throughput of the monochromator is very low. This is illustrated in Figure 13.15(b), where

X-ray crystal optics (a)

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Si(1 1 1) pre-monochromator

High resolution Si(10 6 4) monochromator (b) 5 meV Si(10 6 4) Darwin width ~ 2 µrad

 Si(1 1 1) Darwin width ~ 20 µrad

Figure 13.15 High-energy-resolution monochromator used by Faigel et al. (a) [125], and its corresponding Dumond diagram (b). The solid black area denotes the transmitted photons, while the gray shaded areas denote the photons within the bandwidth (5 meV), which are not transmitted.

the gray shaded areas in the Dumond diagram represent all the “wasted” photons within the 5 meV bandwidth. A significant improvement to this design was proposed by Ishikawa et al. [115]. His design consists of two nested channel-cut crystals arranged in a (+n, +m, −m, −n) geometry, as shown in Figure 13.16(a). He proposed a Si(4 2 2) asymmetric reflection for the first crystal and a symmetric Si(12 2 2) for the second, for use at 14.4 keV. The first reflection has an acceptance angle that is matched to that of the incoming radiation, while the second reflection is a high-order near back-reflection. The third and fourth reflections are simply used to preserve the original beam size and direction. The key here is the use of asymmetry in the first crystal to increase the angular acceptance (see Eqn (37)). Using this approach, Toellner et al. [116] have built and tested several monochromators for use in nuclear resonant scattering. For 14.4 keV incident radiation, they used a asymmetric Si(4 2 2) channel-cut (20◦ asymmetry angle) together with a symmetric Si(10 6 4) channel-cut to achieve an energy resolution of 6.7 meV with an angular acceptance of 22 µrad. The corresponding Dumond diagram is shown in Figure 13.16(b). The asymmetric Si(4 2 2) reflection has an acceptance angle of about 22µrad, which is well matched to the Si(1 1 1) pre-monochromator upstream. The exit beam divergence of the Si(4 2 2), which serves as the incident beam to the Si(10 6 4) symmetric crystal, however, is only about 1.6µrad. The combination of this narrow exit divergence from the Si(4 2 2) and the narrow Darwin width of the symmetric Si(10 6 4) allows for an energy resolution of 6.7 meV. For a 23.8 keV (119 Sn resonance) incident radiation, Mooney et al. [117] used an asymmetric Si(3 3 3) channel-cut with a symmetric Si(5 5 5) channel-cut to achieve an energy resolution of 23 meV with an angular acceptance of 7µrad. This scheme has also been used for inelastic scattering by Macrander et al. [118].

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(a)

Symmetric Si(10 6 4)

Symmetric Si(10 6 4)

Asymmetric Si(4 2 2) 

(b)

Asymmetric Si(4 2 2) acceptance ~ 22 µrad

6.7 meV FWHM

Asymmetric Si(4 2 2) exit divergence ~ 1.6 µrad Symmetric Si(10 6 4) Darwin width ~ 2 µrad Si(1 1 1) Darwin width ~ 20 µrad

Figure 13.16 (a) Schematic of the high-energy-resolution monochromator as proposed by Ishikawa et al. [126]. (b) The Dumond diagram of the Ishikawa-type high-energy-resolution monochromator as built by Toellner et al. [127]. Here, in comparison with Figure 13.3, only a relatively small proportion (shaded gray area) of photons within the bandwidth are rejected. The key here is the asymmetric Si(4 2 2) reflection, which accepts all the photons within the narrow bandwidth from the Si(1 1 1) pre-monochromator and compresses them into a narrow exit divergence (1.6 µrad).

For cases in which the highest energy resolution is required, Toellner [119] and Chumakov et al. [120] have extended the approach by Faigel et al. [114] of using two high-order reflections in a (+m, +n) geometry. Instead of symmetric reflections, they use asymmetry to match the incoming beam divergence with the acceptance of the high-resolution monochromator and thus obtain a better throughput (see Figure 13.17(a)). Using Si(9 7 5) reflections at 14.4 keV, (θBragg = 80.4◦ ) with a miscut (asymmetry) angle of 75.4◦ , Chumakov et al. [120] achieved 1.65 meV energy resolution with an angular acceptance of 4 µrad. Using a higher degree of asymmetry, Toellner et al. [121] achieved 920 ± 110 µeV energy resolution with an angular

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(a)

Asymmetric Si(9 7 5)

Asymmetric Si(9 7 5)

(b) 20 10 0 –10 –20

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–1

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0.30 0.25 0.20 0.15 0.10 0.05

20 Ve r tic 10 0 al an gle –10 (µr ad –20 –2 )

–1

0 eV) y Energ (m

1

0.00 2

Figure 13.17 (a) Schematic of the highest energy resolution monochromators as built by Chumakov et al. [131] and Toellner et al. [132,133]. (b) three-dimensional transmission plot of the 660 µeV energy resolution monochromator built by Toellner [133], see text.

acceptance of 8.6 µrad (79.52◦ miscut angle for the first crystal and 78.4◦ miscut angle for the second). Pushing the degree of asymmetry even further (79.94◦ miscut for the first crystal and a 79.53◦ miscut on the second), Toellner [122] has succeeded in achieving an energy resolution of about 660µeV with an angular acceptance of 411.8 µrad. A three-dimensional transmission plot of this 660 µeV resolution monochromator is shown in Figure 13.17(b). At 23.8 keV, an energy resolution of 140 µeV has been reported [123]. Note that while such a scheme does achieve very high energy resolution, the peak reflectivity through the crystals tends to suffer because of the high degree of asymmetry required. For this reason, the use of a third and fourth reflection to regain the beam shape and direction, as described above, is not usually implemented. Recently, however, Ishikawa [124] has slightly modified this approach by using four separated crystals in a (+, −, −, +) geometry. (Unlike Faigel [114], the first two (and last two) reflections are not identical.) This allows him to use a lower asymmetry in each crystal, recover the beam direction and shape, and still enables him to achieve a 120 µeV resolution at 14.4 keV.

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3.3

General comments on high-energy-resolution monochromators

In general, multi-crystal high-energy-resolution monochromators (as described in Section 3.2) are more versatile than the back-reflection monochromators described in Section 3.1. On the other hand, the back-reflection monochromators have larger angular acceptances and thus can be useful in cases where the radiation source is more divergent. For example, back-reflection crystals (bent, crosshatched or diced) are frequently used as analyzers in inelastic scattering experiments [107,111,125,126]. A few remarks on the technical aspects of high-energy-resolution monochromators are in order here. First, excellent single crystalline material is required. For millielectron volt resolution in the 10–20 keV range, -E/E ∼ 10−7 . Thus, the crystalline material must have -d/d < 10−7 and be dislocation free. Currently, only silicon and germanium are commercially available with these specifications and usable sizes. The temperature stability of the crystals is also an important consideration. At room temperature, silicon has a thermal expansion coefficient, α, of about 2.6×10−6 K −1 . Since -d/d = α-T , temperature stability in the 10–100 mK is required. In cases where there is a high degree of asymmetry and the incidence angle is very small, the crystals must be polished (in a strain-free manner) to reduce any surface diffuse scattering. This demanding requirement has forced the channelcut arrangement to be abandoned for separated crystals so that the reflection surfaces could be well polished. For the case where multiple reflections are required, the angular resolution and stability of the crystals relative to one another can be quite daunting. For example, in the setup used by Toellner et al. [121], the exit beam after the first crystal has an angular divergence of 0.38 µrad and the second crystal has an angular acceptance of 0.55 µrad as a result of the asymmetry. This has forced the development of rotational stages with extremely high precision and stability [127].

4

X-ray phase retarders

Circularly polarized X-rays are routinely utilized in elastic and inelastic magnetic scattering and magnetic X-ray spectroscopy [128,129]. Circularly (or elliptically) polarized X-rays can be obtained in several ways: (1) by using certain X-ray or γ -ray emissions from radioactive sources, (2) by using off-axis (above or below the particle orbital plane) synchrotron radiation, (3) by using special insertion devices at the synchrotron, or (4) by using X-ray phase retarders (XPRs). Sakai and Ono [128] were able to isolate the Compton profile of unpaired electrons from the paired electrons in iron by using circularly polarized 120 keV γ -ray emission from 57 Co. The disadvantage of this technique is the very low flux and the limited choice of available energies. Cooper et al. [130] were able to repeat the measurement using off-axis synchrotron bending magnet radiation. While a wide range of energies was available, the flux here was still low because only a small portion of the total emitted radiation was used. Most of the beam, including the most intense, on-axis central portions, was not used. Special insertion devices [131], such as helical (or asymmetric) wigglers, can be used to generate elliptical or circularly polarized X-rays. Such devices can be very expensive and may adversely impact users who do not require circular or elliptical polarization. Furthermore, since the polarization occurs at the source, it is subject to depolarization from downstream beamline components, such as the monochromator. A comparison between the use of XPRs and special insertion devices for the production of circularly polarized X-rays has been studied by Lang et al. [132].

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X-ray phase plates or XPRs are used to manipulate the polarization states of electromagnetic radiation. In the optical regime, birefringent materials can be used to manipulate the polarization states. Unfortunately, in the X-ray regime, materials have negligible or no birefringent properties, except in near Bragg conditions. This effect is known as diffractive birefringence and was demonstrated by Hart and Lang [133] and Hattori et al. [134]. Skalicky and Malgrange [135] extended this study and showed that elliptically polarized X-rays can be generated from incoming linearly polarized X-rays by crystal diffraction. Baranova and Zel’dovich [136] have studied X-ray birefringence effects for multibeam processes. This section will concentrate on X-ray optics that produce circularly polarized radiation from incoming linearly polarized radiation. Readers who are interested in the production of linearly polarized radiation from unpolarized sources are referred to the article by Hart and Rodrigues [137]. On-axis radiation from a normal undulator is highly linearly polarized in the particle orbital plane. Thus, the use of linear polarizers upstream of the XPR is not necessary, although in many cases they are used to clean up/improve the degree of linear polarization. Dynamical diffraction theory shows that near the Bragg condition (where a single reciprocal lattice vector is excited), wave vectors for σ - and π-polarized radiation inside the crystal are of different lengths and can travel in slightly different directions. In addition, from Eqn (36), the Darwin widths are also different for the two polarization states. Near the Bragg condition, a total of eight possible waves exist inside the crystal: π σ , Kσ , Kπ , Kπ , Kσ , Kσ , Kπ KO,α H,α H,β H,α and KH,β , as illustrated in Figure 13.5. From O,α O,β O,β Section 1.2, it can be shown that σ,π KO,α

σ,π KH,α

σ,π KO,β

σ,π KH,β

    1/2 b χO  1 re λ b  η + η2 + +k 1+ = |C| 1/2 FH FH |b| |b| 2 2 πV       b χO  1 re λ 1/2 η − η2 + |C||b|−1/2 FH FH +k 1+ =− 2 πV |b| 2     1/2 1 re λ b b  χO  η − η2 + = |C| 1/2 FH FH +k 1+ 2 πV |b| |b| 2       χO  1 re λ b 1/2 η + η2 + |C||b|−1/2 FH FH +k 1+ =− 2 πV |b| 2

(47)

where C = 1 for σ -polarization and C = cos 2θ for π -polarization. The degree of circular polarization can be defined as σ Dπ 2DO,H O,H PC =  2   sin δϕ  σ   π 2 DO,H  + DO,H 

