Synchrotron Radiation

An everyday application of special relativity

Jan-Erik Rubensson

Department of Physics and Astronomy, Uppsala University, Sweden

Morgan & Claypool Publishers

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ISBN 978-1-6817-4115-4 (ebook)
ISBN 978-1-6817-4051-5 (print)
ISBN 978-1-6817-4243-4 (mobi)

DOI 10.1088/978-1-6817-4115-4

Version: 20160701

IOP Concise Physics
ISSN 2053-2571 (online)
ISSN 2054-7307 (print)

A Morgan & Claypool publication as part of IOP Concise Physics
Published by Morgan & Claypool Publishers, 40 Oak Drive, San Rafael, CA, 94903 USA

IOP Publishing, Temple Circus, Temple Way, Bristol BS1 6HG, UK

This is to all the synchrotron radiation users of the future.

Contents

Preface

Acknowledgement

Author biography

1 Introduction

1.1 Perspectives

1.2 X-rays, structure, and dynamics

1.3 Outline

References

2 Properties of radiation

2.1 Intensity

2.2 Flux

2.3 Emittance and Liouville’s theorem

2.4 Brilliance

2.5 Polarization

2.6 Coherence

2.6.1 Transverse coherence or spatial coherence or coherence width

2.6.2 Longitudinal coherence or time coherence or coherence length

2.6.3 Orbital angular momentum

2.7 Time structure

References

3 Accelerating charged particles

3.1 Angular distribution at low speed

3.2 Angular distribution at high speed

3.3 Energy distribution

3.4 Emitted power in a bend

3.5 Brilliance calculations

3.6 Polarization of bend magnet radiation

3.7 Time structure of synchrotron radiation

3.8 Coherence of bend magnet radiation

References

4 Insertion devices

4.1 Wigglers

4.2 The undulator

4.3 Weak field, on axis

4.4 Off the axis

4.5 Spectral purity, and the central radiation cone

4.6 Finite magnetic field

4.7 Power

4.8 The importance of electron-beam quality

4.9 Brilliance

4.10 Polarization properties

4.11 Coherence and orbital angular momentum

References

5 Coherent particles

5.1 Free-electron lasers

5.2 Self-amplifed spontaneous emission

5.3 Seeding

References

6 X-ray optics

6.1 Beamlines

6.2 Optical constants

6.3 Absorption and reflection

6.4 Multilayer reflection

6.5 Reflective optics

6.6 Refractive optics

6.7 Diffractive optics

6.8 Dispersive optics

6.9 Optics quality

6.10 Monochromators

6.11 Ray tracing

References

Preface

The success story of synchrotron radiation is almost unparalleled. The synchrotron provides radiation of a quality that facilitates a wide range of experiments, with applications in virtually all branches of natural sciences: from fundamental molecular and condensed matter physics, to magnetism and applied materials science, from electrochemistry to applications in solar cell, battery and environmental research, from pharmaceutics to biology, where structure determination of biological macromolecules may be the most spectacular application. Synchrotron radiation also finds users dedicated to archeology and cultural heritage, and other fields conventionally not associated with natural sciences. As the versatility is realized, new synchrotron radiation facilities are being built all around the world.

Thus, there are an increasing number of synchrotron radiation users who are specialists in scientific disciplines far from basic physics and accelerator technology. Certainly one may perform excellent experiments with the radiation with only little knowledge about the principles by which it is produced. For design of new, innovative types of experiments knowledge about fundamental opportunities and limitations is desirable, and for the personal peace-of-mind it is good to know that the brilliant radiation is not coming out of a big, magic, black box.

The radiation is produced according to the basic laws of physics, and it is fascinating that several of the most fundamental relations are needed to understand the emission process. At the heart of the process is the classical Maxwellian notion that accelerating charged particles emit electromagnetic radiation, and the properties of the radiation can be understood using classical concepts like the Doppler effect. Also, concepts from modern physics like the Heisenberg uncertainty principle are used to understand the properties of the radiation, and in particular, the principles of special relativity. The effects of special relativity are so obvious in synchrotron radiation that it may as well be used to understand special relativity as the other way around. Synchrotron radiation is by now so common that it constitutes an everyday application of special relativity.

The book is intended for the growing number of non-specialist users, and for readers who are interested in how bright x-rays can be produced. It focuses on the basic principles, which are essential for the understanding. Although the book does contain some math, the math is invariably simple and easy to follow. Synchrotron radiation is also simple in the sense that its properties often can be described by (quasi) analytical expressions. The rigorous, not-so-simple derivation of those expressions can be found in more comprehensive textbooks or research papers, to which interested readers are referred. Here, these expressions are accompanied by qualitative arguments, and statements of the ‘it-can-be-shown-that…’ form.

Acknowledgement

This book is based on lecture notes used in the ‘Synchrotron Radiation Course’ given at the master programme at Uppsala University. I am indebted to Svante Svensson who left me his lecture notes to build on when I took over the course many years ago. In my teaching I have also followed David Attwood’s comprehensive Soft X-Rays and Extreme Ultraviolet Radiation, and here I have borrowed from several accounts in that book. I also would like to thank my students over the years for critical remarks and discussions.

Author biography

Jan-Erik Rubensson

The author is physics professor at Uppsala University. Most of his research is based on spectroscopy using synchrotron radiation, and especially, he has been engaged in method development related to resonant inelastic soft x-ray scattering (RIXS). The new synchrotrons allow for substantial refinement of RIXS techniques, which promises progress in fields ranging from fundamental atomic and molecular physics, over electrochemistry and pharmaceutics, to magnetism and correlated-electron systems. Several advanced RIXS facilities are presently being built at the new synchrotrons, and the author is spokesperson for the VERITAS beamline at the MAX IV laboratory. In addition, the author investigates the prospects of time-resolved and non-linear inelastic x-ray scattering at free-electron lasers. His scientific focus is on ultrafast electron dynamics in free molecules, and the coupling between, spin, orbital, and nuclear degrees of freedom in general.

The author is teaching basic mechanics and optics on the bachelor level at Uppsala University, and he has the responsibility for the master-level course in synchrotron radiation.

IOP Concise Physics

Synchrotron Radiation

An everyday application of special relativity

Jan-Erik Rubensson

Chapter 1

Introduction

Some perspectives on synchrotron radiation, and a short history. The use of x-rays for the study of matter.

1.1 Perspectives

Synchrotrons are large-scale facilities where a radiation source with a diameter of some hundred meters is surrounded by specialized equipment which hardly looks meaningful to a first-time visitor. The equipment includes advanced instruments and detectors, with electronics and precision mechanics, the design and construction of which can take several years, but occasionally also improvised solutions to current technical problems or spontaneous setups for testing new ideas, which may include aluminum foil, clamps, paper clips and tape. The funding agency may look at synchrotrons as a place where money is delivered for the amusement of a few scientists who seem more playful than serious, and from where not much of value emanates. The inventor sees the great vision and the beautiful principle realized in the synchrotron, and the accelerator physicist sees all the wonderful optimization challenges. I assume that most readers are users or would-be users, who rather see the synchrotron as a black magic box that gives them radiation with fantastic properties for their experiment. The aim of this book is to give a glimpse inside the black box. Even though the discussion uses a mathematical language, it focuses on conceptual understanding, and not on strict derivations. It is sometimes possible to arrive at quantitative predictions from basic physical principles and rather simple algebra. I hope it can also serve as an introduction to general readers who are interested in how the brightest x-ray sources work.

There are several excellent pedagogic texts describing the principles of synchrotron radiation in more depth [1–6]. I am in debt to David Attwood’s book Soft x-rays and Extreme Ultraviolet Radiation [1], and in several parts I follow his presentations rather closely. On his homepage [7] you can find many excellent lectures.

Oxford University Press has published a Series on synchrotron radiation [8], and Springer has a constantly updated reference handbook entitled Synchrotron Light Sources and Free-Electron Lasers [9].

There are also excellent internet resources, with pedagogic introductions and analyses. An invaluable help concerning quantitative estimates of properties of radiation and its interaction with matter is the home page of Center for X-ray Optics at Lawrence Berkeley National Laboratory [10].

1.2 X-rays, structure, and dynamics

The medical use of x-rays is often emphasized, and its value can hardly be overstated. Their large penetration length has been exploited to unveil internal structure ever since their discovery by Wilhelm Conrad Röntgen in 1895. The role of x-rays in the investigation of structure and dynamics on the atomic length and time scale is not less important. The diffraction of x-rays by crystals, as observed by Max von Laue in 1912, confirmed that the wavelength of x-ray radiation is similar to atomic distances. Thus, x-ray scattering became the prime tool to investigate the structure of materials, and in 1952 the structure of DNA was determined by means of x-ray diffraction experiments, opening the door to atomic-scale biology. X-rays were also decisive for our understanding of the electronic structure of the atom, through the discovery of characteristic radiation by Charles Glover Barkla in 1909, and the systematic investigations of Henry Moseley and others. X-ray spectroscopy continues to uncover the electronic structure associated with materials properties and processes in the material world. There are several excellent books on the 20th century physics history, e.g. Inward Bound: Of Matter and Forces in the Physical World [11] by Abraham Pais, and a brief account of early x-ray history by Alexi Assmus can be found in [12].

Currently time-resolved experiments with short-pulse radiation are refined to reach atomic time scales: the time scale for nuclear dynamics is typically in the femtosecond range, while electron dynamics is rather in the attosecond range. For the probe radiation to have a defined frequency its period has to be substantially shorter than the pulse itself, and thus also such experiments call for x-rays.

With x-rays the properties of matter can be understood in terms of structure and dynamics on the atomic length and time scales. It is in that context that the success of synchrotron radiation can be appreciated. Synchrotrons provide brilliant x-rays with variable and well-defined energy and polarization state.

Synchrotron radiation was first observed in the General Electric synchrotron accelerator (figure 1.1) built in 1946 and announced in May 1947 in a letter [13]:

Figure 1.1. First observation of synchrotron light at the General Electric laboratory 1947 in the 70 MeV synchrotron. The image is taken from [14].

On April 24, Langmuir and I were running the machine and as usual were trying to push the electron gun and its associated pulse transformer to the limit. Some intermittent sparking had occurred and we asked the technician to observe with a mirror around the protective concrete wall. He immediately signaled to turn off the synchrotron as ‘he saw an arc in the tube.’ The vacuum was still excellent, so Langmuir and I came to the end of the wall and observed.

The first attempt to characterize synchrotron radiation over a large energy range was made by Hartman and Tomboulian in 1953 [15]:

An attempt was made … to extend the region of observation to the vacuum ultraviolet. Since a grazing incidence spectrograph was available and could readily be modified for this purpose, it was decided … to proceed directly with the exploration of the soft x-ray region (50 Å to 500 Å).

The same authors followed up with thorough investigation and in their 1956 paper [16] one can find this hand-drawn figure (figure 1.2), which compares the angular distribution of the radiation from an accelerating charge at classical and relativistic speed. As a consequence of special relativity, radiation from a particle at high speed is emitted primarily in the forward direction, giving the radiation fascinating properties. In synchrotrons, electrons (or positrons) are moving very close to the speed of light. Synchrotron radiation is our most common everyday application of special relativity.

Figure 1.2. Accelerating electrically charged particles emit radiation. At slow speed the angular distribution is donut shaped with maximum intensity perpendicular to the acceleration. At relativistic speed the intensity maximum is in the forward direction. The drawing is taken from [15].

In this book I will take a heuristic approach to understanding the basic properties of synchrotron radiation, rather than attempting a formal derivation of the formulae. For those who are theoretically inclined there are several texts. A description of synchrotron radiation was worked out early on in the classical paper by Schwinger [17]. A more recent comprehensive theoretical description is given by Kim [18], and the basics are treated in several textbooks on electromagnetic field theory.

The development of synchrotron radiation techniques since the early days is a success story. The first generation synchrotrons were built for collision experiments, with synchrotron radiation as an unwanted by-product. The usage of the radiation was termed ‘parasitic’. The second generation storage rings were dedicated to and optimized for producing synchrotron radiation, and the third generation sources feature so-called insertion devices, magnetic structures with which the radiation can be tailored for specific purposes. For the next generation the ultimate storage ring is envisioned, for which the performance is limited by fundamental principles only. Closely related is the recent development of x-ray free-electon lasers, which I will describe briefly here.

