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

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 (∞) = − 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

96

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  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

1.0

40

0.8

20

0.6

0

0.4

–20

0.2

–40

0 –40

–20

0

20 ∆ (µrad)

40

60

 (degree)

W.-K. Lee et al.

Reflectivity

402

–60 80

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

X-ray crystal optics

403

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

404

W.-K. Lee et al. 0.35 150 0.3 100 50 0.2 0 0.15 –50

 (degree)

Transmissivity

0.25

0.1 –100 0.05 0 –1000

–150 –500

0 ∆ (µrad)

500

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), 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)

406

W.-K. Lee et al.

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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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