(48)

where Dσ (Dπ ) is the σ (π ) component of the diffracted (DH ) or forward-diffracted (DO ) beam, and δϕ is the phase difference between Dσ and Dπ . |PC | < 1 implies an incomplete circular polarization. Another useful measure of the XPR is the efficiency, which is defined as the ratio of the number of incoming photons to the number of circularly polarized photons. Remember that these measures of performance are not intrinsic to the XPR but depend on the conditions of the experiment, such as the degree of linear polarization and the angular

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divergence of the incoming beam. Nevertheless, the cited performance numbers can be useful for rough comparisons. X-ray phase retarders work by controlling the relative phase and amplitude of each of these waves inside the crystal that make up the external forward-diffracted or diffracted wave. In order to produce circularly polarized radiation from an incoming linearly polarized X-ray, three conditions must be satisfied: (1) the Kσ and Kπ waves inside the crystal must be coherently excited; (2) the Kσ and Kπ wave amplitudes must be equal at the exit surface; and (3) the phase difference between Kσ and Kπ must equal (2n + 1)π/2 at the exit surface. If the first condition is not satisfied, the waves will not interfere and no polarization rotation will occur. This condition is easily satisfied by tilting the diffraction plane relative to the incoming beam linear polarization direction. If either (2) or (3) is not satisfied, incomplete circular polarization (|PC | < 1) will occur. From Figure 13.5 and Eqn (30), XO > 0 for the α-branch and XO < 0 for the β-branch. Thus, the α- and β-waves are 180◦ out of phase. Also, it can be seen that Kσ leads Kπ for the α-branch, whereas Kσ lags Kπ for the β-branch. Therefore, unless the waves from one of these branches is suppressed, the phase difference between Kσ and Kπ from the α-branch will be offset by the phase difference from the β-branch, thereby producing linearly polarized X-rays. There are basically three types of XPRs in use (see Figure 13.18 and Table 13.3). Although the basic principles involved are similar, as described above, the actual realization and operation of these XPRs are quite different. Hirano et al. [138] has written a good review of the different types of XPRs.

Type I XPR

Type II XPR

(a) Incident polarization

Incident polarization

(b) Diffracted beam









Incident beam

Diffracted beam

Incident beam

Transmitted beam

Atomic planes Atomic planes

Type III XPR – Laue case (c)

Type III XPR – Bragg case (d) Incident polarization

Incident polarization 







Transmitted beam Incident beam

Incident beam

Figure 13.18 The different types of XPRs, see Table 13.3.

Transmitted beam

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Table 13.3 Types of XPRs commonly in use XPR type

I

II

III

Material type Operating point Beam Energy range Geometry

High Z At/near peak Diffracted/forward-diffracted >15 keV Laue

Any Near peak Diffracted 6–15 Bragg

Low Z Far from peak Forward-diffracted 6–15 Either

4.1

Type I: high-Z, Laue geometry XPRs

This type of XPR operates at or near the peak reflectivity, where the birefringence is the strongest. The discussion will be based on the diffracted beam case, although either the diffracted or forward-diffracted beams can be used. Recall that, in the Laue geometry, tiepoints on both the α- and β-branches are excited. For the symmetric Laue geometry (b = 1), the phase difference for the diffracted beam between the σ - and π-waves in the crystal (from the α-branch dispersion surface) at the exit surface is given by δϕ ≡ ϕσ − ϕπ =

σ 2πt (KH,α

π − KH,α ) · nˆ

re tλ|FH | = √ V γ O γH

" ησ2

+1−

#

 ησ2

+ cos2 2θ (49)

where ησ is the σ -polarized value of η (C = 1 in Eqn (47)), t is the crystal thickness and nˆ is the unit normal to the surface. As mentioned above, the phase retardation from the β-branch is opposite that from the α-branch. Thus, in order to obtain the required phase shift, it is necessary to suppress the waves from the β-branch. This is accomplished by taking advantage of the anomalous absorption of the β-branch waves and anomalous transmission of the α-waves (Borrman effect), as described in Section 1.2, Eqns (42–43). Therefore, in this type of XPR, absorptive crystals are required and reflections in which all the atoms scatter in phase are best suited. The first practical X-ray phase plate was made by Hart [139]. Using a thin silicon crystal in Laue geometry, he managed to produce elliptically polarized radiation from a linearly polarized incoming beam. Hart chose a 72 µm thick piece of silicon such that the accumulated π σ phase difference between KH,α and KH,α (Si(2 2 0) reflection at 8 keV) was 3π/2. A threequarter wave phase difference was chosen instead of a one-quarter-wave phase difference so that, due to the additional thickness, the waves from the β-branch dispersion surface would be further attenuated relative to the α-waves. Figure 13.19 shows the reflectivity and phase retardation for Hart’s XPR. Despite the added thickness, the β-waves were still insufficiently attenuated and Pendellosung effects are clearly visible. Furthermore, the β-waves also caused σ and Kπ to be less than 3π/2 resulting in elliptical polarization. the phase lag between KH H Hart’s experiment was reproduced by Annaka [140] and Brummer et al. [141] with slight differences and more quantitative analyses of the beam polarization. These early experiments were performed using laboratory X-ray sources with CuK α radiation. A linear polarizer was required upstream of the XPR to convert the unpolarized radiation from the X-ray source to a linearly polarized radiation necessary for the phase plate. The first application of an X-ray phase retarder at a synchrotron was by Golovchenko et al. [142]. Using a Si(2 2 0) (11/4) wave phase plate (Laue geometry, diffracted beam)

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Figure 13.19 Calculated reflectivities for σ - (thin solid line) and π- (dotted line) polarizations and the phase retardation δϕ (bold solid line) for symmetric Laue (2 2 0) reflection of 72 µm thick silicon at 8 keV.

at 17.5 keV, they used the circularly polarized radiation to perform a magnetic-Compton scattering experiment. Here, an upstream linear polarizer was not necessary because the on-axis radiation from the synchrotron was linearly polarized. They report better than 75% circular polarization from the phase plate, with 15% efficiency. Using this type of XPR, Mills [143] has studied the production of circularly polarized X-rays in the 20–55 keV range. Mills’ monolithic crystal design incorporates the Laue phase plate together with a subsequent Bragg reflection from the same crystalline planes. A weak link in the monolith allows for the index of refraction correction between the Laue and Bragg reflections. One advantage of this monolithic design is that the circularly polarized beam is parallel to the incoming (linearly polarized) beam. Consider the dependence of the phase lag, δϕ, on the variable η, which is directly related to -θ. From Eqn (49) t (-θ)2 |FH |

η < cos 2θ :

δϕ ∝

η∼1:

δϕ ∝ t-θ

(50) (51)

Equations (50) and (51) show that within the reflection range (|η| < 1) the phase retardation, δϕ, is strongly dependent on the incident beam divergence (via η). Rays that hit the polarizer at different angles will result in a different accumulated phase retardation. This decreases the degree of circular polarization in the exit beam. Thus, incident beam angular collimation is essential if a high degree of circular polarization is desired. In Hart’s original 72 µm phase plate, the β-waves were insufficiently attenuated, as seen in Figure 13.19. As a result, a portion of the diffracted beam is linearly polarized and the degree of circular polarization decreases. In order to further attenuate the β-waves, one can either increase the thickness of the silicon or use a material with a higher absorption coefficient (higher Z). In either case, the efficiency will decrease due to the increased absorption. Increasing the thickness of the silicon will, however, also increase the phase difference sensitivity to the incoming beam divergence (Eqns (50) and (51)). Therefore, use of a higher

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Figure 13.20 Calculated reflectivities for σ - (thin solid line) and π- (dotted line) polarizations and the phase retardation δϕ (bold solid line) for symmetric Laue (2 2 0) reflection of 180 µm thick germanium at 20 keV.

Z material, such as germanium, is better [144]. This also helps in decreasing the phase difference sensitivity to beam divergence due to the larger structure factor for germanium. Therefore, from Eqn (50), thin crystals of high-Z materials and large structure factors are advantageous. The helicity of the circularly polarized radiation can be changed by rotating either the incoming polarization or the XPR by 90◦ . (Under some circumstances, it is possible to take advantage of crystal symmetry to change the helicity by rotations of less than 90◦ [145].) The effective thickness of the crystal can be changed by rotating about the reciprocal lattice vector. This gives the phase retarder a small amount of energy tunability. For good efficiency and ease of fabrication, this type of XPR is appropriate for energies >15 keV. At lower energies, the losses to absorption become significant and the optimal thicknesses becomes sufficiently thin (tens of µm) to make fabrication a challenge. Although this type of XPR is usually operated at the Bragg peak, Keller and Stern [146] have suggested that, in the case of a three-quarter wave silicon XPR, operation slightly off the Bragg peak may be beneficial. A typical performance of this type of XPR is shown in Figure 13.20. Here, a 180 µm thick piece of germanium is used as a quarter wave plate at 20 keV using the Laue (2 2 0) reflection. The peak reflectivities remain in the 15–20% range, and, for an incidence angular divergence of ±5 µrad, the phase difference remains between 85◦ and 90◦ . (Compare this with Figure 13.19, where a ±5 µrad incident angular divergence will result in a 60◦ variation in the phase difference.) 4.2

Type II: Bragg geometry, diffracted-beam XPRs

In this case, only one dispersion branch for each polarization state (α or β) is excited at each incident angle. In 1984, Brummer et al. [147] demonstrated the polarization effect for X-rays in the Bragg geometry diffracted beam. This type of XPR operates on a different principle than the others since, within the Bragg reflection, there are no real wave-vector solutions to the set of equations describing dynamical diffraction, as discussed in Section 1.2 of this

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Figure 13.21 Calculated reflectivities for σ - (thin solid line) and π- (dotted line) polarizations and the phase retardation δϕ (bold solid line) for a symmetric Bragg (2 2 0) reflection with silicon at 8 keV.

chapter. Equation (47) thus does not apply within this region of total reflection for the Bragg case. Instead, this XPR operates by taking advantage of the following: (1) the reflection widths of the σ - and π-polarizations differ, and (2) the phase of the reflected wave changes by 180◦ across the region of total reflection. This is illustrated in Figure 13.21, which shows the reflectivity and phase difference between the σ - and π-polarized waves for a silicon (2 2 0) Bragg reflection at 8 keV. Within the reflection widths, however, the phase difference between the σ - and π-polarized radiation is usually several times smaller than the interesting value of π/2. In order for the σ - and π-wave amplitudes to be equal, this device must be operated within the narrower π-polarization reflectance curve. Within this range (1 ≤ η ≤ cos 2θ ), the phase difference between the σ - and π-waves is approximately given by
 π 1 δϕ = ϕσ − ϕπ = (52) 1− ησ 2 |cos 2θB | In order to accumulate sufficient phase difference to obtain circularly polarized radiation (π/2), it is necessary to undergo multiple Bragg reflections. By using a five-bounce channelcut crystal, Batterman [148] was able to produce circularly polarized radiation from a linearly polarized incoming beam. Shastri et al. [149] have used this type of XPR to study circular magnetic X-ray dichroism near the Fe K edge (7.112 keV). This type of XPR has the highest efficiency of all the XPRs considered in this section. Shastri et al. [149] quoted a 56% efficiency for their four-bounce channel-cut circular polarizer. The reason is simply that it operates near the peak Bragg reflectivity, which can be very close to unity. Unlike the other types of XPRs, the photoelectric absorption inside the crystal is severely reduced due to the limited penetration of the evanescent waves into the crystal. The helicity of the circularly polarized X-ray can be changed by simply going to a different side of the rocking curve; that is, η > 0 or η < 0. Thus, helicity change can be accomplished by a small rotation of the order of arc-seconds instead of 90◦ as required in the Laue diffracted beam (type I) case. One disadvantage of this approach is that it lacks energy tunability. The multibounce channel-cut crystal works as a quarter-wave plate only