Taking the brilliance of the radiation as a figure of merit, the improvement since the discovery emulates Moore's law for microelectronics integration: whereas computer speed is only doubled in 18 months, synchrotron radiation brilliance is tripled (figure 1.3). The concept of brilliance will be discussed in more detail later, but briefly it reflects the number of photons of a certain wavelength that can be concentrated in a small area (as compared to the number of microelectronic functions in a small area). This development creates overwhelming new opportunities for experiments which rely on x-rays.

Figure 1.3. Until around 1960 the best x-ray sources were x-ray tubes, based on the same principle as used by Röntgen. With synchrotron radiation brilliance is increasing rapidly. Courtesy of O G Shpyrko [20].

There are a large number (>50) of dedicated synchrotron laboratories around the globe, and they are easy to find via their homepages [19]. Many of them have a lot of information and nice animations which describe synchrotron radiation.

1.3 Outline

I will start by introducing concepts which are often used to describe electromagnetic radiation and radiation sources in chapter 2. In chapter 3 we will see how the properties of synchrotron radiation emerge, by considering the emission from particles traveling close to the speed of light. The analysis is simplified when the acceleration and the velocity directions of the emitting particles are perpendicular, and we shall see that a simple understanding only requires acceptance of the most basic assumptions of special relativity. In chapter 4 I will discuss so-called ‘insertion devices’, and especially the undulator, which today is the most common radiation source at synchrotron radiation facilities. In chapter 5 I will briefly describe the basic ideas behind x-ray free-electron lasers. Although this is different from synchrotron radiation facilities, it is closely related. Finally, in chapter 6 I will describe x-ray optical properties and the implications for the optical elements and beamline design.

Basic understanding of the phenomena is at the forefront throughout the book. The large number of equations may be discouraging to many would-be synchrotron-radiation users and generally interested readers with little mathematical background. Mathematics is unavoidable to appreciate the importance of the various parameters for the properties of the radiation. However, the mathematical reasoning in this book is simple, and rarely advances beyond the level of Pythagoras' theorem. In cases where a rigorous analysis and quantitative predictions require more advanced algebra, I will refer to other texts.

References

[1] Attwood D 2007 Soft X-rays and Extreme Ultraviolet Radiation: Principles and Applications (Cambridge: Cambridge University Press)

[2] Willmott P 2011 An Introduction to Synchrotron Radiation: Techniques and Applications (New York: Wiley)

[3] Margaritondo G 2002 Elements of Synchrotron Light for Biology, Chemistrty, and Medical Research (Oxford: Oxford University Press)

[4] Als-Nielsen J and McMorrow D 2001 Elements of Modern X-ray Physics 2nd edn (New York: Wiley)

[5] Hoffman A 2007 The Physics of Synchrotron Radiation (Cambridge: Cambridge University Press)

[6] Mobilio S, Boscherini F and Meneghini C (eds) 2015 Synchrotron Radiation: Basics, Methods and Applications (Berlin: Springer)

[7] http://www.eecs.berkeley.edu/~attwood/

[8] http://ukcatalogue.oup.com/category/academic/series/physics/ossr.do

[9] Jaeschke E, Khan S, Schneider J R and Hastings J B Synchrotron Light Sources and Free-Electron Lasers (Berlin: Springer)

[10] http://www.cxro.lbl.gov/

[11] Pais A 1988 Inward Bound: Of Matter and Forces in the Physical World (Oxford: Clarendon)

[12] www.slac.stanford.edu/pubs/beamline/25/2/25-2-assmus.pdf

[13] Elder F R, Gurewitsch A, Langmuir R V and Pollock H C 1947 Phys. Rev. 71 829

[14] Pollock H C 1983 Am. J. Phys. 51 278–80

[15] Hartman P L and Tomboulian D H 1953 Far ultraviolet radiation from the Cornell synchrotron Phys. Rev. 91 1577

[16] Tomboulian D H and Hartman P L 1956 Spectral and angular distribution of ultraviolet radiaton from the 300-Mev Cornell synchrotron Phys. Rev. 102 1423

[17] Scwhinger J 1949 Phys. Rev. 75 1912–49

[18] Kim K-J 1988 Characteristics of synchrotron radiation AIP Conf. Proc. 184 565

[19] www.lightsources.org/regions

[20] Shpyrko O G 2004 Experimental x-ray studies of liquid surfaces PhD Thesis Harvard University

IOP Concise Physics

Synchrotron Radiation

An everyday application of special relativity

Jan-Erik Rubensson

Chapter 2

Properties of radiation

Before describing the properties of synchrotron radiation we introduce some key concepts, by which electromagnetic radiation in general is described.

2.1 Intensity

The energy density associated with electric, , and magnetic, , fields in vacuum is

where in SI units the vacuum permittivity is , and the vacuum permeability is . In an electromagnetic plane wave

so that

Intensity is defined as the average energy, , per unit time, , traveling through cross section with area, (see figure 2.1). The corresponding volume is . For a harmonic wave in vacuum with amplitude we have , and the intensity is

Figure 2.1. Intensity is the average energy per unit time flowing through the cross section area . The intensity of a plane harmonic wave is given by the average energy contained in a cylinder with area and length .

The speed of light in vacuum is , and is the amplitude of the wave in [Vm−1]. Intensity has the unit [Wm−2] as expected. If you don’t feel comfortable with these considerations you may consult electricity and optics text books. This discussion is covered e.g., in [1].

In a quantum mechanical description the intensity is related to number of photons per time and area. The energy density is related to the number of photons, , and their frequency, :

where is Planck’s constant. Assuming a constant flux of photons, the intensity (average energy per unit time traveling through the area ) is then

The classical equation (2.4) and the quantum mechanical equation (2.6) appear very different. As long as we are doing experiments in the ‘linear regime’, where excitations are made with one-photon-at-a-time, it is convenient to count the number of photons, as opposed to considering the E-field amplitude. Thus, equation (2.6) is a more convenient starting point. For typically photons per second with an energy of 1 keV, the power is 0.16 mW, and focused in a 1 mm^2 spot the intensity is around 160 Wm−2, corresponding to a classical field amplitude in a harmonic wave of around 350 Vm−1.

Intensity is a property of the radiation, and not of the source.

2.2 Flux

Many experiments require monochromatic radiation. Therefore, the spectral purity of the source is an important parameter. One has introduced a number of concepts which are relevant for experiments. On most of these concepts there is no consensus on the terminology, so one has to be careful and look at the definition in each specific case. The flux of a source is normally defined:

with the unit

The notation may be confusing; the intention is that only photons within the 0.1% frequency bandwidth (BW) are to be counted (figure 2.2). In this way ‘flux’ does take the spectral purity into account, e.g. at a nominal photon energy of 1000 eV, only photons/second within the 999.5–1000.5 eV band contributes to .

Figure 2.2. Only photons in 0.1% bandwidth contribute to the flux at a certain photon energy.

2.3 Emittance and Liouville’s theorem

The usefulness of a source is also critically dependent on its size and the angular divergence of the emitted radiation. The spatial extensions are often given as the standard deviation of a distribution, and , in the horizontal and vertical direction, respectively. Similarly, the angular distribution of the flux is characterized by the standard deviation of the intensity around a nominal direction, and , with respect to the horizontal and vertical plane, respectively. These measures contain an implicit assumption of a statistical distribution of the intensity. In addition, since the two dimensions are separated, the solid angle defined by gets the unit rad^2 rather than sterad. This is convenient as long as small angles are considered.

The emittance in the horizontal and vertical directions is defined (figure 2.3):

Figure 2.3. Vertical emittance is defined by the extension of the source and the angular divergence in the vertical direction.

It is immediately realized that small emittance is desirable for a radiation source. As intensity is inversely proportional to the cross section area, decreasing spatial extension implies increasing intensity. Also, one can realize that a source is more useful if all the photons go in the same direction, than if they are isotropically distributed. No focusing optics is needed to increase intensity if we have a hypothetical point source, which radiates in one direction only. This would imply zero emittance, which is unphysical: for any number of photons per second the intensity would be infinite. At the other extreme, it is difficult to design an efficient optical system if the source is large and radiates in all directions.

These notions are quantified in Liouville’s theorem, which states that the emittance can only be conserved or made larger in any optical system. We will come back to this in the optics section. Here, it suffices to understand that a small emittance is important. The concept of emittance is equivalent to etendue, which is used in other contexts.

2.4 Brilliance

The importance of emittance motivates the introduction of brilliance (or spectral brightness), which is defined as flux/emittance. This is a central concept for characterization of radiation sources:

in the common units:

This is the common notation and unit in which various radiation sources are compared. We shall see that the synchrotron provides outstanding brilliance for most of the electromagnetic spectrum.

2.5 Polarization

The electric field of a harmonic electromagnetic wave propagating in the positive direction of a Cartesian coordinate system can be written:

where and are the amplitudes of the electric field in the and directions respectively, λ_0 is the wavelength of the radiation, and is the phase shift between the two components. If the radiation is linearly polarized. If , maps an ellipse perpendicular to the direction, and the radiation is elliptically polarized. If also the radiation is circularly polarized; the magnitude of is constant and it rotates perpendicular to the direction. According to the most common convention, implies right-handed polarization and implies left handed polarization.

The polarization state of a coherent wave described by equation (2.11), with well-defined wavelength and phase at each , is thus easy to determine. In principle, any electromagnetic wave propagating along can be described as a superposition of waves of the same type as (2.11). In practice, however, the polarization often changes so rapidly that it is not meaningful to measure the polarization state of the wave, and such radiation is termed unpolarized. Usually, sources emit partially polarized radiation, comprising both unpolarized components and components with well-defined polarization state. It was shown by George Gabriel Stokes in the nineteenth century that partially polarized radiation can be characterized by four measurable quantities: the Stokes parameters. The time-integrated measurements include the total intensity (), the difference between linearly polarized intensity in and directions (), at 45 degree angle (), and between left- and right- circularly polarized radiation ().

2.6 Coherence

Coherence, the phase order, is a property of the radiation, rather than of the source, and a coherent fraction can always be filtered out from any source, as illustrated in figure 2.4. A source which emits radiation for which this filtering process does not overly reduce the power is beneficial if coherent radiation is wanted.

Figure 2.4. Illustration of coherence, taken from Soft X-rays and Extreme Ultraviolet Radiation [2] courtesy of A Schwalow.

The radiation from a filament of a light bulb (figure 2.4a) has limited phase order. Using a pin hole (figure 2.4b) a spatial (or transverse) coherence is achieved, and using a color filter (figure 2.4c), which chooses only one wavelength temporal (or longitudinal) is achieved. With both (figure 2.4d) spatially as well as temporally coherent light has been filtered out, albeit at the expense of most of the original intensity.

2.6.1 Transverse coherence or spatial coherence or coherence width

Figure 2.4b illustrates the fact that wave fronts propagating from an aperture are determined by diffraction, thereby providing transverse phase order. The central diffraction maximum after radiation passes an aperture is a maximum precisely because wavelets from all positions in the aperture are in phase over the spherical wavefront. Thus, within the central maximum there is perfect spatial coherence. To find the coherence width we consider the diffraction from a circular pin-hole in a screen which is introduced perpendicular to the beam. We recall the criterion for the angle to the first intensity minimum,

where is the wavelength of the radiation and is the diameter of the aperture. At the wavelets interfere destructively, and the coherence width corresponds to a smaller angle, and the transverse coherence at a distance from the aperture is lt (figure 2.5), and it it is safe to put a lower limit to the transversal coherence:

We see that this discussion is related to the emittance concept. The radiation from a source with small ( and ) compared to the wavelength has a large coherence width. Diffraction puts an upper limit to the emittance: no perfectly parallel beam can be emitted from a source with finite .

Figure 2.5. Transversal coherence determined by diffraction in an aperture is ensured in a radiation cone with an opening angle which is a fraction of the angle to the first diffraction minimum.

2.6.2 Longitudinal coherence or time coherence or coherence length

If the color filter (figure 2.4c) lets through monochromatic radiation with one well-defined wavelength, , also the phase along the beam is well-defined, and the coherence length is infinite. In practice, however, it must be a band pass filter letting through radiation in an interval from to . In figure 2.6 we see two harmonic waves at an instant, where (red) and (blue) are in phase at length = 0. Because they have different wavelengths their phase relation changes along the beam until they, after , are completely out of phase. Thus

Figure 2.6. The waves with wavelength (red) and (blue) are in phase at length = 0, and gradually goes out of phase. The length along the beam where phase order is maintained is the coherence length.