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within a very narrow range of X-ray energy. In this case, δϕ ∝ -θ . Thus, good angular collimation of the incoming beam is necessary. For good reflectivity of the π-waves, the incidence angle must not be too close to 45◦ , while it cannot be too small either, otherwise the phase retardation per reflection (Eqn (52)) will be very small. With currently available crystalline materials, this type of XPR is appropriate in the 6–15 keV range. 4.3

Type III: low-Z, transmitted beam XPRs

An interesting type of XPR was suggested by Dmitrienko and Belyakov [150–152]. By looking at the forward-diffracted beam (instead of the diffracted beam), they showed theoretically that, in both the Laue and Bragg geometries, angular positions relatively far away (|η|  1) from the reflectivity peaks are excellent operating points for producing circularly polarized radiation. In this region, the diffractive birefringence is not as large as near the reflectivity peak, but it varies very slowly with the incoming beam direction. Thus, the detrimental effect of incoming beam divergence is reduced. In addition, away from the reflectance peak, perfect crystalline structure is not required for the birefringence effect. Crystals with some mosaic structure can still be used [150]. The phase difference between the σ - and π-polarized waves inside the crystal is given by Bragg case: δϕ = ϕσ − ϕπ =

σ 2πt (KO

π − KO ) · nˆ

re tλ|FH | =± √ V |γO γH |

σ 2πt (KO

π − KO ) · nˆ

re tλ|FH | =± √ V |γO γH |

Laue case: δϕ = ϕσ − ϕπ =

# "  2 2 2 ησ − cos 2θ − ησ − 1 # "  2 2 2 ησ + 1 − ησ + cos 2θ (53)

where the positive sign is for the α-branch, and the negative sign is for the β-branch. Here, the phase retarder is operated relatively far away from the total reflection range, and only one dispersion branch (both polarizations) contributes significantly to the forward-diffracted waves. At low incidence angles, the α-branch dominates, and, at high incidence angles, the β-branch dominates. Thus, use of high-Z materials and/or thick crystals is not necessary to suppress the β-waves via anomalous absorption. In fact, from Eqn (44), the absorption coefficient approaches the normal value µ0 in this region. There is no anomalous absorption or transmission, and thus, for good transmissivity, low-Z materials are preferred. Recall that on the α-branch, Kσ > Kπ , while on the β-branch, Kπ > Kσ . Helicity change is accomplished by going to different sides of the reflectance curve, which also means activating different dispersion branches. In both the Bragg and Laue cases, far from the reflectance peak (|η|  1), Eqn (53) reduces to δϕ = ±

re tλ|FH | sin2 2θB π re λ3 [FH FH ] sin 2θB t = ± √ 2 γH 2V |γO γH |ησ (π V )2 -θ

(54)

Equation (54) clearly shows the 1/-θ dependence in the phase difference. This type of XPR thus has a relatively weak dependence on the incoming beam divergence. The transmissivity

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0.25

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0 ∆ (µrad)

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1000

Figure 13.22 Calculated reflectivities for σ - (thin solid line) and π- (dotted line) polarizations and the phase retardation δϕ (bold solid line) using Eqn (55) for symmetric Laue (2 2 0) reflection of 1 mm thick diamond at 10 keV.

and phase difference for a 1 mm thick diamond XPR at 10 keV near the the diamond Bragg (2 2 0) reflection is plotted in Figure 13.22. This type of XPR was built by Hirano and co-workers in 1991 [153]. Using a 62 µm thick silicon crystal in symmetric Bragg (2 2 0) reflection at CuK α radiation, they reported PC ∼ 0.9–0.96. The calculated transmission of their XPR was about 36%. Subsequent publications by the same group show that this phase plate can be used as a circular polarizer in the 7.7–8.8 keV range with PC > 0.9. This group has studied the performance of transmission-type XPRs in both the Laue and Bragg geometries with silicon, diamond and LiF crystals [154–157]. Measured transmissivities ranged from 5% to 25%. Taking advantage of its weak dependence on the incoming beam divergence, Giles and co-workers [158–160] have been able to incorporate this type of XPR into an energy dispersive synchrotron beamline. Here, the angle between the XPR diffraction plane is not at the ideal 45◦ to the dispersive monochromator diffraction plane [160]. Instead, small variations of the angle are used to achieve a nondispersive arrangement between the XPR and the bent (dispersive) monochromator. They have used diamond XPRs in transmission in both the Laue and Bragg geometries in the 6.4–8.6 keV range. Even with this energy dispersive setup, they still reported a PC ∼ 70–80% [158–160]. This group has also confirmed that mosaic crystals can be used for this type of XPR. Using beryllium crystals with a mosaicity of about 80 arcsecs, they managed to produce circularly polarized X-rays with PC ∼ 63% in their energy dispersive beamline [161,162]. As mentioned above, for good efficiency, low-Z materials are best suited for this type of XPR. From Eqn (54), it can be seen that the required thickness increases rapidly with increasing energy. For a given energy, thicker crystals are advantageous because, from Eqn (54) and Figure 13.22, thicker crystals would push the operating point (|-θ |) to larger values where the phase retardation is less sensitive to incoming beam divergence. The loss to absorption does not increase by much, as can be seen in Figure 13.22. Currently, good quality low-Z materials, such as Be, LiF and diamond, that are of a usable size are limited in thickness to about 1 mm. Therefore, at present, this type of XPR is limited in use to energies <15 keV.

X-ray crystal optics 4.4

405

General comparisons of various XPRs

In the higher energy range (>15 keV), type I (Laue geometry, diffracted beam) XPRs are commonly used. Here, the choice of high-Z materials, such as germanium, tend to give a better degree of circular polarization because of their higher attenuation of the β-branch radiation. The efficiency depends on the required PC , which dictates the required crystal thickness. The cited efficiencies from the literature for this type of XPR range from 5% to 25%. In the 6–15 keV range, Bragg diffracted beam XPRs (type II) or low-Z forward-diffracted beam XPRs (type III) can be used. Bragg diffracted beam XPRs are more efficient (cited efficiencies of ∼56%), compared to the low-Z forward diffracted beam XPRs (cited efficiencies ∼5–25%) . However, low-Z forward-diffracted beam XPRs tend to produce beams with much better PC due to its weaker dependence on the incoming beam divergence. Both types have the ability to change beam helicity via a small angular rotation. The advent of synchrotron radiation and insertion devices have clearly contributed greatly to the performance of XPRs. For one, on-axis synchrotron radiation is linearly polarized in the orbital plane. As described above, these XPRs require linearly polarized X-rays as input. Thus, an additional linear polarizer, which would reduce the beam intensity, is not required (although in some cases, where a high degree of circular polarization is required, a linear polarizer is used to improve the degree of linear polarization). Second, the degree of circular polarization obtainable depends on the incoming beam divergence. Thus, the advent of undulators, with their small beam angular divergence, greatly improves the performance of these XPRs.

5

Crystal focussing optics

The higher brilliance and smaller beam sizes available at third-generation synchrotron radiation sources have also greatly benefited the development of X-ray focussing optics [163,164]. Many focussing schemes, such as Kirkpatrick–Baez mirrors [165–173], capillary optics [174–181], and Fresnel zone plates (FZP) [182–194] have seen a marked improvement in performance, partly due to the advantageous characteristics of undulator radiation and partly due to improvements in optics fabrication techniques. In other cases, third-generation sources have made possible the development of long-proposed optics, such as compound refractive lenses [195–203]. In this section, crystal-based focussing optics, namely, sagittal focussing and Bragg–Fresnel lenses (BFL) will be discussed in detail. 5.1

Crystal sagittal focussing

The original impetus for sagittal (horizontal) focussing of synchrotron radiation came about due to the large horizontal angular divergences of bending magnet radiation. Bending magnet beamlines typically have angular acceptances of a few milliradians. At distances of 30–60 m, the unfocussed beam size is a few centimeters wide, which is usually much larger than the sample or vertical beam sizes. Thus, it is desirable to focus the beam horizontally. Sparks et al. [204] were the first to investigate sagittal focussing using crystals for synchrotron radiation. They showed that with a cylindrical bend, the angular errors in the diffraction plane (-θ) are -θ =

ψ 2 (1 + M)(3M − 1) 8M 2 sin θB

(55)

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where the source magnification, M = source distance/focus distance and 2^ is the horizontal angular divergence. These errors decrease the efficiency of the crystal. Rays with angular errors larger than the Darwin widths would not be reflected. For a cylindrical bend, the minimal angular errors are achieved by having a source magnification M ∼ 1/3. By using conical instead of cylindrical bends, it is possible to relax the M ∼ 1/3 constraint [205]. The horizontal (-x) and vertical (-y) blurring of the focus spot, due to the cylindrical approximation, are given by Ice and Sparks [206]: -x =

F1 ψ 3 (1 + M)2 4M 2 θ 2

-y =

F1 ψ 2 (1 + M) 2θ

(56)

where F1 is the distance to the source. The main problem with sagittal focussing is that of anti-clastic bending: a bend in one direction induces a (often undesirable) bend in the transverse direction. This is due to the non-zero off diagonal terms of the material elasticity tensor, and for isotropic materials, is quantified by the Poisson ratio. Sparks tackled this problem by the use of stiffening ribs behind the thin bent crystal [207]. However, the disadvantage of this approach is that the area under the stiffening ribs is flat and thus, does not focus. The focus spot size is therefore limited by the width of the ribs. Batterman [208], and Mills [209] used an alternate design whereby narrow slots (which produce weak links for bending) are cut into the diffraction surface. The diffraction surface is then a polygonal approximation to the cylinder. The diffracting part of the crystal is flat, and its width limits the focus size. Variations of these two approaches have been used by several groups [210–212]. An alternative solution was suggested by Kushnir et al. [213]. He showed that if the dimensions of the bent crystal was chosen such that the length/width ratio were large (>7 for the case of Si(111)), or such that the ratio was close to the “golden value” (2.36, for Si(111)), there would be no anti-clastic bending in the middle portion of the crystal. In such an approach, the stiffening ribs or weak-link slots are not required and the focus spot size can be smaller. Another challenge in sagittal focussing is the ability to achieve a uniform bend radius. It can be shown that for a given bending moment, the radius of curvature is proportional to the third power of the crystal thickness [214]. Thus, small errors in crystal fabrication can lead to large local differences in the bend radius and severely degrade the focussing. Due to the excellent angular collimation of the undulator radiation (∼102 times smaller than bending magnet radiation in the horizontal direction), the angular errors in the sagittal crystals and the blurring of the focus size are much (104 –106 times) smaller (see Eqns. 55 and 56). Furthermore, since the beam size is correspondingly small, the parts of the crystal for which one has to maintain the correct shape and be free of anti-clastic bending is also small and thus, much easier to achieve. Schulze et al. [215] have taken advantage of this and made a relatively simple sagittal focussing crystal for an undulator beamline by cutting a thin web in a thick crystal. The thick sides of the crystal act as the stiffening ribs and sufficiently reduce the anti-clastic bending in the middle of the thin web. A horizontal focus spot size of 20 µm (FWHM) was achieved. 5.2

Bragg–Fresnel lenses

Bragg–Fresnel lenses (BFL) are reflection optics that combine Bragg reflection from a single crystal or multilayer structure together with the principle of Fresnel zones [216]. They were first proposed by Aristov and collaborators [217–219]. BFL are closely related to FZP, which consist of concentric rings of two alternating materials. Although FZP are not crystal-based

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F rn

f

Focal point

Fresnel zone plate

Figure 13.23 Schematic of a zone plate illuminated by a source at infinity.

optics, it is useful to describe them here, given their close relation to BFL. Given a source at infinity and its image at the focal point of the FZP, the radii of the rings are chosen so that the optical path from the source to the image through two successive rings differs by λ/2. From Figure 13.23 λ 2

(57)

F 2 = f 2 + rn2

(58)

F =f +n

rn2 = nf λ +

n 2 λ2 4

(59)

where rn is the radius of the nth ring, f is the focal length of the primary (first order) focus of the FZP, and λ is the wavelength of the incident radiation. Equation (59) determines the radii of the Fresnel zones. For typical X-ray zone plates, nλ/f 1, and one can approximate  rn = nf λ (60) The area of a ring is then   2 A = π rn+1 − rn2 = πf λ

(61)

All zones have the same area and contribute equally to the amplitude of the transmitted wave. The contributions at the image point from two successive rings have a relative phase of π due to the optical path difference, see Eqn (57). If the rings are made of the same material, these contributions interfere destructively. In an amplitude FZP, the alternating zones are made of transmissive and absorbing materials. The contributions from all the transparent rings are then in phase and interfere constructively at the image, thus resulting in focussing. In a phase FZP, the alternating materials are chosen so that the radiation will acquire an additional relative phase shift of π in going through two adjacent rings. The total relative phase shift between two successive rings is then 2π , and focussing is obtained at the image resulting from the constructive interference of the contributions from all the zones.

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Image formation for Fresnel zone plates follows the same rules as for refractive lenses [220,221]. The object distance p and image distance q are related by the thin lens equation 1 1 1 + = p q f

(62)

From Eqn (60), f is given by f =

r12 λ

(63)

where r1 is the radius of the first zone. Higher order foci occur at distances f/m, for odd m > 1 [222]. The efficiency of a zone plate is defined as the ratio of the intensity in the mth order focus to the intensity incident on the optic [223]. The ideal first-order efficiency of an amplitude FZP is 1/π 2 ≈ 10%. For an ideal phase FZP, the amplitude at the image is twice that of an amplitude FZP, and the first-order efficiency is then 4/π 2 ≈ 40%. In both cases, the efficiency of the mth order focus will be reduced by 1/m2 compared to the primary focus [220,222,223]. The intensity of the undiffracted zero order is given by the amplitude at infinity of the plane wave that propagates from the FZP. In the case of an amplitude zone plate, the amplitude at infinity is 1/2 of the incident amplitude, since the alternating transparent and absorbing rings have the same area and the contributions from all transmissive zones are in phase. The intensity of the zero order of an amplitude FZP is then at least 25%. For an ideal phase FZP, there is a relative phase shift of π between sections of the wavefront that pass through adjacent zones. The contributions at infinity from two adjacent zones interfere destructively, and there is no zero-order amplitude. Thus an important advantage of the phase FZP is the absence of undiffracted zero-order background [223]. The resolving power δθ (Rayleigh criterion) of the zone plate is given by [220,224]: δθ = 1.22

λ 2rn

(64)

The spatial resolution δ is then δ = f δθ = 1.22

fλ 2rn

(65)

From Eqn (60) 2 f λ = rn+1 − rn2

(66)

f λ = (rn+1 + rn )(rn+1 − rn ) ≈ 2rn -rn

(67)

⇒ δ = 1.22-rn

(68)

The spatial resolution of the zone plate is approximately equal to the width -rn of the narrowest zone. For higher order foci m, the resolution is δ/m [221].

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(b)

Figure 13.24 Scanning electron microscope images of BFLs used at the ESRF: (a) linear BFL; (b) circular BFL. (Reproduced with permission from [246].)

The geometrical design parameters of the FZP are determined by the choice of resolution δ, wavelength of operation λ, and primary focal length f : smallest zone width zone plate diameter number of zones

δ 1.22 fλ 2rn = -rn

-rn =

n=

rn2 fλ

(69) (70) (71)

BFL couple Bragg reflection with FZP by having the Fresnel zone structures fabricated on the reflection surface itself. The ideal BFL consists of elliptical Fresnel zones [225–227], and focusses the X-rays by modulating both the amplitude and the phase of the wavefront. More generally used are the linear BFL in sagittal focussing geometry [228], where the linear Fresnel zones are parallel to the plane of Bragg diffraction, and the BFL with circular zones in Bragg back-reflection geometry [229], see Figure 13.24. The geometrical parameters of the linear and circular Fresnel zones are given by Eqn (60). The linear BFL focusses the incident beam into a line, while the circular lens produces 2D focussing. In these two configurations, the BFLs act purely as phase optics, thus resulting in a higher focussing efficiency compared to elliptical lenses. The maximum theoretical first-order efficiency for a phase BFL is 40%; typical experimental efficiencies range from 25% to 35% [230,231]. For elliptical lenses, the typical measured efficiency is 10–16% [226,232]. The phase modulation in a BFL is achieved by the depth of the grooves that form the Fresnel zones. The phase difference -φ between two sections of the wavefront that reflect from adjacent zones is given by [219,233] 4π hd|χO | (72) λ2 where h is the depth of the zones and χO is the zeroth-order Fourier component of the crystal polarizability. As is the case for the transmission Fresnel zone plates, the phase difference is set to π, so that the contributions from adjacent Fresnel zones interfere constructively. The depth hπ that is required is then given by -φ =

hπ =

λ2 4d|χO |

(73)

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The crystal polarizability is given by (see Eqn (20)) |χO | =

re λ2 FO πV

(74)

where FO is the structure factor and V is the unit-cell volume. Combining Eqns (73) and (74) hπ =

πV 4dre FO

(75)

The depth hπ of the groove of the Fresnel zones is then independent of the wavelength and, for a given crystal, depends only on the d-spacing of the desired Bragg reflection. For Si(1 1 1), the required depth is 1.26 µm. For a phase lens, the efficiency η is given by [223,234] η=

2(1 − cos -φ) m2 π 2

(76)

where m is the order of Fresnel diffraction. For a linear BFL, the phase shift -φ between adjacent zones is independent of the energy, and the lens will perform equally well over a wide energy range, with a focal distance that depends on the energy as given by Eqn (63). Circular BFLs are designed for near back-scattering geometry, where λ = 2d, and thus will only approach their maximum theoretical efficiency at one energy. Operation of a circular BFL at discrete energies corresponding to higher orders of back-scattering is possible, with a reduction of the focussing efficiency. In contrast, the geometrical parameters of an elliptical BFL, that is, the radii of the ellipses, depend on the energy [227]; elliptical lenses will only focus at the energy for which they were designed. The standard fabrication procedure for single-crystal or multilayer BFLs comprises three main steps: electron beam lithography to generate a mask, mask transfer by optical lithography or metal sputtering, and reactive ion etching of the pattern into the BFL substrate [235–237]. As is the case for FZPs, the resolution of the lens will be determined by the width of the narrowest zone. In most cases, this smallest dimension is 0.25–0.5 µm. Bragg–Fresnel lenses can be used to focus either white or monochromatic X-ray radiation. Tests have been carried out at the ESRF to determine the thermal stability and resistance of BFLs subject to the undulator X-ray beam. There was no change in performance for a circular Si BFL under a power load of 100 W with a heat flux of 12 W/mm2 [234]. The lens was in air and not cooled, and its temperature stabilized at 420◦ C. In a different experiment, a contact-cooled elliptical multilayer BFL was shown to be stable under the undulator white beam [238]. Linear silicon single-crystal BFLs have been tested for energies ranging from 2 to 100 keV, on undulator, wiggler and bending magnet beamlines [231,237,239–242], achieving linear foci of 1–5 µm, with efficiencies of 25–35%. Focussing by linear multilayer BFLs in the range of 8–14 keV has also been demonstrated, with an efficiency of 25% [243]. Some of the applications that have been implemented using linear BFLs are microfluorescence analysis [243,244], transmission microscopy [243], and high-pressure powder microdiffraction [245]. A linear lens has also been used as the first crystal in a double-crystal diffractometer, thus providing high-resolution information on the lattice distortions of a semiconductor second crystal [241,246]. Two-dimensional focussing has been demonstrated using two linear BFLs in a Kirkpatrick–Baez arrangement [236,247,248] and by meridional bending of a linear

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3 µm

5 µm

Figure 13.25 X-ray image of a free-standing 0.5 µm gold grid taken at 9.5 keV using a Ge circular BFL in back-scattering geometry. The gold grid was supported by a mesh with 15 µm pitch size and 3 µm bars. The gold 0.5 µm width grating is clearly resolved in the open areas and underneath the 3 µm mesh bars. (Reproduced with permission from [253].)

BFL [249]. Recently the use of linear BFLs in the development of hard X-ray phase contrast microtomography has been proposed [250]. Circular BFLs fabricated on Si and Ge single crystals have been used for two-dimensional focussing of white and monochromatic undulator X-ray beams, resulting in focal spots down to 0.7×0.7 µm2 and efficiencies up to 25–30% [229,234,246,251]. The white undulator beam at the ESRF has been characterized using one of these lenses [234,252]. Monochromatic beam applications include a small angle X-ray scattering camera capable of a 1.5×2 µm2 focussed spot with an efficiency of 25% [230]; a submicron fluorescence probe [251]; and a phasecontrast imaging microscope with a resolution of less than 1 µm [253], see Figure 13.25. Scanning microscopes have also been developed using elliptical BFLs fabricated on multilayer substrates. The microscopes operate in transmission or fluorescence mode, typically in the range of 8–14 keV, with focal spots ranging from 1 µm in diameter to 6 × 6 µm2 [226,227,232,238,254]. Recent developments in the field of Bragg–Fresnel optics include efforts to improve the resolution and the flux delivered by the lenses. The resolution of a BFL is determined by the width of the outermost Fresnel zone (see Eqn (68)), while the efficiency of the lens will be affected by the uniformity of the thickness (depth) of the zones. The smallest linewidth that can be achieved with uniform zone thickness is limited by the technical challenge of etching the narrow, deep outside zones at the same faster rate as the wider central zones. To overcome this disparity in etching rates, Li and collaborators have fabricated BFLs on GaAsbased heterostructures [237]. The semiconductor heterostructures were epitaxially grown and incorporated a built-in AlAs/AlGaAs etch stop layer under the GaAs top layer. The top layer had the exact thickness required for a (1 1 1) GaAs BFL. The etching process can then be

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allowed to proceed until the thinner outside zones are etched, while the central wider zones will only be etched to the etch stop layer, thus ensuring the correct depth of the zones. This fabrication technique shows promise for developing BFLs with uniform zone thickness and narrower outer zones. Compound elliptical multilayer lenses have been fabricated in an attempt to increase the flux at the focus. In these lenses, two elliptical patterns are etched on the multilayer substrate, an inner and an outer ellipse, designed in such a way that the third-order focus of the outer lens coincides with the first-order focus of the inner lens. The result is a larger aperture and thus an increase in the flux delivered at the focus [227,254]. One such lens has shown an increase in focussed intensity by a factor of 2 at 12 keV over a non-compound lens [227]. In summary, crystal-based diffraction focussing optics, that is, sagittal focussing crystals and BFLs, have demonstrated the capability to focus hard X-rays over a wide energy range and with high efficiency. They form part of the tools available to microprobe and microimaging applications at third-generation synchrotron radiation sources.