If we assume that we can write

and therefore

Thus, after wavelengths the phase order is lost in a beam with wavelengths in a range of around , and the coherence length can be defined

Coherence length is determined by spectral purity only and is thus associated with flux and brilliance, as defined above.

2.6.3 Orbital angular momentum

A well-defined phase does not imply that phase must be the same throughout a cross section of a beam, as in a conventional plane wave. Instead the phase can be controlled, e.g. to change smoothly when measured in a circle around the center of the beam (figure 2.7). Such a beam of photons possesses orbital angular momentum (OAM), which notably is different from polarization properties, which are associated with the spin of the photons. Although, the interaction between matter and OAM beams has been explored for some time, it is only during recent years that it is being discussed in the x-ray range.

Figure 2.7. Illustration of helical phase fronts for an electromagnetic wave, of different orbital angular momentum quantum numbers . A conventional plane wave has , and a phase change by per wavelength gives the photons an OAM of . The Figure is reproduced from an article in the Proceedings of the Royal Society of London by Miles Padgett [3].

2.7 Time structure

The duration of an electromagnetic wave, , can be manipulated from being very long, where for all practical purposes one can talk about continuous wave radiation, to very short. Today femtosecond pulses are generated on a regular basis, attosecond pulses are used in frontier experiments and zeptosecond pulses are being considered. Heisenberg uncertainty (or Fourier analysis) puts a limit to the energy (or frequency) definition in the pulse, ,

where . Thus, for femtosecond pulses with we get , and for attosecond pulses with we get . Only if the photon energy is much larger than are ‘frequency’ and ‘wavelength’ meaningful concepts. Differentiating and using equation (2.15) we get for :

and therefore

Combining this relation with equation (2.14) we get the relation between pulse duration and coherence length:

Note that this expression parallels the equation for spatial coherence, achieved due to spatial constraints (2.13). The angular spread in diffraction can also be derived using the Heisenberg uncertainty relation for momentum and space, whereas the energy–time relation is central for equation (2.17). Unfortunately, decreasing the duration impacts the coherence only when the pulse is so short that the wavelength is as long as the pulse, and increasing duration does not necessarily improve . Rather, increasing the coherence length, e.g. by introducing a color filter, increases the duration of a short pulse, in accordance with equation (2.17).

References

[1] Hecht E 2001 Optics 4th edn (New York: Addison Wesley)

[2] Attwood D 2007 Soft X-Rays and Extreme Ultraviolet Radiation: Principles and Applications (Cambridge: Cambridge University Press)

[3] Padgett M 2014 Light’s twist Proc. R. Soc 470 20140633

IOP Concise Physics

Synchrotron Radiation

An everyday application of special relativity

Jan-Erik Rubensson

Chapter 3

Accelerating charged particles

In the classical world the source of electromagnetic radiation is accelerating charged particles. This is familiar from accelerating electrons in antennas, bremsstrahlung, and accelerating ions in plasmas, as well as blackbody radiation in light bulb filaments. Although quantum physics indicates that a transition between discrete states is the source, rather than acceleration in the classical sense, it is still charged particles which emit radiation as they change state.

3.1 Angular distribution at low speed

Using Maxwell’s equations it can be shown that the intensity, or average power per unit area, at a distance r radiated by a charge e that changes momentum at the rate is

where e is the charge, is the rest mass of the particle, and is the angle between the direction of acceleration, , and the direction of the emitted radiation. Thus, the emitted power reaches a maximum in the direction perpendicular to the momentum change, and is zero along the momentum change direction. This gives the classical donut shape (figure 3.1). The intensity of the emitted radiation is proportional to the inverse square of the distance to the observer, just like for a spherical wave.

Figure 3.1. The angular distribution of electromagnetic radiation from an accelerating electric charge moving at slow speed. The pattern is rotationally symmetric around the axis of momentum change, and the figure represents the typical donut shape.

3.2 Angular distribution at high speed

Synchrotron radiation is emitted by particles at very high speed. At typically 1 GeV the energy of the electron is around 2000 times its rest energy and the speed is . Special relativity is thus central for deriving its properties, and many features follow from the angular distribution. As we have seen already in chapter 1, special relativity folds the radiation in a narrow cone in the forward direction, which contributes to give the radiation the desired low emittance (see chapter 2). The effect can be understood by the same arguments that many physics textbooks use to introduce the concept of time dilation.

Assume that a particle is traveling with the speed v, and that it emits light perpendicular to its path. Light travels a distance during the time , measured in the rest frame of the particle. As the light and particle paths are perpendicular, no Lorentz contraction applies, and thus an observer in the laboratory frame measures the same perpendicular distance, . The total light path length measured in the laboratory frame is, however, much longer, as the speed of the particle during emission must be taken into account. In a classical world that would have implied a larger light speed, but according to the postulate of special relativity the speed of light is the same in both frames; therefore the time intervals must be different. According to Pythagoras’ theorem (figure 3.2), the light path length measured in the laboratory frame, , is

We see directly in figure 3.2 and from equation (3.2) that this implies that , and it is also immediately realized that the direction of the light and particle paths becomes almost the same if the speed of the electron, v, approaches the speed of light. The angle between the two,

Figure 3.2. A light pulse is emitted from a moving charge, perpendicularly to its path. When the light pulse has moved the distance in the particle rest frame, it has moved the distance in the laboratory frame. As is the same in both frames, Pythagoras’ theorem can be used to derive time dilation, and to understand the effect that fast-moving particles radiate primarily in the forward direction.

is fully determined by time dilation. Thus, both the angular distribution of light emission from a fast particle and time dilation is a direct consequence of the assumption that the speed of light is the same in all frames of reference.

The explicit expression for time dilation indeed follows directly from Pythagoras’ theorem (3.2), e.g. by solving for :

where we have introduced the relativistic factor,

with being the total energy of the particle. The last step in equation (3.6) is also due to basic special relativity.

When the speed of the electron approaches the speed of light becomes large. For synchrotrons where the electron energy is given an energy of some GeV the range is typically . Recalling that the rest energy of the electron, , and the electron energy in the specific storage ring, , can be easily calculated from equation (3.6).

The angle between the path of light which in the rest frame of the electron is emitted perpendicular to the path of the electron, is thus emitted in the laboratory frame with the angle

from the path of the electron. As is a large number this is a small angle, and it is said that the light intensity is folded into the forward direction. Although the argument above is essential for the understanding of the angular distribution, we know from equation (3.1) that the radiation from an accelerating charged particle is more complicated than just being perpendicular to its path.

Typically, the bend-magnet field in a synchrotron is applied perpendicularly to the particle path, giving rise to a Lorentz force, which changes the momentum of the particle in the direction perpendicular to both the velocity of the particle and the magnetic field

At classical particle speeds it will thus radiate in a donut shape (equation (3.1)) where half the power is emitted in directions forward of a plane perpendicular to the path, and half the power is emitted in the backward direction (see figure 1.2).

To derive a quantitative expression for the angular distribution the Lorentz transformation of the classical angular distribution to the laboratory frame must be performed. For emission which is emitted in non-perpendicular directions it becomes somewhat more complicated than the derivation of equation (3.7). Only for emission in the perpendicular direction can one disregard Lorentz contraction, and in general one must apply the full Lorentz transformation of velocities to the classical radiation pattern. Although this is not done here, we note that the result agrees with what we expect from our simplified consideration: the donut shape for slow particles is folded into the forward direction when the speed increases, and half the intensity is emitted in a cone with the opening angle

This basic effect of special relativity is directly beneficial as the angular spread is important for the emittance (chapter 2). If one can control the position and momentum of the electron it leads to a small emittance, allowing for optics that can produce small foci. In the following we will show how this effect also determines the energy spectrum.

3.3 Energy distribution

We have seen that photons primarily are emitted in a cone in the forward direction, but the speed of the electrons is not much less than the speed of light. Thus, electron and photon follow each other closely in a straight path, and as the electron path is bent, photons continue straight ahead. While radiation is generated in the bend the direction of the radiation cone changes rapidly, and the radiation from a circulating electron simulates light from a lighthouse. The light cone sweeps by an observer very fast. This time is so short that the Heisenberg uncertainty principle implies a very large energy spread.

In figure 3.3 the electron is traveling from left to right along the red path, which is circular with the radius R. At point A the cone of radiation with the opening angle first starts to radiate in the horizontal direction, and at point B it stops. A downstream observer observes a light pulse generated in this bend. It is straightforward to calculate the duration. The time it takes for the electron with speed v to travel along the bend from A to B is

Figure 3.3. A fast electron, emitting radiation in the cone, is moving along the red path with radius R. In the direction of the blue arrow the first light is emitted at A and the last light is emitted at B. Light from A reaches an observer on the blue line just before the light from B, the pulse duration becomes very small, and the energy spread very large.

The time it takes for the photon to travel in the straight path from to is:

As the photons generated at reach the observer slightly before photons generated at B. The time difference gives the observed pulse duration.

It is immediately realized that while , is very small (the factor of 2 is conventionally introduced, and it depends on how the pulse duration is defined).

This simple derivation is performed entirely in the laboratory reference system, without relativistic transformations. So far, textbook time dilation is the only relativistic effect considered. It is, however, convenient to express the duration in relativistic terms, and we define the parameter

When we get a useful relation between and :

Using this relation in the expression for the pulse duration we get

With Heisenberg’s uncertainty principle

we get the energy spread

To relate this energy spread to the magnetic field, , rather than the radius of curvature of the path, we use the expression for the Lorentz force in the case of centripetal acceleration, and the correspondence to Newton’s second law with relativistic momentum, . In the case where the force is perpendicular to the velocity, the speed, , does not change with time, and therefore also is time-independent (in the general case , which complicates the derivation). Only the direction of changes, and the change in momentum with time takes the form

For centripetal acceleration

and thus

The equation of motion for acceleration in the direction perpendicular to the particle path and a magnetic field B, applied in a direction perpendicular to the beam is given by the Lorentz force (3.8),

and therefore

This is the Larmor radius, and we see that relativity makes it times larger than in the classical case. This can be thought of as due to the relativistically increased mass of the particle. Thus, special relativity makes things experimentally more difficult, as a large B field is needed to make a small bend radius. Combining equations (3.17) and (3.22) we see that the energy spread expressed in terms of magnetic field is proportional to rather than :

With realistic values, and , . The energy spread goes well into the x-ray range.

Note that apart from the relativistic forward folding, the derivation of the energy spectrum from a bend with radius is performed in the laboratory frame without explicit relativistic considerations. It is only when relating the bend to the magnetic field that relativistic momentum must be taken into account.

Exact expressions for the intensity distribution over photon energies and emission angle can be derived. The mathematical description of these distributions involves modified Bessel functions (see e. g. [1]). The normalized expressions for on-axis flux, , and vertically integrated flux, , are plotted in figure 3.4. Both show a very wide energy distribution extending up to high energies where there is a fairly sharp cut-off. Both functions are characterized by one important parameter: the critical energy, . At this energy, half the radiated power on-axis is at higher energies, and half is at lower. The expression for the critical energy is almost identical with the equation derived by simply considering the pulse duration and the Heisenberg uncertainty relation:

Figure 3.4. The normalized expressions for on-axis flux, , and vertically integrated flux, . The figure is taken from [2].

The shape of the distribution functions indicates that the simplified analysis captures the essential physics. Note that the angle-integrated energy distribution of intensity is different from the on-axis distribution. This difference can also be intuitively understood from our simple consideration: off-axis, the observed speed of the particle, which is the crucial variable, is achieved by a projection on the plane. Lower speed implies longer sweeping-by times and hence, lower energy. Therefore, lower energies are emphasized as a function of angle in the vertical plane.

In practical units the critical energy can be written:

Apart from natural constants, the energy spectrum is fully determined by the energy of the electrons, , and the magnetic field B.

3.4 Emitted power in a bend

The total emitted power is also significantly enhanced due to relativistic effects. This was a big problem for the early accelerator developers: particles lose energy as they emit radiation during the acceleration associated with the bending of the beam path. For the synchrotron radiation users this is instead an important asset.

The essence is that time intervals in the laboratory system correspond to much smaller time intervals in the rest system of the particle. While a particle can circulate a synchrotron for one day, as measured in the laboratory system this corresponds to less than a minute in the rest frame of the electron.