6

Conclusion

The latest generation of synchrotron radiation sources has compelled researchers to rethink and re-evaluate all aspects of X-ray optical components from their fabrication to their implementation. The fact that high brilliance and high power densities go hand-in-hand with radiation from undulator beams has, by necessity, increased the complexity of first optical elements. Complicated cooling schemes (cryogenics, inclined geometries with pin-post heat exchangers, etc.) and exotic materials (i.e. large, perfect, synthetic diamond crystals) are the rule rather than the exception at third-generation sources. Thermal issues aside, delivery of the full beam brilliance (and coherence) to the sample requires high-quality surfaces not only on reflection optics, such as mirrors, but also on diffractive optics. Attention to the surface finish on single-crystal components, such as monochromators, is also crucial if the coherence of the X-ray beam is to be preserved. Clearly these stringent specifications necessitate increased vigilance of fabrication and manufacturing techniques. On the other hand, the extraordinary collimation (comparable to the Darwin widths of perfect single crystals of diamond, silicon, and germanium) and small source size of these beams have made some old ideas easier or perhaps feasible for the first time. The high collimation of the beam has permitted the routine use of perfect (or sometimes near-perfect) crystal XPRs to manipulate the polarization state of radiation. (Because of its low absorption, diamond is now often used as the phase retarder, a spin-off of the search for large, high-quality diamonds for high-heat-load optics.) Advances in synchrotron radiation sources have been a perfect marriage with recent advances in microfabrication techniques. This combination has resulted in improvements of effectiveness for BFLs. The small beam size means that submillimeter-sized zone plates can collect a sizable fraction of the beam, increasing the number of photons in the focal spot over that from nonundulator sources. Improvements in fabrication techniques have led to both smaller feature sizes and more complicated designs that result in tighter beams and more efficient operations, respectively. It is important to point out that the optical components discussed here are not proof of principle devices but are used routinely on many beamlines at third-generation sources. In fact, it is not unusual to see many of these elements combined on one beamline to produce a state-of-the-art instrument. On an undulator beamline at the APS, for instance, cryogenically

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cooled silicon is used to monochromate the beam after which it is passed through a phase retarder, to produce circularly polarized X-rays, and finally focussed with a BFL for magnetic X-ray microdiffraction experiments. These new X-ray optical components are truly providing unique and novel instrumentation to maximize the potential of third-generation synchrotron radiation sources.

Acknowledgment We wish to thank Dr Sarvjit Shastri for providing Figures 13.19–13.22 and Dr Thomas Toellner for providing Figure 13.17(b). This work is supported by the US Department of Energy, Basic Energy Sciences – Materials Sciences, under contract #W-31-109-ENG-38.

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[238] Chevallier, P., P. Dhez, F. Legrand, M. Idir, G. Soullie, A. Mirone, A. Erko, A. Snigirev, I. Snigireva, A. Suvorov, A. Freund, P. Engström, J. Als Nielsen and A. Grübel, Nucl. Instrum. Meth. A354, 584 (1995). [239] Aristov, V. V., Yu. A. Basov, A. A. Snigirev and V. A. Yunkin, Nucl. Instrum. Meth. A308, 413 (1991). [240] Snigirev, A., ESRF Newsletter No. 22 (November 1994) p. 20. [241] Snigirev, A. and V. Kohn, SPIE Proc. 2516, 27 (1995). [242] David, C. and A. Souvorov, Rev. Sci. Instrum. 70, 4168 (1999). [243] Chevallier, P., P. Dhez, F. Legrand, A. Erko and B. Vidal, Inst. Phys. Conference Ser. 130, 617 (1993). [244] Kuznetsov, S. M., I. I. Snigireva, A. A. Snigirev, P. Engström and C. Riekel, Appl. Phys. Lett. 65, 827 (1994). [245] Hanfland, M., D. Häusermann, A. Snigirev, I. Snigireva, Y. Ahahama and M. McMahon, ESRF Newsletter No. 22 (November 1994) p. 8. [246] Snigirev, A., Rev. Sci. Instrum. 66, 2053 (1995). [247] Bonse, U., C. Riekel and A. A. Snigirev, Rev. Sci. Instrum. 63(1), 622 (1992). [248] Aristov, V. V., Yu. A. Basov, Ya. M. Hartman, C. Riekel and A. A. Snigirev, Inst. Phys. Conference Ser. 130, 523 (1993). [249] Hartman, Ya., A. K. Freund, I. Snigireva, A. Souvorov and A. Snigirev, Nucl. Instrum. Meth. A385, 371 (1997). [250] Hartman, Y., V. Kohn, S. Kuznetsov, A. Snigirev and I. Snigireva, Nuovo Cimento D 19, 571 (1997). [251] Snigirev, A., I. Snigireva, P. Engström, S. Lequien, A. Suvorov, Ya. Hartman, P. Chevallier, M. Idir, F. Legrand, G. Soullie and S. Engrand, Rev. Sci. Instrum. 66, 1461 (1995). [252] Hartman, Ya., E. Tarazona, P. Elleaume, I. Snigireva and A. Snigirev, Journal de Physique IV (C9) 4, 45 (1994). [253] Snigirev, A., I. Snigireva, P. Bösecke, S. Lequien and I. Schelokov, Op. Comm. 135, 378 (1997). [254] Firsov, A., A. Svintsov, A. Firsova, P. Chevallier and P. Populus, Nucl. Instrum. Meth. A399, 152 (1997).

14 Metrological applications Terubumi Saito

1

Introduction

In the fields related to metrology, the primary concern is to measure as absolutely and as precisely possible, different types of quantities, such as length (including wavelength), angle, power, fundamental constants, and so on. To achieve this goal, all experimental conditions that are relevant to resultant uncertainty should be optimized. For measurements utilizing short wavelength radiation sources, radiation from insertion devices seems to be very attractive since it has many advantages over traditional radiation sources. First of all, its intensity is much higher than conventional lamps or X-ray sources. It is even higher than synchrotron radiation from a bending magnet of an electron storage ring. The result is a better signal-to-noise ratio and a reduction of measurement uncertainty. Second, the radiation from insertion devices has some unique properties. For example, its angular distribution is well collimated, and its radiation is polarized in a state that is dependent on the angle of emission. In principle, both the absolute angular power distribution and the polarization are calculable with information that is added to the basic parameters used to calculate synchrotron radiation intensity. Such information includes electron beam emittance and actual magnetic field distribution of the insertion device. Among the metrological fields that utilize these unique properties, this chapter reviews the application of insertion devices to interferometry and to detector calibration.

2 2.1

Application to interferometry Need for interferometry and beam splitters

Interferometric technique is very useful in obtaining precise measurements of length and certain other geometric parameters, for high resolution spectroscopy by use of the Fast Fourier Transform (FFT) method, and for determination of the optical constants of materials. For interferometry, the radiation source should be spatially and temporally coherent so as to obtain high-contrast fringe formation. Compared to other sources such as spatially filtered radiation from a laser-produced plasma, the undulator radiation from a low emittance ring is currently the most feasible source for interferometry in the extreme-ultraviolet (EUV) region because of a much higher coherent power for a given coherent length and a spectral bandwidth [1]. For geometrical measurements, the need for higher resolution of length necessarily requires the use of shorter-wavelength radiation for interferometry. One typical example can be found in the field of EUV lithography where resolution of the order of 0.1 nm for a wavelength of

T. Saito 50

50

s-polarization 45° Transmittance

Reflectivity (%)

40

30

26.6%

30

27.0%

20

40

Reflectivity

10

20

Transmission (%)

422

10 13.4 nm

0 11

12

13 14 15 Wavelength (nm)

16

0 17

Figure 14.1 Measured reflectance and transmittance of the beam splitter (after [4]).

13 nm will be required to test the shape and surface roughness of reflective optical components [2,3]. In addition, since the optics in the VUV/soft X-ray region usually consists of phasesensitive interference coatings such as multilayers, it is essential that the final evaluation of the system be done at the wavelength where the system is expected to be used. For FFT spectroscopy and optical constant determination, the use of wavelengths of interest is essential as well. Therefore, efforts have been made to extend the available spectral range for interferometry. The biggest problem when extending the available spectral range to the shorter wavelength is the difficulty in achieving good beam splitters, especially in the VUV region. Despite the difficulties involved, possibilities of fabricating VUV/soft X-ray beam splitters with multilayer structures have been shown by several groups, as tabulated in Table 14.1 [4]. For example, Haga et al. reported successful fabrication of a Mo/Si multilayer beam splitter with a 10 × 10 mm self-standing area and a flatness of 1.1 nm (rms) [4]. Figure 14.1 [4] shows the measured reflectance and transmittance of the beam splitter; both reach 27% at a wavelength of 13.4 nm for an angle of incidence of 45◦ . 2.2

Geometrical measurement

Another approach to achieve beam splitting is to use a grating beam splitter as adopted in the phase-shifting point diffraction interferometer (PS/PDI) [5]. Attwood et al. proposed at-wavelength interferometry by which optical systems are tested at the wavelength in the EUV at which they are expected to operate [2]. Since the wavelength for the interferometer is approximately 50 times shorter than that used in visible interferometry, a configuration that is inherently immune to vibration and has long-term stability is required. In addition, optical path differences should be shorter than the coherence length to obtain fringe visibility. A type of interferometer called a common path interferometer is known to satisfy these requirements. In this type of interferometer, the reference and the measurement wave fronts travel in essentially the same path through the interferometer, including the optical

Multilayer

Mo/Si (26 pairs) Mo/Si (13 pairs) W/C (15 pairs) Mo/C (35 pairs) Mo/Si (40–80 pairs) Mo/Si (6 pairs) Mo/Si (8–12 pairs)

Reference

Stearns et al. (1986) Ceglio (1989) Susini et al. (1988) Khan Malek et al. (1989) Nomura et al. (1992) Nguyen et al. (1994) Da Silva et al. (1995)

Table 14.1 Soft X-ray beam splitters (after [4])

Si3 N4 (30 nm) Si3 N4 (30 nm) Polypropylene (2 µm) SiC (300 nm) Self-standing SiN (150 nm) SiN (100 nm)

Membrane ∼20 ∼13.4 0.3 6 81 ∼15 20

<5 × 15 mm 10–20 mm2 25 × 12 mm 10 × 10 mm 8 mm diameter 2.5 × 2.5 mm 12 × 12 mm

∼4 ∼45 0.03 0.45 7 ∼10 15

Peak Transmittance reflectivity (%) (%)

Size

λ = 20.8 nm, θ = 0.5◦ λ . 13 nm, θ = 0.5◦ λ = 12.4 nm, θ = 79.05◦ λ = 1.33 nm, θ = 78.5◦ λ = 12.8 nm, θ = 45◦ λ = 13.6 nm, θ = 45◦ λ = 15.5 nm

Remarks (λ = wavelength, θ = angle from normal incidence)

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T. Saito

MO Water-cooled Varied beam defining retractable M1 lineplane apertures M2 mirror plane spherical space mirror grating mirror 166° 8.0 cm period undulator

EUV/soft X-ray photoemission microscope

M3 retractable plane mirror

Exit slit

166° M4 bendable

M5 plane

EUV interferometer

M6 bendable

Entrance pinhole

Coherent optics and zone plate photoemission (to 1 keV)

Figure 14.2 The undulator and beamline used to provide spatially and spectrally coherent power at EUV wavelengths (after [2]).

component or system under test. Shearing interferometers are in this class of common path interferometers. They have the characteristics that the reference and the measurement wave fronts are equally affected by aberrations of the optical component or system under test. They produce an interference pattern by interfering the aberrated wave front with a sheared duplicate of itself. A grating lateral shearing interferometer that can operate with a minimal coherence length is one of many kinds of shearing interferometers and is adopted in the PS/PDI system. An undulator beamline at ALS to provide spatially and spectrally coherent photons for this experiment together with some branch lines is shown in Figure 14.2 [2]. A setup of the PS/PDI for a wavelength of 13.4 nm in the measurements of a 10× Schwarzschild objective and the relevant optics is shown in Figure 14.3 [2]. The undulator radiation passing through the object plane pinhole illuminates the grating, which serves as a phase-shifting beam splitter. In the image plane focused by the test optics are two pinholes for spatial filtering: the larger one is for the zeroth order and the other is for the first order produced by the grating. Although both orders contain information regarding the optical system under test, the beam passing through the pinhole for the first order becomes a spherical reference beam because the diameter of the pinhole is smaller than the diffraction-limited resolution of the test optics. A phase shift between the test (zeroth order) beam and the reference beam is achieved by a lateral translation of the grating beam splitter. The test and reference wave fronts overlap at the surface of the CCD camera where their mutual interference is recorded. In their recent result with the 10× Schwarzschild objective, they reported a reference wave front rms error of 0.04 nm, or λ/333, for λ = 13.4 nm by this technique. 2.3

Optical constants determination

To determine complex refractive indices of thin foils in the soft X-ray region, a new interesting interferometric technique using two undulators in series was proposed by Dambach et al. [6] and was shown to work successfully. Figure 14.4 is a schematic picture of their experimental setup. A sample foil is placed between the two undulators, U0 and Ur , on the orbit of the electron beam. Therefore, the sample foil should be semi-transparent not only for photons but also for electrons. The position of the downstream undulator, Ur is fixed and generates

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EUV CCD camera Image-plane spatial filter (sub-0.1 µm pinhole) 10X Schwarzschild objective, 0.08 NA

From undulator

Coarse grating (phase-shifting beam splitter) Object-plane spatial filter (sub-1 µm pinhole) 45° multilayer turning mirror

K-B prefocusing mirrors Shutter

Optical table

Figure 14.3 At-wavelength interferometry setup of a 10× Schwarzschild optic multilayer coated for 13.4 nm wavelength using spatially coherent undulator radiation and a phaseshifting point diffraction interferometry (after [2]). Detector

U0

Ur

e– u

to

Spectrometer

e–

Foil

df

Wave r

Exit slit

Wave 0 O

Grating

d

Figure 14.4 Schematic picture of the interferometer used to measure complex index of refraction of thin self-supporting foils. Parameters: Period of the undulator λu = 1.2 cm; number of periods n = 10; undulator parameter K = 1.1; magnetic length Lu = 14 cm; maximum and minimum distance between undulators d = 33 and 17 cm, respectively; distances between foil and middle of the reference undulator df = 8.9 cm; between the foil and grating 10 m; between the foil and detector 16 m (after [6]).

reference wave trains while the upstream undulator, U0 , can be moved along the electron beam axis and produces another wave train that passes the sample and undergoes a phase change caused by the sample. These two wave trains are separated along the axis by the distance, -, which is a function of the distance, d, between the two undulators, and the observation angle, θ with respect to the direction of the electron beam. The grating spectrometer enables the two wave trains to overlap at the detector or, in other words, works as a Fourier analyzer of the wave trains. From intensity oscillation data as a function of the spacing, d, of the two undulators, the researchers determined the complex index of refraction n(ω) = 1 − δ(ω) − iβ(ω) for a carbon foil and for a polymide foil, as shown in Figure 14.5. It should be

426

T. Saito (a)

6 Carbon

 (10–3)

5 4 Polyimide

3 2 1 0

(b)

3

Carbon

 (10–3)

2 1 Polyimide

0

–1 –2

275

280

285

290

295

300

305

310

315

h (eV)

Figure 14.5 Results of the interference experiment at the K-absorption edge of carbon: (a) absorption β(ω), and (b) dispersion δ(ω) as measured width a (65 ± 2) µg/cm2 carbon and a (35 ± 2) µg/cm2 polyimide foil (after [6]).

noted that the fine structures, such as the π ∗ resonance at 285 eV and the transition to the σ ∗ antibonding band at 293 eV, were observed not only for the absorption spectra but also for a real part of the index, δ(ω), for the first time, by this technique. However, it should also be pointed out that this technique has some restrictions. First of all, the sample must be thin enough to allow both photons and electrons to pass through. The latter requirement necessarily results in the reality that the undulators cannot be installed in an electron storage ring but can only be installed in a single path electron beam from an injector. In fact, the interferometer was installed in an 855 MeV electron beam from the Mainz Microtron. To prevent electron beam transmission through the sample, a magnetic chicane placed between the two undulators may be used, as suggested in their paper.

3 3.1

Application to radiometry Insertion devices as primary standard radiation sources

At some national standards laboratories [7–13], the synchrotron radiation emitted from an electron storage ring has been used as a primary radiation source, especially in the VUV region, because its spectral power at a point (spectral irradiance) can be calculated based on the Schwinger equation from the parameters of the electron energy, the electron current, the magnetic field strength of the bending magnet and the geometric parameters (distance between the tangent point of the source and the observation point, and the observation solid angle). Strictly speaking, it is realized that the finite source size and angular spread of the

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electron beam (emittance) result in a broadening of the angular distribution of the radiation and, hence, a decrease in the power in the horizontal plane, which becomes more significant in the shorter wavelength region [7]. Wiggler/undulator radiation is essentially produced by the same principle as that for synchrotron radiation from bending magnets. It only differs from synchrotron radiation in that such an insertion device has a periodic magnetic structure. Therefore, total power radiated by an undulator or a wiggler integrated over a full spectral region and over a full angle is simply given by the same equation [14]: PT = N π cZ0 I eγ 2 K 2 /(3λu ) In practical units, PT [kW ] = 0.633E 2 [GeV] where N is the number of magnetic periods, Z0 the impedance of vacuum (=377.), I the beam current, e the electron charge, c the velocity of light, λu the magnet period, γ the electron energy, E, divided by the rest mass energy, and K the deflection parameter. However, angular and spectral power distributions are strongly dependent on the deflection parameter, K, which determines the degree of electron beam deflection, and, therefore, determines the degree of interference among the rays in an insertion device. The radiation intensity of a wiggler having a large K(1) with M magnetic poles of alternating polarity is simply M times that of a single bending magnet with the same field strength as that of the wiggler magnetic pole. The spectral power density of an undulator having K  1 is concentrated in a certain spectral region due to interference phenomena, and its spectral distribution is, therefore, very different from the distribution of synchrotron radiation from a bending magnet. The spectral bandwidth of the fundamental component of the undulator radiation is theoretically equal to 1/N , where N is the number of periods. Following a few preliminary observations [15–19], a number of experimental characterizations of insertion devices were reported on polarization characteristics [20–26], spectral distribution [20,25,27–34], wavelength tunability [27,34], and their angular dependence [20,27–29,31–33]. For the purpose of detector calibration, a theoretical model [35–39] to predict the actual spectral power density of the radiation and its verification by comparison with absolute measurements [25,27–34] are essential. Since measurements are usually conducted for a limited portion of the radiation, knowledge of the angular power distribution and of the dependence of the polarization on the emission angle [40– 42] is also important. Especially for undulator radiation, spectral distribution strongly depends on electron beam emittance and imperfections of the periodic magnetic structure such as variation in magnetic field strength, misalignment of magnets, and so on. However, if all the necessary information such as the actual magnetic field distribution and electron beam emittance are described well, it is expected that undulator radiation can also be used as a primary source with calculable spectral distribution. For wiggler radiation, Yanagihara et al. made absolute measurements using photocalorimetric devices [43,44]. Figure 14.6 shows one of the devices they used. They determined the power absorbed by the device from the following formula: Q = ka -T / l

428

T. Saito Absorber

Thermocouples

5 mm

50 mm Heat sink

Figure 14.6 Cross section of the photocalorimetric device for the measurement of wiggler radiation power. The device is made of copper of 99.9% purity. The heat sink is cooled with water (after [43]).

Power density (W/mm2)

8 Present Kim

6

Gap = 30 mm 4 2 60 mm 0

–5

0 Verticle distance (mm)

5

Figure 14.7 Power density distribution of X-radiation from the BL-28 multipole wiggler of the Photon Factory measured at a distance of 14.1 m for a magnet gap of 30 mm (circles) and 60 mm (triangles). The solid and dashed curves show the densities calculated using Kim’s formula and a more exact formula, respectively. All data are normalized to a ring current of 300 mA (after [43]).

where k, a and -T / l are the thermal conductivity of copper, the cross section of the copper rod, and the temperature gradient produced over a distance l along the rod, respectively. One of the comparisons between the measurement results and the calculations is shown in Figure 14.7. The measurement was made by shifting an aperture of 0.8 mm (vertical) × 1.6 mm (horizontal) in front of the absorber. The measured peaks are lower than those calculated for a zero-emittance beam by about 15 and 24% for the magnet gap of 30 and 60 mm, respectively, while the measured FWHM are larger than those calculated by about 10% in both cases. Taking into account the finite positron beam emittance (shown by the dashed lines in the figure), the researchers also reported that the calculated power density gives only a small decrement of at most 3% from that calculated for the zero-emittance beam, and they cannot explain this difference. The measurement uncertainty was estimated to be approximately 5% including that of about 2% attributed to the assumed thermal conductivity of copper. They stated that some other sources such as error due to the non-zero reflectivity of copper, energy loss by photoelectron emission, and thermal radiation loss are negligibly small compared to the total uncertainty. As a source of the remaining discrepancy, they suspected misalignment of the wiggler magnets as well as radiation loss due to Cu Kα X-rays.

Metrological applications

(a)

1014

429

on-axis

(b)

(c)

Brightness (photons/s/0.1% bw mrad2)

1013 1012 1011 1014

x = 80 µrad, y = 0

1013 1012 1011 1014 1. 2.

x = 0, y = 80 µrad 3. 4. 5.