We first consider the classical case: from equation (3.1) one can derive the total power emitted in the classical case:

Apart from natural constants the total emitted power is thus determined by . For centripetal acceleration in a bend with radius we have as before (3.19):

and the kinetic energy of the particle is,

so that

Plugged into (3.26) this becomes

We see that the radiated power is proportional to the square of the kinetic energy of the particles. When the particle is moving at very high speed, the acceleration of the particle measured in a system which follows the particle path can be plugged into equation (3.26). To see what that implies in the laboratory system we can make the corresponding transformation of the acceleration. Considering that acceleration of the particle is perpendicular to its average path, the Lorentz length contraction becomes irrelevant and the relativistic transformation of acceleration depends again only on time dilation:

Time intervals measured in the rest frame of the particle are much smaller than in the laboratory frame. This implies that the acceleration in the rest frame of the particle, , is much larger, than the acceleration observed in the laboratory frame, . Using the relativistic expression for momentum , and the fact that so that is constant, we have:

and the emitted power is accordingly

The centripetal acceleration measured in the laboratory frame is for particles with , and we get the expression for the total power:

Thus, the radiated power is proportional to the fourth power of the energy of the particle when its speed is close to speed of light, while at low speed it is proportional to the square of the energy only. Apart from the particle energy and natural constants, only the radius of curvature at the bend influences the radiated power. If we want the expression in terms of magnetic field instead of curvature we use the relativistic Larmor frequency (3.22) and :

Thus, as the field needed to make a bend increases with energy, reducing the energy dependence of the power to at high speed.

A tacit assumption in the ‘derivation’ of the total power expression is that the energy of the particle is constant. This is obviously impossible because the particle radiates part of its energy. It is a safe approximation if the radiated energy is small compared to the initial energy of the beam. The energy lost per revolution in a circular orbit is

where we again have used that . In convenient units we have

In practice the radius of curvature is in the meter range and the energy of the beam is in the GeV range so we see that only a small fraction of the energy is lost per revolution, and the approximation made above is justified. Note, however, that a revolution in e.g. a storage ring with circumference m is accomplished in approximately microsecond, and thus the particle would radiate all its energy on the order of a second. The situation is similar to early atomic models which assumed a circular electron path around the nucleus. The energy would be radiated and the atom would not be stable. In the synchrotron the power corresponding to the radiation loss is provided by radiofrequency cavities, in which an electric field accelerates the particle to compensate for the energy loss.

With an average electron current of , corresponding to incoherently radiating particles per second passing a bend, equation (3.33) gives a total power of:

or from equation (3.34):

In practice, a storage ‘ring’, has consecutive bends and straight sections, where the particles do not radiate. As the particles radiate only in a small fraction of the path, the average power expression is modified. If L is the total length of the straight sections,

which since this gives a significantly smaller average power than given by equation (3.39). Still, the power needed to maintain a 100 mA beam of particles of GeV energies in a ring with radius of curvature in the meter range is very large. For the 3 GeV ring of MAX IV the synchrotron radiation power will be on the order of 500 kW. To reduce the power lost in radiation from particles of a certain energy in storage rings, increased size (R and/or L) seems to be the only possibility. For LEP2 in CERN where , m, and a particle current of mA corresponds to a power of 24 MW. Another way to reduce synchrotron radiation power is to use heavier particles. This is because we have used , where is the rest energy of the electron, to make the estimates based on equations (3.37) and (3.38). For a proton the energy loss is smaller than as it is proportional to the inverse fourth power of the rest energy.

For protons the radiation energy loss is no problem for accelerator physicists. At LHC one expects that a proton current which is two orders of magnitude larger than the electron current in LEP, at an energy of 7 TeV in a ring with smaller radius than LEP, will give a radiation power loss of only around 7 kW. Thus, only light particles (electrons and positrons) are considered when generation of radiation is the main purpose.

3.5 Brilliance calculations

We have seen that the energy spectrum, and the angular distribution of synchrotron radiation from a bend can be predicted in terms of only a few controllable parameters. In [3, 4] the on-axis photon flux dF per horizontal and vertical angle, in a certain frequency range is given explicitly by

in the unit photons/s/mrad^2/0.1 % BW.

This expression is very similar to the expression for brilliance (2.8), and it is indeed closely related to brilliance in an ideal storage ring, where position and momentum of the electron are controlled. On the home page of Center for X-ray Optics at the Lawrence Berkeley National Laboratory one finds a calculator [5] that allows for optimization of the radiation properties by variation of the crucial parameters in this equation.

This calculator is general, in the sense that it shows how varies as one leaves the horizontal plane ( ). As expected the photon flux decreases, because the observer leaves the forward direction. At angle of the peak value is half of the on-axis value, and also the energy position of the maximum decreases significantly. This decrease can be intuitively understood from our simple derivation: off-axis, the ‘observed speed’ of the particle is achieved by a projection of the motion in the plane. This implies longer sweeping-by times, and hence smaller energy spread. Therefore, lower energies are emphasized as the angle from the horizontal plane increases.

3.6 Polarization of bend magnet radiation

Synchrotron radiation has well defined polarization. The on-axis radiation is linearly polarized in the horizontal plane. This can be understood using a classical antenna analogue; as the direction of acceleration and the observer are in the horizontal plane no vertical components need to be considered. As the classical radiation doughnut is relativistically folded forward, indeed vertical components enter at small angles from the horizontal plane, but on axis they cancel. In the CXRO ‘flux calculator’ one can choose between s (horizontal) and p (vertical) polarization. Indeed no p polarized radiation is predicted on axis. As increases the flux of p polarized radiation increases to a maximum and decreases as the angle further increases.

In figure 3.5 we see how the vertical polarization maximum depends on the photon energy, relative to the critical energy. At 10% of the critical energy the maximum is already around , and as the angle increases both the vertical and horizontal components decrease and become equal around , where the intensity of both is on the 1% level, relative to the on-axis intensity of the horizontal component.

Figure 3.5. The angular dependence of the intensity of horizontal and vertical polarization, where is the photon energy and is the critical enegry. The figure is taken from [2]. It is adapted from [6].

The phase relation between the two components is fixed so that the radiation in general is elliptically polarized. This can also be understood from classical arguments. A charge at constant classical speed in a circular path radiates circularly polarized light perpendicular to the plane of motion, and obviously the helicity is opposite above and below the plane. Simultaneously, in phase, it radiates linearly polarized light in the plane of motion. In all other directions the radiation is elliptically polarized. For fast particles the radiation pattern is folded into the forward direction, but these notions are still instrumental for the understanding: the radiation is indeed elliptically polarized with opposite helicity above and below the horizontal plane. At angles where the vertical and horizontal components have the same magnitude the radiation is circularly polarized. The ability to vary the polarization properties by exploiting synchrotron radiation emitted at various angles has proven to be very useful, especially for applications in magnetism, where polarization-dependent scattering gives important information about magnetic moments.

3.7 Time structure of synchrotron radiation

The time structure of synchrotron radiation is set by technical constraints. It is in principle impossible to make a continuous current in a closed path as the path is deterministically defined by the electric and magnetic fields. Electrons have to be kicked in by an injection system, and energy lost in synchrotron radiation has to be refilled in high-frequency cavities. The electrons come in bunches with on the order of electrons, and the radiation correspondingly comes in flashes. Note that time-averaged properties are given in equations (3.40) and (3.41). In modern synchrotrons the bunches often have a duration 20–100 ps, and they are separated by around 2 ns. Thus, the peak power is around 100 times larger than given by the time-averaged equations above, and most of the time the synchrotron is dark. It should be noted that the duration of the flashes due to the passing-by time of a bunch is very large compared to the ‘flash’ associated with a the passing-by time of a single electron, which determines the energy spectrum.

For most experiments synchrotron radiation is used as if it were continuous wave radiation. For experiments requiring pulsed radiation, e.g. experiments relying on time-of-flight measurements, various patterns of bunches can be provided. If only one electron bunch circulates the ring (so-called ‘single-bunch mode’), the separation between the flashes, determined only by the circumference of the ring, is on the order of microseconds. In this range, mechanical choppers can be used to further extend the separation between pulses.

For experiments requiring high temporal resolution, e.g. pump-probe experiments, the bunch duration can be modified, the challenge being to reduce the duration. With ‘low momentum compaction’ factor (low-alpha mode) operation of the storage ring pulses with duration down to 1 ps can be delivered. With short-duration laser pulses a slice of the electron bunch can be selected, and the duration of the corresponding radiation can be below 100 fs. Such modifications reduce the brilliance significantly, and thus call for high-luminosity experiments.

3.8 Coherence of bend magnet radiation

The spatial coherence is determined by how well the motion of the electron in the beam can be defined. We have seen that an electron radiates within a cone with an opening angle of . If the position and momentum is well defined during the passing of the bend, the coherence within the radiation cone is spatially coherent. As the energy spectrum is intrinsically broad, however, extensive filtering is needed to increase the coherence length. We will come back to coherence in the context of undulators.

References

[1] Jackson J D 1999 Classical Electrodynamics (New York: Wiley)

[2] Center for X-Ray Optics 2009 X-ray data booklet (Berkeley, CA: Lawrence Berkeley National Laboratory) http://cxro.lbl.gov/x-ray-data-booklet

[3] Kim K-J 1988 Characteristics of synchrotron radiation AIP Conf. Proc. 184 565

[4] Attwood D 2007 Soft X-rays and Extreme Ultraviolet Radiation: Principles and Applications (Cambridge: Cambridge University Press)

[5] http://henke.lbl.gov/optical_constants/bend2.html

[6] Green G K 1976 Spectra and optics of synchrotron radiation Brookhaven National Lab. Report BNL 50522 (New York: Brookhaven National Laboratory)

IOP Concise Physics

Synchrotron Radiation

An everyday application of special relativity

Jan-Erik Rubensson

Chapter 4

Insertion devices

The wiggler is effectively flexible multiple bend magnets, and in the undulator the bends radiate coherently.

4.1 Wigglers

Insertion devices introduce magnetic field structures to manipulate the electron path, thereby tailoring the emitted radiation. From bend magnet radiation we know that the parameter that determines the energy spectrum, the critical energy, is set by the energy of the particle, and the magnetic field, , according to equation (3.25):

the total power according to equation (3.38)

and the on-axis photon flux by equation (3.41)

Introducing a B field in a straight section, to make the beam wiggle (figure 4.1), leads to bend-magnet type radiation, where B can be chosen to place the energy distribution at a desired energy (critical energy, 3.25) and emitted total power (3.38). The magnetic field can be very high with superconducting magnets, which are often used to produce hard x-ray radiation from low-energy storage rings where conventional bending magnet radiation reaches the soft x-range only.

Figure 4.1. A wiggler is effectively consecutive bend magnets, which make the electron wiggle in its path.

A periodic B field structure with N periods simulates consecutive bend magnets (figure 4.2). The maximum flux is, correspondingly, times the maximum flux of a bending magnet with the same B field.

Figure 4.2. Using the ‘flux calculator’ at the homepage of CXRO, wiggler radiation can be simulated. Instead of a bend with a typical field of 1 T (red) one can use superconducting magnets with a field of 7T (blue), moving the critical energy from around 2.4 keV to well above 16.8 keV, and using 5 periods (green). the total flux increases by a factor of 10. In this example the on-axis flux is calculated for a 1.9 GeV ring with a current of 400 mA.

The polarization can also be influenced by various magnetic structures. Wiggling in the vertical plane gives vertical polarization on axis. Wiggling with components in both planes gives elliptically polarized radiation and also circularly polarized radiation can be produced. These options are more often taken advantage of in undulators, which will be the topic of the next subsections.

4.2 The undulator

As wigglers, undulators are based on periodic magnetic structures. The crucial difference is that the magnetic field is weaker. Somewhat counter-intuitively this makes the undulator more powerful than the wiggler in many respects. The energy distribution becomes completely different from bend magnet radiation: the intensity is strongly peaked at certain energies: undulator harmonics. We will derive the undulator equation, which gives the energy for the undulator harmonics, in terms of a few adjustable experimental parameters.

4.3 Weak field, on axis

The undulating electron can be seen as a radiating dipole. As a classical electron moves with speed through the magnetic structure, with the period length , its undulation frequency, and thereby the frequency of the radiation is set by

The period length of the magnetic structure is typically in the cm range, and even at the classical frequency is in the fclassical ≈ 10^9−10^10 Hz range i.e. microwave frequencies. As the emitter is moving against the observer with high speed there will be a large Doppler shift towards large frequencies. We consider special relativity.

When the electron travels at relativistic speed, , it sees the magnetic structure Lorentz contracted, , and therefore the emitted frequency, in the rest-frame of the electron, is

This relativistic contraction shifts the frequency into the range of visible light. As the electron is moving towards the observer in the laboratory frame the radiation is Doppler shifted. For an observer on-axis the relativistic Doppler shift gives the measured frequency

In the first step of equation (4.3), the factor is due to time dilation; as , time intervals measured in the laboratory frame are larger than time intervals measured in the electron rest frame, frequencies measured in the laboratory frame are smaller than in the electron rest frame. Thus, the blueshift due to the Lorentz contraction of the magnetic structures is cancelled by time dilation in the Doppler formula, and the final result is the same as for classical speeds.

An alternative way of analyzing undulator frequency is to assume that electrons radiate coherently from each part of the undulator. Analogously to the derivation of bend magnet energy spread via geometrical arguments (figure 4.3) entirely in the laboratory frame we find a geometric condition for constructive interference. This derivation follows the classical textbook arguments for the Doppler effect. After the electron emits radiation in one bend this radiation travels the distance . During the same time the electron, only travels a distance which is shorter, since it has lower average velocity, v, in the forward direction.

and thus:

Figure 4.3. After being emitted in one bend, the radiation travels a distance , while the slower electron travels the distance .

If a multiple of the wavelength equals the difference between the distances the radiation and the electron has passed, (where is a positive number), the radiation emitted in the next period will be in phase with the radiation emitted in the first period. We get , and thus

We see that for equation (4.3) and (4.5) are identical. The picture of a fast-moving oscillating charge is equivalent to the picture of coherent radiation from each undulator period.

Even if the results are the identical for relativistic and classical speeds, we can simplify by rewriting the equations when :

One can identify the relation between the wavelength of the radiation emitted in the forward direction and the undulator period, which takes the simple form for the first harmonic:

As the relativistic factor is a big number, the Lorentz contraction and the Doppler shift takes the wavelength from the cm to the Ångström range.

4.4 Off the axis

Since the emitted frequency is determined by the Doppler shift, it is obvious that the emission off axis must have a lower frequency. The angle between the average direction of the particle velocity and the direction of observation. is given by the (classical) Doppler shift formula (equation (4.3) modified):

This equation explicitly describes how the frequency decreases with increasing angle. To write it in a more convenient form we only consider small angles, where , and high speeds, where , so that

The wavelength of the radiation , emitted at an angle from the axis can thus be identified:

From the point of view of constructive interference between coherent radiation from different undulator periods, the wavelength can be be derived by geometrical considerations. (figure 4.4).

Figure 4.4. When the radiation is emitted at an angle the condition for constructive interference applies for wavelengths which are longer than for the radiation emitted on axis.

If the condition for constructive interference is fulfilled for on-axis radiation with wavelength it is fulfilled for radiation with wavelength at angle . Geometry gives :

where we have used for small and we get

which directly leads to equation (4.10)

4.5 Spectral purity, and the central radiation cone

Before discussing the magnetic-field dependence in the next section, we consider the spectral purity of the undulator radiation. The notion that the oscillation of the charge sets the frequency immediately also gives the spectral width. In concordance with Fourier analysis, it is given by the number of oscillations, i.e. the number of undulator periods, N:

This expression gives the ‘natural’ spectral width on axis. For realistic values, e.g. , the spectral with at 100 eV is thus around 1 eV, and the longitudinal coherence . These conditions apply not only on axis, but in the ‘central radiation cone’, where the angular variation is sufficiently small not to compromise this ‘natural‘ width. The wavelength changes are given by equation (4.10). Changing the angle from to , the angle defining the central radiation cone, leads to a wavelength shift from λ:

and, as we have

Combining equations (4.11) and (4.13) we get the angle of this ‘central radiation cone’, where the wavelength spread is set only by the number of undulator periods

This has implications when considering brilliance and coherence, and it also poses a restriction on the electron beam: not to compromise the spectral purity in the undulator radiation the angular spread also of the electron beam must be smaller than

4.6 Finite magnetic field

So far we have assumed that the magnetic field is very small. This may seem an academic exercise as very small magnetic implies very low photon flux. As increases, so does the amplitude of the oscillation of the electron, which influences also the wavelength of the emitted radiation. We assume a coordinate system where the net motion of the particle is in the direction, and the excursions are in the plane. Increasing amplitude implies an increasing component of the velocity, and as the speed (and therefore also ) is constant, this in turn implies a decreasing component of the velocity. As the undulator wavelength is determined by the relativistic contraction of the undulator period and the Doppler effect the decreasing z component of the velocity is crucial.

From the relativistic factor we derive an expression for :

Solving for we get

As expected, decreases when increases. We determine due to a field with only a component in the direction, varying with according to

As the velocity component along is totally dominating, the Lorentz force gives rise to an acceleration in the direction. Using relativistic momentum, and the fact that is constant in time, the equation of motion can be written

Time integration of this equation gives:

and therefore the component of the velocity is

where we have introduced the dimensionless magnetic deflection parameter:

Apart from natural constants it contains the amplitude of the oscillating magnetic field, and the undulator period. We will come back to its interpretation after analyzing the component of the velocity and its impact on the wavelength of the radiation.

Combining equation (4.16) and equation (4.20) we get

The Doppler effect and the Lorentz contraction, determining the energy spectrum, is dependent only on the component of the velocity. The average decrease due to the finite component is achieved by averaging the function over a period. As the result of this averaging is we have

where we have introduced the effective axial relativistic factor:

The observed wavelength will be governed by this factor, rather than the strict relativistic factor. Using this parameter in equation (4.10) we get the undulator equation:

Considering interference between emission from the bends (figure 4.5) the effective axial slowing-down of the electron due to the excursion is taken into account in an analogous way as before. During the time when the photon travels the distance at speed c the electron only travels at the average speed :

This can be rewritten with (4.23)

using that for small x.

When lB is a multiple of the wavelength λm we have constructive interference, and

which gives the corresponding contribution to the undulator equation. When taking the angular dependence into account it is given by (4.25) for the first harmonic:

The wavelength of the radiation is determined by the energy of the electron (γ) and the undulator period (λu), which are normally rather difficult to vary in an experiment. The magnetic field amplitude (B_0) is conveniently controlled as the gap between the magnetic poles in a permanent magnet undulator is varied, or the current in an electromagnet undulator. Thereby, the wavelength is varied, and as expected an increasing field increases the wavelength. In practical units we get:

where

From this picture, the occurrences of higher harmonics are easily understood. To derive higher harmonics from the oscillating-dipole picture is more involved. As we have seen already, the oscillation of the velocity component in the -direction (equation (4.16)) leads to an oscillation of the velocity component in the -direction (equation (4.23)). This oscillation along the direction has twice the period of the oscillation in the direction. Thus, the radiation has twice the frequency, and it is this oscillation that gives rise to the second harmonic. Note that, since the acceleration is directed along the undulator, the second harmonic does in principle not radiate on-axis.

Figure 4.5. After being emitted in one bend, the radiation travels a distance , while the slower electron travels the distance .

To get to higher harmonics one has to analyze the motion of the electrons in detail. Here we make some brief remarks only. Using equation (4.20), and the fact that we see that the angle that the electron makes with the axis is

Thus, the maximum excursion angle is

Obviously, the electron excursion angle lies within the radiation cone, within from each bend if . If this is the case, the radiation from each bend can interfere, and we have undulator radiation. If the radiation from each bend does not overlap () this coherence is lost, and the radiation can rather be analyzed as wiggler radiation. In the intermediate region, higher harmonics get more intense as increases. Since the number of cycles increases with , the bandwidth gets correspondingly smaller.

Obviously, to get pure undulator radiation should be kept small. Unfortunately, small also implies that the emitted power is small.

4.7 Power

The flux and brilliance of undulator radiation is derived in the book by David Attwood [1]. A method similar to the one used to derive the power of bend magnet radiation can be used. The difference between the acceleration in the frame moving with the the particle and in the laboratory frame leads to a relativistic increase in total emitted power. In contrast to the acceleration due to a bend magnet the acceleration is varying with time.

The velocity component in the direction which is perpendicular to the average direction of the electron beam is given by equation (4.20)

where we set . Integration gives the coordinate for motion in the direction as a function of time

In the electron ‘rest frame’ moving in the direction, time dilation, enters, and the position in this frame is

where we use the axial relativistic factor defined in equation (4.24), . Again, we exploit that only time dilation matters, as no Lorentz contraction applies in the direction, and we assume that the displacement about the average in the direction is small. The acceleration in the direction in this frame is achieved by taking the second derivative with respect to .

To get the total power, this expression can be plugged into the expression for total emitted power (3.32), in analogy to the derivation of the total power of bend magnet radiation. For the average power, again the averaging of the function gives , and we have

More relevant is the power emitted in a narrow cone in the forward direction, the central radiation cone. This derivation is performed in [1]. Equation (4.37) is used to derive the average power per unit solid angle in the ‘rest’ frame following the electron, and this expression is then transformed into the laboratory system. It is shown that the average power in the first harmonic with a spectral width emitted by an electron in the central radiation cone is an expression which has a very similar form to equation (4.38):

where the multiplicative factor , a correction for finite , is close to unity for and decreases monotonically so that . Explicitly

where

Apart from natural constants the power depends on the square of the electron energy. For small magnetic fields (small ) it is almost independent of as . Comparing equations (4.38) and (4.39), we see that total power emitted (in the first harmonic) approximately is achieved by multiplying equation (4.39) by a factor of .

The number of independently radiating electrons in an undulator with length, , when the current is is given by

If we assume that the electrons are radiating independently, this can be used in equation (4.39):

Apart from natural constants the average emitted power depends only on dimensionless and , the electron current and the undulator period. In convenient units:

For a typical electron current of A in a 3 GeV ring with an undulator of cm, we get for a power of W. This may seem surprisingly low, when we compare to the huge power emitted in bend magnet and wiggler radiation. Remember, however, that the analysis is for the first harmonic of the central radiation cone, of radiation with a bandwidth of . The brilliance of undulator radiation is outstanding.

4.8 The importance of electron-beam quality

The brilliance of undulator harmonics is ultimately determined by the quality of the electron beam. According to the undulator equation the frequency of the emitted radiation and an uncertainty in electron energy, implies an uncertainty in the emitted frequency:

In modern synchrotrons the frequency spread due to the uncertainty in the electron beam energy is much smaller than the width of the fundamental undulator harmonic determined by the number of undulator periods .

A larger challenge is the control of the electron beam emittance. In particular, this is the case for the emittance in the horizontal direction. Not to compromise the spectral purity in the undulator radiation the angular spread of the electron must be smaller than . For a 3 GeV ring and an undulator with 100 periods this angle is around 17 μrad. The new sources are improving, and when the emittance of the radiation is determined by diffraction, rather than the limited emittance of the electron beam the storage rings are called ultimate [2]. Assuming no angular spread, diffraction gives coherent radiation within an opening angle of

which for a typical beam diameter of μm gives spatially coherent radiation within a few μrad around 1 keV, and fractions of rad in the hard x-ray range.

The finite effective perpendicular size of the electron bunch also corresponds directly to a spread in the observation angle . This causes spectral broadening as the wavelength of the emitted radiation is given the undulator equation (4.25). With electron beam sizes around 100 μm in modern synchrotrons, and typical distances to the first optics of around 10 m, this correspond to a divergence of μrad. The finite bunch diameter in a modern synchrotron does therefore not have a big impact on the radiation the undulator spectrum, but μm is too large for appreciable spatial coherence.

In a new-generation machine, like the 3 GeV ring of MAX IV, the widths of the Gaussian distribution,

are planned to be around 50 μm and 6 μm in the horizontal and vertical direction, respectively, thus entering a range with substantial spatial coherence over a large energy range.

A large challenge is the control over the angular spread. If the electron beam passes the undulator at an angle, , the Doppler shift decreases, so that the observed wavelength is larger than the perfect-beam on-axis wavelength , both because the observation angle is instead of and because the electrons see the period instead of :

The change in observation angle is dominating, while the prolonged effective electron path through the undulator gives a smaller correction. It is straighforward to show that

For this relative energy spread to be less than we get the following restriction for angular spread of the electron beam:

and we see that the right hand side is equivalent to the angle for the central radiation cone (equation (4.14)). Assuming a Gaussian spread of the electron angle in a dimension perpendicular to the beam,

with a width of , one can estimate the spectral shift and broadening due to finite electron angle. It can be shown that if the width of the angular distribution is increased from zero to , the maximum of the photon energy distribution decreases by a factor of two, the peak is shifted slightly to lower energies, and a low-energy tail develops. In current synchrotron sources the angular spread is on the order of 10–20 μrad, and thus, there is a challenge to decrease the angular spread.