100

200

1013

6. harm.

1012 1011

300

Photon energy (eV)

Figure 14.8 On- (a) and off-axis (b), (c) brightness of the vertical undulator of U-2 at BESSY for K ≈ 0.95(gap = 68 mm). The solid line indicates the calculated spectrum including the emittance effects; the + symbols indicate the measured data (ring current 100 mA) (after [29]).

For undulator radiation, the possibility of its use as a primary radiation source was studied with BESSY undulators [28,29]. The researchers made absolute measurements in the soft X-ray region from 50 to 500 eV using a calibrated pinhole transmission grating spectrometer with relative uncertainties between 5% (at 100 eV) and 20% (at 500 eV). The spectrometer diffraction efficiencies were determined experimentally for the first through the fourth order. This kind of spectrometer is also useful for undulator radiation diagnostic purposes. In their calculation of the undulator spectrum, they consider the effects caused by magnetic field errors, fringe fields (based on measured magnetic field distribution), emittance, polarization, and a finite aperture. Figure 14.8 compares their experimental results with calculations for both on-axis and off-axis brightness [29]. Compared to the theoretical predictions [28], which do not use the actual magnetic field distribution, apparently improved agreement between the calculations and the measurements within their claimed uncertainties of 10% were obtained. The detailed structures in the measurements, some of which were assigned to interference between the natural third harmonics and the contribution due to the third harmonic of the fringe magnetic field, are well reproduced by the calculation. Similar comparisons between measurements and calculations based on the measured magnetic field profile and the measured beam emittance were also made for undulator A with 72 periods at APS [32]. Their absolute measurements were made in the photon energy range from 4 to 110 keV by using both a helium gas-scattering spectrometer and a silicon crystal spectrometer with maximum uncertainties of 18.9 and 18.2%, respectively. Figure 14.9 compares the results obtained using the gas-scattering spectrometer. As is clearly seen in the figure, the calculation reproduces the measurement spectrum well within the uncertainties.

T. Saito

Brilliance (ph/s/0.1% bw/mrad2/ mm2/mA)×1016

430

2.5 2.0 1.5 1.0 0.5 0

5

10

15

20 25 30 Energy (keV)

35

40

Figure 14.9 Measured on-axis spectral brilliance (solid line) using the gas-scattering spectrometer, and calculated on-axis spectral brilliance (dotted line) of the undulator radiation from a 7 GeV electron beam at a gap of 15.8 mm (K = 1.61). The calculation included the measured magnetic field of the undulator, the measured electron beam emittance (εx = 6.9 nm rad, εy = 0.2 nm rad) and the design value for the electron beam energy spread (0.1%). The calculated spectral brilliance was further convolved with energybroadening functions. The measured spectrum has been corrected for the Compton shift (after [32]).

The above two examples demonstrate that since the influences of errors of magnetic fields [45,46] and of finite beam emittance on the undulator spectrum are significant, the use of a calculation model that takes into account the actual magnetic field distribution and the beam emittance is essential if undulators are to be used as primary standards. 3.2

Undulators as intense VUV sources for thermal detectors

Besides the property of acting as a primary source, undulator radiation has some other attractive features. One of the biggest advantages over other sources is its high intensity and tunability. An attempt to use undulator radiation as an intense radiation source in the UV region for detector calibration will be discussed below in some detail. Detector calibration in the UV and VUV region is usually achieved by using a thermal detector, whose responsivity is constant with wavelength, to extend the spectral region of a primary standard detector available in another spectral region [47]. In general, a drawback of this method is a low signal-to-noise ratio in the measurement due to low sensitivity of the thermal detectors. Therefore, the use of this technique was, in most cases, limited to discrete spectral points where intense line spectra from a discharge lamp are available. T. Saito et al. [48] used undulator radiation to perform detector calibration based on an Electrical Substitution Radiometer (ESR) operated at room temperature [48]. They calibrated a Si photodiode, a GaP photodiode, and a CsTe phototube against an ESR operated at room temperature using undulator radiation monochromatized by interference filters. The optics used in the calibration are shown in Figure 14.10. The source is undulator radiation from a 15-period polarizing undulator that is installed in the NIJI-II storage ring [49] at the Electrotechnical Laboratory. The undulator is the crossed and overlapped type of polarizing undulator proposed by Onuki [50]. This type of polarizing undulator is capable of

Metrological applications

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Undulator

Multichannel spectrometer 22.5° Plane mirror Quartz window

Quartz plate

ESR Aperture Detector to be calibrated stop Interference filter

Figure 14.10 Optical arrangement for detector calibration using undulator radiation (after [48]).

Responsibility (A/W)

0.25 0.20 Si PD 0.15

GaP PD

0.10 0.05 CsTe PT 0 200

300 400 Wavelength (nm)

500

Figure 14.11 Calibration results for Si, GaP photodiodes, and CsTe phototube using monochromatized undulator radiation based on an ESR. In addition, the lines show other calibration results based on the ETL responsivity scale (after [48]).

producing brilliant, quasi-monochromatic, wavelength-tunable and polarized radiation of any ellipticity. Chapter 6 in Part 1 contains the details of the polarizing undulator. The undulator radiation is deflected by an Al-coated plane mirror and is passed through a quartz window. Since the undulator spectrum depends on the observation angle, an aperture stop of 2 mm in diameter defining the observation angle is placed behind the quartz window. Interference filters are used in the present calibration to improve spectral purity. The electron energy of the storage ring and the magnet gap of the undulator are varied from 199 to 260 MeV, and 80 to 92 mm, respectively, depending on wavelength. The power of the radiation transmitted through the quartz plate, which is used for detector calibration, is measured using the ESR. Figure 14.11 shows the calibration results using monochromatized undulator radiation measured with the ESR. In addition, the lines indicate other calibration results based on the ETL responsivity scale at the time using conventional light sources (quartz halogen lamp and deuterium lamp) and a monochromator. The ETL responsivity scale was established in the wavelength range above 250 nm based on silicon photodiode self-calibration using a thermopile as a detector whose responsivity is wavelength-independent. It is found that the calibration results using undulator radiation for a Si photodiode, a GaP photodiode, and a

432

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CsTe phototube agree with the existing scale with uncertainties of within ±4, ±5 and ±17%, respectively.

References [1] Attwood, D., “Undulator radiation for at-wavelength interferometry of optics for extreme-ultraviolet lithography,” Appl. Opt. 32, 7022–31 (1993). [2] Attwood, D., E. Anderson, P. Batson, R. Beguiristain, J. Bokor, K. Goldberg, E. Gullikson, K. Jackson, K. Nguyen, M. Koike, H. Medecki, S. Mrowk, R. Tackaberry, E. Tejinil and J. Underwood, “At-wavelength metrologies for extreme ultraviolet lithography,” FED Journal 9, 5–14 (1998). [3] Goldberg, K. A., P. Naulleau, C. Sang Lee Bresloff, C. Henderson, D. Attwood and J. Bokor, “High-accuracy interferometry of extreme ultraviolet lithographic optical systems,” J. Vac. Sci. Technol. B, Microelectron. Nanometer Struct. 16, 3435–9 (1998). [4] Haga, T., C. K. Tinone, M. Shimada, T. Ohkubo and A. Ozawa, “Soft X-ray multilayer beam splitters,” J. Synchrotron Rad. 5, 690–2 (1998). [5] Medecki, H., E. Tejnil, K. A. Goldberg and J. Bokor, “Phase-shifting point diffraction interferometer,” Opt. Lett. 21, 1526–8 (1996). [6] Dambach, S., H. Backe, T. Doerk, N. Eftekhari, H. Euteneuer, F. Goergen, F. Hagenbuck, K. H. Kaiser, O. Kettig, G. Kube, W. Lauth, H. Schoepe, A. Steinhof, T. Tonn and T. Walcher, “Novel interferometer in the soft X-ray region,” Phy. Rev. Lett. 80, 5473–6 (1998). [7] Suzuki, I. H., “Comparison between measured and calculated absolute intensities of ultra-soft X-rays of synchrotron radiation,” Nucl. Instrum. Meth. 228, 167–71 (1984). [8] Kostkowski, H. J., J. L. Lean, R. D. Saunders and L. R. Hughey, “Comparison of the NBS SURF and tungsten ultraviolet irradiance standards,” Appl. Opt. 25, 3297–306 (1986). [9] Riehle, F. and B. Wende, “Establishment of a spectral irradiance scale in the visible and near infrared using the electron storage ring BESSY,” Metrologia 22, 75–85 (1986). [10] Anevski, S. I., A. E. Vernyi, V. S. Panasyuk and V. B. Khromchenko, “Improvements of the TROLL-2 synchrotron and new developments,” Nucl. Instrum. Meth. A308, 35–8 (1991). [11] Arnold, D. and G. Ulm, “Electron storage ring BESSY as a source of calculable spectral photon flux in the X-ray region,” Rev. Sci. Instrum. 63, 1539–42 (1991). [12] Fox, N. P., P. J. Key, F. Riele and B. Wende, “Intercomparison between two independent primary radiometric standards in the visible and near infrared: a cryogenic radiometer and the electron storage ring BESSY,” Appl. Opt. 25, 2409–20 (1986). [13] Tegeler, E., “Determination of the spectral radiance of deuterium lamps using the storage ring BESSY as a primary radiometric standard,” Nucl. Instrum. Meth. A282, 706–13 (1989). [14] Kim, K. J., “Characteristics of synchrotron radiation” in X-Ray Data Booklet, pp. 4–10, D. Vaughan (ed.) (Lawrence Berkeley Laboratory, Berkeley, 1986). [15] Motz, H., W. Thon and R. N. Whitehurst, “Experiments on radiation by fast electron beams,” J. Appl. Phys. 24, 826 (1953). [16] Alferov, D. F., Y. A. Bashmakov, K. A. Belovintsev, E. G. Bessonov and P. A. Cherenkov, “The undulator as a source of electromagnetic radiation,” Par. Accel. 9, 223 (1979). [17] Nikitin, M. N., A. F. Medvedev and M. B. Moiseev, “Interference of synchrotron radiation,” Sov. Tech. Phys. Lett. 5, 347 (1979). [18] Kitamura, H., S. Tamamushi, T. Yamakawa, S. Sato, Y. Miyahara, G. Isoyama and H. Nishimura, “Observation of undulator radiation I. Operation studies and visual observation,” Jpn. J. App. Phy. Part 1, Regular Papers and Short Notes 21, 1728–31 (1982). [19] Bazin, C., M. Billardon, D. Deacon, Y. Farge, J. M. Ortega, J. Pe’rot, Y. Petroff and M. Velghe, “First results of a superconducting undulator on the ACO storage ring,” J. Phys. Lett. 41, L547–50 (1980).