In the 3 GeV ring of MAX IV also the angular spread is on the order of a few μrad, so that the horizontal emittance is estimated to be 0.33 nm rad, which reduces to 0.2 nm rad as insertion devices are added. The vertical emittance is much smaller, implying that the radiation in the central radiation cone has a high degree of spatial coherence.

4.9 Brilliance

Obviously the brilliance (spectral brightness) of a radiation source, given by equation (2.10):

is crucially dependent on the electron beam emittance. Using the the expression for average power ( of equation (4.44)) in the first harmonic and the size and divergence distribution of the electron beam, this quantity can be estimated. The convenient units for brilliance are

The unit conversion is done by dividing the power (energy/time) by the energy/photon, and using that to achieve . In realistic cases, where , the constraint to is achieved by multiplying equation (4.39) by .

As is demonstrated by David Attwood [1], the on-axis brilliance (spectral brightness) in the units of equation (4.53) can be estimated:

This formula is accurate as long as the undulator condition is fulfilled. We see that emittance of the electron beam is crucial. This being the main motivation for the quest for low emittance in the storage rings. Experiments only requiring a large photon flux are not necessarily helped with low emittance. In practice it is difficult to simultaneously maintain a low emittance and a high current. This is one of the missions of the accelerator physicist.

4.10 Polarization properties

In a planar undulator, where the electron is forced to accelerate in one plane, the polarization in the first (and all odd) harmonic(s) is linear in that same plane. For the even harmonics it is more complicated, and we get a variation from horizontal to vertical linear polarization as we go off axis.

Considering the first harmonic, several solutions have been found to control and vary the polarization. Early on, a scheme (figure 4.6) based on crossed planar undulators was devised by Kwang-Je Kim [3]. Horizontally or vertically polarized radiation can be achieved by using the first or the second undulator, respectively, and by varying the gaps any relation between the components is feasible, thus giving planar polarization in any direction (2.11). The phase shifter introduced between the undulators makes it possible to combine the radiation between the two components to any kind of elliptically polarized radiation, and also left as well as right circularly polarized radiation.

Figure 4.6. The principle of crossed planar undulators. The figure is taken from the paper by Kwang-Je Kim [3].

Several other schemes have been suggested, such as the ‘Figure-8 Undulator’ [4] for fast switching between left and right circularly polarized radiation. A common scheme used today is the so-called Apple II [5] undulator (figure 4.7), in which four blocks of magnetic arrays are shifted relative to each other to give flexibility in influencing the electron path.

Such undulators thus constitute high brilliance sources with the options to choose wavelength as well as polarization in wide ranges.

Figure 4.7. The Apple II undulator uses four blocks of permanent magnets which can be shifted (left) so that the beam undulates so that full control of the polarization state is achieved. The view from the beam direction (right) indicates the magnet field direction. The figure is from [6].

4.11 Coherence and orbital angular momentum

The spatial coherence of synchrotron radiation has so far only been used at the expense of extensive filtering, as the coherent fraction has been relatively low. This is rapidly changing with the advent of the new generation synchrotron radiation sources, where the emittance is kept low so the coherent fraction is large.

It is expected that this ‘new’ feature of sycnhrotron radiation will be used in novel experiments. In particular the opportunity to control and use the orbital angular momentum of the photon beam opens up. Dependence of scattering cross sections on orbital angular momentum has been predicted, and it has been demonstrated [7] that the 2nd harmonic off-axis radiation of helical undulators carries orbital angular momentum.

References

[1] Attwood D 2007 Soft X-rays and Extreme Ultraviolet Radiation: Principles and Applications (Cambridge: Cambridge University Press)

[2] http://onlinelibrary.wiley.com/doi/10.1111/jsy2.2014.21.issue-5/issuetoc

[3] Kim K-J 1985 Applications of Circularly Polarized Radiation Using Synchrotron and Ordinary Sources ed F Allen and C Bustamante (New York: Springer) ch 2

[4] Shirasawa K, Hiraya A, Tanaka T and Kitamura H 2004 Optimization of asysmmetric figure-8 undulator as circularly polarized light source Phys. Rev. St Accel. Beams 7 020702

[5] Sasaki S, Miyata K and Takada T 1992 Japan. J. Appl. Phys. 31 L1794

[6] Schmitz-Antoniak C 2015 Rep. Prog. Phys. 78 062501

[7] Bahrdt J, Holldack K, Kuske P, Müller R, Scheer M and Schmid P 2013 Phys. Rev. Lett. 111 034801

IOP Concise Physics

Synchrotron Radiation

An everyday application of special relativity

Jan-Erik Rubensson

Chapter 5

Coherent particles

In free-electron lasers the electrons move together as a supercharge, implying extraordinary peak brilliance and ultrashort duration.

5.1 Free-electron lasers

Free-electron lasers (FEL) working in the x-ray range have a peak brilliance up to 8 orders of magnitude larger than the brightest synchrotrons, and the pulse duration is on the femtosecond scale. This gives a wealth of new experimental opportunities, which are presently being explored. FELs are based on long undulators, so that several of the considerations in the previous chapter are valid, e.g. the undulator equation for the wavelength of the emitted radiation can be applied. The key to achieving very high brilliance is to make the electrons move coherently. The total emitted power emitted by a bunch with independent accelerating electrons is proportional to the number of electrons and the square of the elementary charge (e.g. equation (4.35))

If the electrons in a bunch move as one supercharge, the power is instead

and thus we get an enhancement by a factor , which can be on the order of in a bunch.

To achieve this, a bunch control is necessary, which is impossible in closed-loop synchrotrons. It can, however, be achieved in one-pass linear accelerators combined with long undulators. Today there are a number of operating facilities, and several are planned and under construction. X-ray facilities like the Linac Coherent Light Source [1] at SLAC in Stanford, SACLA [2] at Hyogo, Japan, SwissFEL [6] and the European XFEL [3] at DESY in Hamburg can be of several kilometers length (figure 5.1), whereas XUV/soft x-ray facilities like FERMI [4] and FLASH [5] are more compact. Many of these facilities have excellent home pages with introductory material and instructive animations.

Figure 5.1. A map of the European XFEL [6]. The facility is 3.4 km long. Copyright European XFEL.

5.2 Self-amplifed spontaneous emission

The development at x-ray FEL facilities is largely due to the realization that the bunch structuring necessary to make the electrons emit coherently is achieved by the very radiation that the electrons themselves emit. The conventional undulator radiation analyzed in the previous chapter is denoted ‘spontaneous emission’ in this context. When this radiation interacts with the electrons it structures the electron bunch into microbunches, and as these move concertedly through the undulator they generate self-amplified spontaneous emission (SASE). Although this term suggests that a quantum approach is needed, the amplification due to self-organization in microbunches can be understood using classical-physics arguments. The derivation can be done in the lab frame, without even invoking relativistic transformations.

In figure 5.2, the path of an electron passing through an undulator is depicted in the thick blue line. We recall that the emitted radiation in the forward direction has a much shorter wavelength than the undulator period. According to the on-axis undulator equation,

and the electron and the radiation travel at almost the same speed. Assuming a planar undulator, the radiation is polarized so that it has components only in the direction,

Figure 5.2. The thick blue line indicates the path of the electron, while the thin blue line is the electric field of the generated radiation. Two electrons in a bunch, propagating from left to right, are depicted at three different positions. After the initial position (left) they travel (middle) and finally (right) along the undulator.

The work, , exerted by the corresponding force on the electrons as they travel a distance :

Thus, only while , work is exerted on the electrons by the field. At the left position of the red electron in figure 5.2, and the radiation field . Therefore, the work exerted on this electron is negative, , according to equation (5.4). As discussed in connection with figures 4.3–4.5, constructive interference between radiation from undulator periods requires that while the radiation emitted in the first bend travels the distance along the undulator, the electrons travel the distance . Thus, traveling one undulator period implies a phase shift corresponding to one wavelength of the radiation. For the right position of the red electron in figure 5.2, where again , this implies that also here , and therefore . For the middle position in figure 5.2, where the photons have travelled the electrons have travelled less, implying a corresponding phase shift. At this position, for the red electron, but now also , so that also at this position. This suggests that a net negative work is exerted on the red electron as it travels through the undulator, and as the E field is opposite, a net positive work is exerted on the green electron as it travels through the undulator. This implies a gain in energy for electrons at one position in a bunch and a loss in energy for the electrons which are apart. This interaction between the electric field and the electrons tends to structure the electron bunches to microbunches. These are moving as a supercharge through the undulator, giving the enhanced power, according to equation (5.2). As this process proceeds through a long undulator the power of the emitted radiation increases exponentially.

In a more rigorous way this ‘resonance condition’ is derived in the introduction of The Physics of Free Electron Lasers by Saldin et al [7]. The authors consider the interaction in a helical undulator, where the electron, with the velocity perpendicular the direction direction is determined by the excursion angle , and it varies with the undulator period :

The electric field is perpendicular to the direction and has the wavelength :

The rate of energy exchange is achieved by taking the time derivative of equation (5.4):

With and we can write

where can be interpreted as the angle between the electric field and the transverse velocity of the particle. For effective energy conversion this angle should be constant along the undulator:

and, again, while :

and thus,

We recognize this as the the first term in the undulator equation. This implies that the criterion for effective energy exchange is fulfilled for the very wavelength which is spontaneously emitted in the undulator, and it is this interaction that leads to microbunching. For the description of the gain specialized books should be consulted. Here we note that the emitted power increases exponentially to saturation over typically tens of meters of an undulator (figure 5.3). As expected from equation (5.3), the increase corresponds to the number of electrons in the bunch.

Figure 5.3. As microbunches are formed the emitted power increases exponentially, until saturation where microbunches are fully formed. The data (open red circles) were obtained at the first stage of the SASE FEL at DESY [8]. Simulations have been performed with simulation code FAST.

The radiation from a SASE FEL is fully polarized and transversely coherent. On the other hand, compared to conventional optical lasers the longitudinal coherence is rather poor, which is a consequence of the start-up from shot noise. As the microbunching is initiated in a stochastic fashion, the SASE pulses vary stochastically, regarding both energy and time structure (figure 5.4).

Figure 5.4. Typical temporal (left) and spectral (right) structure of the radiation pulse from a SASE XFEL at a wavelength of 1 Å. The red lines correspond to averaged values. The figure is taken from the DESY homepage [9]. Simulations have been performed with simulation code FAST.

For many experiments this implies that each pulse must be measured and characterized, together with the actual measurement data, a task which puts high demands on detectors. The readout speed is already an issue at repetition rates around 10–100 Hz, and the challenge increases with the repetition rate. At the European XFEL in Hamburg the repetition rate is planned to be 27 kHz. In addition it becomes a challenge for data handling, as a huge amount of data is produced when multidimensional data sets are stored for each pulse. New statistical methods to analyze the data are developed and exploited for FEL experiments, e.g. covariance mapping [11].

5.3 Seeding

To improve stability, and the longitudinal coherence various seeding schemes (figure 5.5) have been developed, with the aim to initiate the amplification in a controlled manner. The vision is to produce fully coherent x-rays, for which the energy bandwidth is limited only by the finite duration of the pulse.

Figure 5.5. Schematic layout for self-seeding. Courtesy of S Reiche at PSI Villigen [10].

At FERMI in Trieste, seeding with conventional lasers and high harmonic generation [12] is now being exploited in the XUV/soft x-ray range. Pulses with a bandwidth of , and energy fluctuations in the 1 meV range are available below 100 eV. This is still around one order of magnitude from the Fourier limit, but it is a large improvement compared to SASE pulses.

Self-seeding is an option which is now implemented in the harder x-ray region at the LCLS in Stanford, and is planned for most upcoming facilities. Here a first undulator produces ‘spontaneous’ undulator radiation, which is monochromatized and used as seed for the second undulator, where FEL amplification proceeds.

Several variations of these seeding schemes are now being suggested and tested at various FEL facilities. The development is pursued in close collaboration between users and accelerator physicists.

The goal thus depends on experimental needs, and in principle photon pulses can be tailored within large ranges. Peak power can be maximized and duration minimized for experiments requiring ultrafast processes. As beam damage is severe at high intensity, short pulse duration is advantageous where pertinent processes can occur before sample destruction. The reachable intensities allow for non-linear x-ray physics. Series of a few femtosecond pulses with controllable delay, and with different photon energy make ultrafast pump-probe experiments feasible. High average power facilitates studies of dilute samples and the exploitation of processes with low cross section.

References

[1] https://portal.slac.stanford.edu/sites/lcls_public/aboutlcls/Pages/default.aspx

[2] http://xfel.riken.jp/eng/

[3] http://www.xfel.eu/

[4] https://www.elettra.trieste.it/FERMI/

[5] http://flash.desy.de/

[6] https://www.psi.ch/swissfel/

[7] Saldin E, Schneidmiller E V and Yurkov M V 2000 The Physics of Free Electron Lasers (New York: Springer)

[8] Ayvazyan V et al 2002 Exponential growth and saturation at the TTF free-electron laser Eur. Phys. J. D 20 149–56

[9] TESLA Technical Design Report 2001 DESY 2001-11, ECFA 2001-209, TESLA Report 2001-23, TESLA-FEL 2001-05 (Hamburg: DESY) www.desy.de/

[10] Reiche S

[11] https://en.wikipedia.org/wiki/Covariance_mapping

[12] Lewenstein M, Balcou Ph, Ivanov M Yu, L’Huillier A and Corkum P B 1994 Phys. Rev. A 49 2117

IOP Concise Physics

Synchrotron Radiation

An everyday application of special relativity

Jan-Erik Rubensson

Chapter 6

X-ray optics

Refraction index is just below unity, and absorption is large. Grazing incidence reflection leads to long beamlines.

6.1 Beamlines

Synchrotrons and FELs provide high-brilliance x-rays. To make use of the radiation, beamlines are being designed with the purpose to exploit the properties of the radiation in various experiments. Typically this involves redirecting the beam and focusing at a sample position, including various filtering processes, where, e.g., a narrow frequency bandwidth, well-defined polarization, or coherence can be chosen. This requires advanced x-ray optics, which will be discussed in this chapter.

X-ray optics differ from optics for visible light because the refractive index is very close to unity, and because absorption plays a large role for all optical components. Especially in the soft x-ray range, this makes conventional refractive optics impractical. We shall see that reflection with a few degrees grazing incidence angle in most cases is the method of choice.

6.2 Optical constants

Crucial optical processes are absorption, transmission and reflection, and the materials properties pertinent for these processes are summarized in the complex refractive index

where n is the conventional index of refraction and k is termed the extinction coefficient. In the x-ray range and are commonly introduced. It can be shown that the parameters to a good approximation can be described by

where except for the classical electron radius, m, and the wavelength of the radiation , the materials properties are deduced from the atomic density , and the atomic scattering factors and , which comprise information about the interaction cross section between radiation and atoms. Although this approximation neglects electronic structure due to interaction between atoms, and is thus inaccurate in the vicinity of core resonances, it gives excellent estimates of the properties in large energy ranges. Atomic scattering factors as a function of energy are tabulated [1] on the homepage of Center for X-Ray Optics, and the refraction indices as a function of energy can be plotted (figure 6.1).

Figure 6.1. Output for Si_3N_4 from the CXRO home page: δ and β vary somewhat at the core edges, but the overall trend is a monotonous steep decrease from around 10−2 at 100 eV to 10−6 at 10 keV [2].

6.3 Absorption and reflection

The absorption coefficient, , defined in the Beer–Lambert law (figure 6.2), describes how intensity is attenuated from an intensity to as a beam goes a distance, , through a material with density :

The attenuation length, , is defined as the distance where

One can show that these entities are related to the atomic scattering factors

The attenuation length of soft x-rays is on the order of 100 nm for most materials, and for decreasing wavelength the attenuation length increases to the order of millimeters for energies around 30 keV.

Figure 6.2. The parameters of the Beer–Lambert law.

Reflection and transmission at an interface can be derived by considering that the E field is continuous. This implies that the angle of reflection, , must equal the angle of incidence, , (see figure 6.3)

Figure 6.3. Reflection and transmission at an interface. The continuity requirement gives the law of reflection, Snell’s law and Fresnel’s equations.

and that the angle of transmission, , is related to by Snell’s law,

The Fresnel equations, which give the intensity relations between the beams can also be derived by assuming continuity at the interface, and the ratio between the amplitudes of the reflected radiation, , and the incident radiation defines the reflectivity. For reflectivity of radiation polarized perpendicular to the scattering plane, , and in the plane, , as defined in figure 6.3. If absorption is neglected we have

The ratio between the reflected and and incident intensities is achieved by squaring the amplitude ratios. The result for the case where and is shown in figure 6.4.

Figure 6.4. Typical reflectivity in the soft x-ray range for radiation polarized perpendicular (blue) and parallel (red) to the scattering plane. Absorption is neglected.

This is a typical case in the x-ray range, where the index of refraction for all materials is close to and smaller than unity. A number of conclusions can be drawn from this result. At normal incidence () only around of the incident intensity is reflected. At grazing incidence, larger incidence angles, however, the reflectivity grows and approaches unity before , for both polarization directions. When a beam passes an interface from a medium with index of refraction , to a medium with index of refraction, , and , Snell’s law implies that , and that there must be a limit where . The corresponding angle of incidence, , is the critical angle for total reflection, beyond which no transmission is possible. Snell’s law gives

and, thus

Consider radiation incident on a surface of a material where from vacuum where . If we define the critical incidence angle relative to the surface, instead of the surface normal, Snell’s law gives

and therefore

In our example () implying , which indeed corresponds to the incidence angle () where the intensity ratio approaches unity in figure 6.4.

The analysis neglects absorption, which can be included in the Fresnel equations by using the complex index of refraction. This makes the equations more involved, and the principal effect, with realistic numbers, is a smearing out of the variations predicted when absorption is neglected. It can be shown that the intensity ratio beyond the critical angle is not unity if absorption is taken into account, but:

This implies that if or , i.e. if the deviation from unity of the real part of the refraction index is very different from the deviation of the imaginary part from zero, the reflectivity beyond the critical angle is indeed close to unity. We see that this is also the case when , i.e. when absorption can be neglected. If, however, , we get .

One can also show that for normal incidence ,

Thus, the reflectivity at normal incidence increases if absorption is taken into account, and in cases where , , whereas as we saw earlier (6.6) , so that reflection and absorption measurements give the same information.

It is also obvious from the Fresnel equations (figure 6.4) that the reflectivity of radiation polarized in the scattering plane is very low at a certain incidence angle, the Brewster angle. If absorption is neglected in-plane reflectivity vanishes completely, as the numerator in equation (6.10) goes to zero. This happens when the angle between reflected and transmitted radiation is , , and we have

and thus,

as for x-rays the Brewster angle is

This can be used for manipulating and measuring the polarization state of the radiation, but at an expense of intensity, since also the perpendicular component is weak. In our example the intensity ratio is . At grazing incidence, where the reflectivity is larger, the two polarization directions have very similar reflectivity.

6.4 Multilayer reflection

Multilayered mirrors are constructed to enhance reflectivity. Interference between reflections at the various interfaces is exploited, and a Bragg type condition applies. In figure 6.5, where refraction is neglected, we see that the path length difference for reflections in adjacent interfaces is . Given that the real part of the index of refraction is close to unity this is very close to the optical path length difference, and since the condition for constructive interference is

Figure 6.5. Materials with different optical constants (like W and B_4C) are combined to make multilayers, which enhance reflectivity in certain wavelength ranges. Note that this is a simplified figure, where refraction is neglected.

Often the angle measured relative to the surface is used and, as we have . If refraction is considered the the angle relative to the interfaces is different from the incidence angle, and the optical path, , is not the same as the path in vacuum. The expression becomes more complicated, and one can show that

where is the average deviation from unity of the index of refraction in the multilayer system. As intuitively expected, the deviation from the simple expression is large at small and large .

Typical periods are in the nm range, implying that the Bragg angle, , for hard x-rays ( nm at 5 keV) is small. At small refraction is important, and it is realized, that this angle must be larger than the critical angle for total reflection for exploitation of multilayer reflection. In the hard x-ray range the critical angle for total reflection is only a fraction of a degree which hampers many optical schemes. With multilayers, reflectivity can be enhanced to almost unity at few degrees glancing angle, as is demonstrated by the Optics group at ESRF in figure 6.6.

Figure 6.6. The Optics group at ESRF demonstrates that 200 layers of Ni/B4C give almost unity reflectivity for hard x-rays at around 2 degrees glancing angle [3].

In the soft x-ray range, where the wavelength and the period match, multilayers are used to achieve reflectivity at near-normal incidence. The ratio between reflected and incident intensity can be around 0.5 near 100 eV, and drops to a few percent when approaching 1 keV. Working at the Brewster angle, , multilayers are used as polarizers.

Multilayer reflection is simulated on the CXRO homepage [2], where also the influence of imperfections in the interfaces are taken into account.

6.5 Reflective optics

For many purposes focusing of x-ray radiation to small spots is necessary, and curved mirrors working in grazing incidence provide a natural choice. Unlike optical systems for visible light, grazing incidence optics does not have a symmetry axis, and astigmatism is severe. For reflection in spherical surfaces point-to-point focusing in the scattering plane is achieved when the entrance and exit arm have the same length. According to the Rowland criterion, a spherical mirror with radius positioned tangentially to a circle with radius focuses a point source situated on that circle to a point on the same circle. Simple geometry (see figure 6.7) shows that for an incidence angle the distance from the mirror both to the source and the focus is

Figure 6.7. The Rowland criterion. Focusing in a spherical mirror images a point source on a line.

For a concave spherical mirror at normal incidence, , we see that : the rays from the source are focused on the source point, and we realize that there is spherical symmetry so that the same is true for focusing in the sagittal plane. When , decreases and simultaneously the symmetry is lost and rays in the sagittal plane increasingly fall at smaller glancing angles on the Rowland sphere, and there is almost no focusing effect perpendicular to the scattering plane. In other words, a point is focused to a line in the sagittal plane, and astigmatism is typical in grazing incidence optics.

Kirkpatrick-Baez mounting (figure 6.8) is a common way to achieve focussing in both directions quasi-independently: two mirrors mounted with the surfaces perpendicular to each other for horizontal and vertical focusing.

Figure 6.8. Kirkpatrick-Baez mounting. The figure. is taken from the ESRF home page [3].

A single mirror with a toroidal surface, with a small radius for sagittal focusing can also be used. This puts high demands on the mirror quality and has some limitations in the imaging precision.

Often demagnification optics is desired, as large intensity and small spots on the sample are advantageous in many experiments. In principle, spherical surfaces are limited to unit magnification, and the natural choice is elliptical surfaces. Rays from a source in one of the focal points of the ellipse will be focused in the other (figure 6.9), as follows directly from Fermat’s principle and the definition of an ellipse. By mounting the mirrors non-symmetrically so that entrance and exit arms are very different the corresponding (de-)magnification is achieved. Using ellipsoidal mirrors, focusing in both the scattering and the sagittal planes is achieved. This, however, puts very high demands on both the manufacturing of the mirrors and the positioning precision. It is a challenge to reach foci on the order of microns. The precision reached today allows for spot sizes down to 100 nm.

Errors tend to be unforgiving, and the spatial resolution deteriorates as one leaves the focus conditions. Elliptical optics is therefore not practical for imaging extended objects.

In general, imaging of extended objects requires more involved optics. One scheme, devised by Hans Wolter, combines an elliptical mirror with a pre-mirror, with e.g. a hyperbolic surface, making a virtual image of the source in one of the foci of the ellipse. Wolter optics is now common in x-ray telescopes, and in figure 6.10 we see how it is combined with gratings in an imaging x-ray spectrometer. The grating is used to disperse the radiation in the direction perpendicular to the plane of the paper.

Figure 6.9. Elliptical surfaces allow for magnification and demagnification.

Figure 6.10. Wolter optics, featuring a plane-hyperbolic and a plane-elliptic mirror used to image the source on a detector in the plane of the figure. In the perpendicular plane a cylindrical grating is used to disperse the radiation. This imaging spectrometer is a concept developed by Joseph Nordgren, and it is the basis for a spectrometer which will be used at the European XFEL

Figure 6.11. The focusing principle of a tapered capillary. The figure is taken from a paper by Balaic and Nugent [4].