Metrological applications

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Index

aberrations 424 absorption coefficients 381 acceptance 14 Acetylcholinesterase 306 α-actinin 305 adaptive optics 389 angle integrated flux 89 angle integrated power 48 angle integrated spectral flux 74, 99 angular distributions 120 angular representation 49 angular spectral flux 73, 83, 98 anticlastic bending 406 Apple-1 223 array of current 152 asymmetric wigglers 125, 233 installed in the HASYLAB 234 proposed by Goulen, Elleaume and Rauox 234 back-reflection monochromators 390 bacteriorhodopsin 313 beam splitter 421–2, 424 bending magnet 108 bending magnet radiation 109 BESSY 230 Betatron coupling 21 Betatron oscillations 9 biomolecules 336 Biot–Savart 161, 205 Bose degeneracy 351, 360, 367–8 Bragg (111) crystals 298 Bragg-Fresnel lenses 406 Bragg geometry 378 Bremsstrahlung 33 brightness 52, 373 brilliance 51–3, 55, 60, 62, 92, 100, 144, 285, 373 broadband resonator 27 bronchography 324 bunching 260 bunch length 17 calibration 427, 430–1 d-10-camphorsulfonic acid 340 capillary optics 405

CCD detector 300 CD image of d-10-camphorsulfonic acid film 344 CD spectroscopy 336, 340 central cone 85–6 Chasman–Green 19 Chasman–Green lattice 19 chromaticity 13 circular devices 248 circular dichroism measurement 336–43 circular dichroism (CD) spectroscopy 336 circularly polarized radiation 248 circular polarization 246, 248 degree of circular polarization 220, 232, 234 degree of circular polarization of the first harmonic radiation 221 classification of polarizing ID 214 closed orbit 22–3 cobalt–iron 151 coercive field 150 coercive force 149 coherence 63, 360–1 coherence length 66 coherent harmonic generation 259 coherent motion 351 collective effects 26 complex refractive indices 424 compound refractive lenses 405 Compton backscattering 286 computed tomography 322 correspondence principle 357 Coulomb scattering 33 coupled bunch instabilities 29 coupling coefficient 22 Courant–Snyder invariant 9 critical frequency 112 critical wavelength 112 crossed and overlapped undulator, 218–22 cryogenic optics 385 cryogenics 207 crystal focussing optics 405 crystal optics 369 crystals are bent 327 current leads 209 cytochrome bc1 complex 311

436

Index

damping partition numbers 17 damping times 17 dark state 367 Darwin widths 378 deflection parameter 79, 203, 371 degree of polarization 247 detector 328 detuning 275 diagnostics 323 diamond optics 293, 298, 387 dichromography 325–6 diffraction enhanced imaging (DEI) 323 diffraction property 237 digital subtraction angiography (DSA) 325 dipole 7 dispersion 10 dispersive geometry 382 double bend achromat 19 double helix undulator 216 universal double-helix proposed by Alferov, Bashmakov and Bessonov 218 DSA 326 dual-photon absorptiometry (DPA) 324 Dumond diagrams 382 dynamic acceptance 13–14 dynamical diffraction 374 dynamical diffraction, dispersion surfaces 377 effective dose 328 electric field in the frequency domain 97 electromagnetically induced transparency (EIT) 367 electromagnetic modulator 247 electromagnet insertion devices 199 electromagnet undulators 153 electromagnet undulators with crossed and retarded magnetic fields 227–9 planar type (of APS and BINP) 229 proposed by Chavanne et al. 228 proposed by Nahon 228 electron beam density 61 electron energy spread 62 electron trajectory 47, 79, 96, 243, 248 ellipsoidal undulator 69, 96 elliptically polarizing undulator 246 elliptical (magnetic field) wigglers 230–3 electromagnet type 233 planar type proposed by Maréchal, Tanaka and Kitamura 232 proposed by Bessonov 230 proposed by Yamamoto and Kitamura 231 elliptical wigglers 128 emittance 14, 17, 58, 61, 427–30 end-station instrumentation 299 energy loss 17 EPU 246 equations of motion 3–4, 6 error is minimum 105 far field 41, 44, 47 Fermi edge singularity 358 Feymann, R. P. 42 field errors 101 field integral errors 182

field integral measurement 179 field termination 169 field vector 45, 56, 58, 70, 72 figure of merit 249 figure-8 undulator 243 filament monoenergetic electron beam 38 finite difference 158 finite element 157, 160 flipping coil bench 180 fluxgate integrator 182 Fourier limit 278 free electron laser 255 Fresnel zone plates 405 β-functions 8, 18 Gaussian beam 55, 65 geometrical measurement 422 grating 422, 424–5 half-odd-integer 246 Hall probe 176 head–tail instability 31 heat-load optics 384 helical undulator 248 helium vessel 207 Helmholtz coil 181 high brilliance beamline ID2 294–5 high-energy-resolution monochromators 390 higher harmonics 237 high temperature superconducting 209 Hill’s equation 8 horizontal emittance 17 horizontal magnetic field 243 human studies 324 Huygens principle 49 hybrid 166, 172 image processing 331 impedance 27 importin-β 311 inclined geometry 387 inelastic diffraction 367 infrared 268 insertion devices 23 integral approach 158 Integral method 161 interaction field integral 169 interferometer 422 interferometry 421 intermediate state 357 intrabeam scattering 31 intravenous coronary angiography (ICA) 323–4 intrinsic coercivity 149 in-vacuum undulator 196–7 irrationally inclined line 238 kappa diffractometers 300 K-edge subtraction 322 KES 325 Kirkpatrick-Baez mirrors 405 Kramers–Heisenberg formula 357 Laue (100) crystals 298 Laue diffraction 314

Index Laue geometry 327, 378 Laue point 379 Liénard–Wiechert 40 lifetime 14, 32 linear polarization 246 line scan system 326 Liouville’s theorem 14 liquid nitrogen vessel 207 lithography 421 lobe structure 86 longitudinal acceptance 15 longitudinal instabilities 28 longitudinal motion 10 Lorentz equation 3 Lorentz force 41 Lorentz point 379 loss factor 28 macromolecular crystallography 293 magnetic circular dichroism (MCD) 336, 368 measurement 344–5 spectroscopy 336 magnetic field computation 151 magnetic field measurement 175 magnetic field sensors 176 magnetic force 190 magnetic lattice 7 magnetic materials 148 magnet row phase 251 mammography 323 Maxwell equations 40 MCD signal 345 MCD spectrum of YIG 347 mechanical and magnetic shims 183 medical applications 322 metrology 421 microbeam radiation therapy 322 microcrystals 313 microdiffractometer 297 microscopic imaging of circular dichroism 341 microwave undulator 251 momentum compaction 11 momentum spread 17 monochromator 327 multilayer 422 multipole shimming 185 multireflection monochromators 392 multiwavelength anomalous diffraction (MAD) 293 mutual intensity 52, 65 myoglobins 315 nested monochromators 393 NIJI-II 291, 337 NIKOS system 326 NMR probe 177 nondispersive geometry 382 non-linearities 12 non-sinusoidal magnetic field 246 nuclear Bragg scattering 362 nucleosome core particle 310 one-dimensional scatterer 237 Onuki-type polarizing undulator 218, 230, 337, 344

437

optical constant 421–2, 424 optical klystron 263 optical resonator 258–9 optics of microscopic imaging system 338 permanent magnets 149, 154 phase contrast imaging 323 phase errors 104, 183 phase shifter 336 phase shimming 190 phase space analysis 130 phonon 364–5 photoactive yellow protein 314 photodiode 430–1 photoelastic modulator 336 photon activation therapy 322 photon beam divergence 75 photon beam size 56, 77 phototube 430, 432 physical aperture 14 pin-post heat exchanger 389 pixel array detectors 301 planar Hall effect 177 planar tye polarizing undulators 222–6 installed in UVSOR and HiSOR 226 proposed by Diviacco and Walker 224 proposed by Elleaume 223 proposed by Sasaki et al. 224 planar undulator 69, 79, 246 Pochels cell 336 polarization 51, 62, 86, 113, 121, 427, 429 polarization modulation spectroscopy by polarizing undulator 336–48 polarization vector 246 polarizing undulator 214–36 polarizing undulators with interference effect (crossed undulator), 229–30 proposed by Kim 229 proposed by Moissev, Nikitin and Fedosov 229 polarizing wigglers 214–36 potential well distortion 29 power density 49, 93, 100, 116 Poynting vector 41 ProDC 302 profiles of the mirobeam of the undulator radiation 340 pseudo-helical trajectory 251 pulsed stretched wire 178 pure permanent magnets 166, 170 QPU 237 Quadriga beamline ID14, the ESRF 298 quadrupole 7 quantum beat 362 quantum lifetime 32 quasi-crystal 238 quasi-periodic lattice 237 quasi-periodic undulator 237 Rabi frequency 351, 367 radiated power 42 radiation excitation 16 radiation field in the frequency domain 81 radiation field in the time domain 80 radiation power 116

438

Index

radiation spectra 242 radiometry 426 Raman shift 353–4, 356, 361, 365–6 rational harmonics 242 rectangular lattice 239 relative permeabilities 149–50 remanent field 149–50 resistive wall 27 resonance fluorescence 360 resonances 20 resonant photoemission 353 responsivity 430–1 retarded time 39 rotated inclined geometry 388 sagittal focussing 405 SASE 259–60, 283 Schwartschild-type mirror 338 second harmonic radiation 346 selective coronary angiography 325 sextupole 7, 13 shimming 25, 182 shims 185 single peak undulator 295 skew quadrupole 22 skin entry radiation dose 328 slope error 385 small angle approximation 46 small signal gain 262 soft magnetic material 150 soft materials 164 source size 143 spectrum shimming 187 spectrum through a slit 88 Spring8 227 square lattice 238 squeezed-coherent motion 351 SRW computer code 51 Stokes parameter 62, 114, 219, 337 stretched scanning wire 180 structural genomics 294 studies with patients 329 sudden approximation 349 Super-ACO 228 superconducting high field wigglers 201 superconducting insertion devices 201 superconducting short period undulators 202 superconducting wiggler 205 support structure 190, 193, 195 synchrotron frequency 12 synchrotron oscillation 11 tandem type undulators 226–7 proposed by Elleaume (Helios) 227 proposed by Hara et al. 227 tapered undulator 102 TERAS 219 theoretical minimum emittance 19 therapy 322

thermal conductivity 428 thick electron beam 58 three-dimensional fields 156 time-resolved macromolecular crystallography 314 total flux 54 total photon flux 116 Touschek effect 34 transverse impedance 30 transverse instabilities 30 transverse mode 282 transverse wake field 29 trichromator 295 triple bend achromat 19 Troika concept 299 tropinone reductase II 305 tunability of the spectrum 93 tune 9 tune diagram 21 two-dimensional fields 151 two-dimensional CD imaging 336–44 two-line ionization chamber 328 ultraviolet (UV)-CD 336 undulator radiation, phase space 373 undulator radiation, power envelope 373 undulator radiation, source divergence 370 undulator radiation, source size 370 undulator radiation, spatial distribution 371 undulator radiation, spectrum 371 undulators 69 with crossed and retarded magnetic fields 214–36 uniformly magnetized volume 162 UV 270 vacuum chamber 26 vacuum ultraviolet (VUV)-CD 336 VEPP-2M, ID 217 vertical emittance 17 vibrating wire 178 visible 270 VUV 282 wake field 26 wave front 422, 424 wavelength tunability 297 wave packet motion 349–50, 357–8, 367–8 wave train 425 wiggler radiation 108, 118 wigglers for circularly polarized radiation 125 Wigner distribution 54 Wigner distribution function 51 X-ray 282 X-ray phase plates 397 X-ray phase retarders 396 X-ray scattering 237 zone plate 364