Waveguide capillaries provide an alternative and cost-effective means to focusing x-rays. X-rays can be focused using tapered glass capillaries (figure 6.11), and two almost parallel plates. The idea is based on one or multiple reflections, against a medium with lower refraction index, just like in ordinary fiber optics. As we have seen, any material has a lower refraction index than vacuum in the x-ray range. With the tapering of a capillary a beam cross section can be reduced to the size of the small final opening, which practically can be in the micron range. This is, however, at the expense of increased divergence.

6.6 Refractive optics

Recently refractive lenses for x-rays have begun to be used, and are now common in the hard x-ray range. Since all materials have a refraction index lower than unity a lens which is positive for visible light is negative for x-rays, and since the refractive index deviates only a small number from unity the focus distance is typically very large. According to the lens maker’s formula for thin lenses with concave radii and we have

which gives for , and

To get a reasonable focusing action several consecutive lenses are used to make a compound refractive lens (CRL) (figure 6.12). With lenses a focal length of is achieved, and it has been demonstrated that a beam of photons with 21 keV energy can be focused down to microns using a CRL comprising 63 individual Al lenses. For soft x-rays refractive lenses are impracticable due to the small attenuation length, which implies unrealistically thin lenses.

6.7 Diffractive optics

Zone plates (figure 6.13), also familiar from conventional optics, are based on Fresnel diffraction. They are often used in x-ray microfocus applications, such as x-ray microscopes. For spatially and longitudinally coherent radiation hitting perpendicularly to the plate it works just like a lens with the focal distance, f, determined by the condition for interference between beams scattered by successive zones. Pythagoras’ theorem gives (see figure 6.13) if rm is the radius of the mth zone

Figure 6.12. Refractive lenses for hard x-rays. The Figure. is taken from [5].

Figure 6.13. Zone plate principle. Spatially and longitudinally coherent beams impinge perpendicularly on a plate with non-transparent and transparent circular zones with radii, rm (left). As the beam proceeds, the paths from consecutive zones differ by a multiple of the wavelength for constructive interference (right).

so that to a good approximation

One challenge in manufacturing zone plates is the precision needed for the outermost zones, where has to be well defined to get a very good focus. This is improving over the years and foci below 100 nm are now routinely achieved. Disadvantages are that this optical element reduces flux as it uses the coherent properties of the radiation, and that the focal distance depends on the wavelength of the radiation. With the increased spatial coherence at the new generation storage rings such zone plates will find increased use.

6.8 Dispersive optics

To achieve high energy resolution (or temporal coherence) a principle based on interference is typically exploited, most often diffraction of the radiation in periodic structures. Fourier analysis sets the limit for the resolution that one can achieve. If the beam is diffracted in a periodic structure with periods the resolving power is:

where is the order of diffraction.

When the wavelength of the radiation compares to atomic distances (1 Å corresponds to 12.4 keV), Bragg diffraction in crystals is the obvious choice. At these wavelengths one also uses the fact that reflectivity can be neglected: radiation scatters appreciably at large angles only if the condition for constructive interference between rays scattered at different scattering planes is fulfilled. The same figure (figure 6.5) as for multilayer reflections can be used to understand the interference condition, with interfaces replaced by atomic crystal planes. If the incidence angle is measured relative to the surface, the same condition,

applies, where is now the distance between the scattering planes. In principle, one must correct for refraction also in this case, but often crystals for x-ray optics are used at large in ranges where is small so that , and this is a small correction only.

In the hard x-ray range it is easy to find scattering planes where the Bragg condition is fulfilled at large . A table of commonly used crystal planes is found in the CXRO yellow booklet [6]. One of the shortest of a practical crystal are the () planes of α-quartz, Å. One of the largest for any natural crystal are the () planes of beryl, Å. Thus one can easily cover the corresponding energy range with natural crystals, and one can substantially increase the energy range towards higher energies by decreasing α, and by using more exotic high-index scattering planes. Monochromatization is achieved also in the 8–100 keV region, putting higher demands on the precision of the positioning of the optical components as the energy increases.

In principle one can also further extend the range towards lower energies using so-called Langmuir–Blodgett crystals and multilayers. Such artificial crystals have imperfections that limit their usefulness, and absorption is also a severely limiting factor. At Å (830 eV) a typical penetration length is around Å. If Å, this means that only around 400 layers contribute to the interference. This limits the resolving power according to equation 6.26.

At high energies, the attenuation length is much larger than the distance between the planes, and essentially perfect crystals can be used, Thus, the resolving power can become very large. X-ray scattering experiments have been performed with a resolution of 1 meV at an energy of around 20 keV, corresponding to a resolving power of . The limiting factor is not intrinsic, i. e. it is not dependent on how many contributing layers, but rather practical, and set by how well the scattering angles can be controlled and measured. The quality of the crystals, of course, becomes crucial at some point, but as long as one can use materials like Si, out of which large, almost perfect crystals can be made this is no limitation.

For low x-ray energies, the reflection grating is the dispersive element of choice. Assuming parallel coherent incident radiation, it can be seen directly in figure 6.14 that the condition for constructive interference is:

Figure 6.14. For constructive interference in a diffraction grating the scattering from adjacent grooves in the periodic structure must have an optical path length difference of mλ. The paths d sin α and d sin β are the path length difference for incoming and outgoing beams, respectively.

Although this equation looks very much like the equation for Bragg diffraction the phenomenology is quite different. In the Bragg case the scattering angle is defined relative to the lattice planes, and the scattering angle is the same as the incidence angle. Intensity is scattered only if the interference condition is fulfilled. At other angles (or wavelengths) ‘nothing happens’, except transmission and absorption. In the reflection grating there is always a direction where a wavelength is diffracted. For , and the grating acts as mirror. In principle all orders of diffraction can be used for dispersion, and the gratings are always optimized for certain angles, and wavelength ranges.

The groove density (grating constant) cannot for practical reasons be much larger 2000 grooves/mm. At larger groove density the quality of the grooves, and the reflection efficiency of the gratings decrease. The distance between the grooves, can thus not be much smaller than 0.5 micron. One would think that the gratings would separate wavelengths on this same length scale, but in fact the grating works for much shorter wavelengths due to the projection of the grating structure on the beam cross section.

We consider the intensity from slits, with perfect reflectivity and incident intensity of monochromatic radiation, .

The phase difference between adjacent slits is

A principal intensity maximum is achieved when the denominator is zero, and when for . The first time this happens on the side of a principle maximum is at and therefore the corresponding width of the peak is

Assuming that the incidence angle is fixed, we can differentiate the expression for , and relate the phase variation to the variation in the angle

This corresponds to an angular broadening due to the finite number of grooves:

According to Rayleigh’s criterion for resolution, two peaks are resolved when the maximum of one coincides with the diffraction minimum of the other. This corresponds to half the angular broadening:

Now, the angular variations relate to the wavelength variations through the grating equation, which we differentiate:

By combining the last two equations we get, as expected:

The best resolving power achieved with a grating in the soft x-ray range is around 0.5 meV at 60 eV (). For first-order diffraction this would imply that at least 120 000 lines of the grating must be illuminated. With 1 micron groove spacing the grating must have a length of at least 120 mm. The challenge in manufacturing gratings is appreciated, e.g., by considering the typical total length of the grooves. With 10 cm groove length, 120 000 grooves implies a total length of 12 km.

Compared to the resolving power achievable with crystal diffraction in the hard x-ray range, gratings will never be able to compete. In practice there is no competition since the two techniques work in different wavelength ranges.

Exotic schemes for separating wavelengths include interferometry. In the hard x-ray range interferometry can be performed using amplitude splitting beamsplitters in so-called channel cut single crystals. (figure 6.15).

Figure 6.15. In single-crystal resonators x-ray pulses are trapped, and the transmitted radiation has a smaller bandwidth than the incident radiation [9].

For VUV/soft x-rays wavefront-splitting beamsplitters, rather than amplitude-splitting beamsplitters, are necessary due to the large absorption. Attempts [7] to use this principle for Fourier spectroscopy with grazing incidence optics have been made, and emission up to 60 eV has been observed.

For high resolution absorption measurements in the VUV range, a scheme has been devised by a group [8] at the SOLEIL synchrotron. After passing through sample gas the synchrotronn beam is reflected off two reflecting prisms that slide relative to each other. By precision motion of these reflectors absoprtion spectra with a resolution of can be recorded around 25 eV energy.

6.9 Optics quality

The fast development of synchrotron radiation techniques puts new demands on optical components. X-ray mirrors for grazing incidence are needed for many purposes, and the quality requirements can be appreciated by considering the simple example of reflection of a ray from a hypothetical point source by a mirror. Assume typical entrance and exit arms, both with a length of r = 5.00 m, and an angle of incidence of α = 88.50°, corresponding to a glancing angle of θ = 1.50° = 26 mrad. It is realized that the angular position of the mirror needs to be defined much better than that for accurate imaging. An error of Δθ = 1 μrad changes the position by Δl = r2Δθ ≈ 10 μm perpendicular to the beam direction 5 m downstream. Often an accuracy closer to 1 μm is wanted, and in our hypothetical arrangement this implies a corresponding angular accuracy of 0.1 μrad or 5.7 · 10−6 degrees. This implies unusual requirements for positioning and stability, and moreover it implies extraordinary demands on the definition of the mirror surface. Thus, to maintain micron accuracy 5 m away from the mirror the deviation from the nominal surface figure must not be larger than 0.1 μrad. This is often specified as the slope error, and is defined as a deviation with a wavelength on the order of the mirror size. Assuming a nominally flat mirror with a length of 0.2 m, a slope error of 0.1 μrad, implies that the radius of curvature anywhere should be larger than 1000 km, or in other words, the (long-wavelength) deviation from a flat surface should not be more than 5 nm. If focusing is attempted, a spherical or cylindrical mirror surface with the same accuracy is needed. In our example, equation (6.20) gives a mirror radius of ρ = 188 m. A micron source would give rise to micron spot only if the positioning error and slope error of the mirror were better than 0.1 μrad.

Another figure of merit for a mirror is the short-wavelength deviations from the perfect surface, which increases diffuse reflection and decreases the desired specular reflection intensity. This surface roughness constitutes a stochastic variation which today can have a standard deviation as low as one atomic distance.

6.10 Monochromators

The function of a beamline is often to monochromatize and focus the synchrotron radiation at a sample position. To this end it is advantageous to use as few optical elements as possible, since intensity is reduced by each reflection. The idea to combine the focusing properties of spherical mirror (figure 6.7) with the diffraction properties of a grating (figure 6.14) goes back to Henry Rowland and the 19th century. To a good approximation radiation of various wavelengths is focused on the Rowland circle, with a simple geometry, analogous to the concave mirror (figure 6.16).

Figure 6.16. A simple spherical grating monochromator, combining the diffraction properties of the grating (Figure 6.13) with the focusing properties of a spherical mirror (figure 6.7)

The distance between the source and the grating is

and the distance to the image is

A simple monochromator principle would be a fixed spherical grating mounted tangentially to the Rowland circle, with the distance to the source according to equation (6.35). The energy of the radiation would be chosen by moving the sample along the Rowland circle according equation (6.36).

A movement Δz along the Rowland circle corresponds to a change in the diffraction angle

where we used that θ + 2β = π. The diffraction angle is related to the wavelength via the grating equation (6.26) which in differentiated form gives

We see that a large grating radius, ρ, large groove density, and high orders of diffraction, m, must be used to maximize space separation of wavelengths.

This simple example in which only one optical component is used illustrates the constraints when considering a beamline for x-rays. In practice there are several demands that complicate matters, which motivates introduction of additional components

It is convenient to have a fixed position for the sample and instead scan the energy by moving the grating. In principle the grating should move along the Rowland circle, implying accurate motion which includes both translation and rotation. Only in small energy intervals, small deviations from the Rowland focusing are allowed and a simple rotation is sufficient.

It is often desired that a monochromatized beam, and not the spatially spread spectrum impinges on the sample. To this end an exit slit is introduced after which the sample is positioned. In such a case, however, the sample will not be in focus, thus limiting intensity. If the experiment requires high intensity additional refocusing optics after the exit slit is needed

In this geometry the definition of the source point on the Rowland circle is crucial, and in practice also an input slit is required. As this input slit must be situated at rather large distance from the synchrotron radiation source, also refocusing optics before the input slit is required to achieve reasonable throughput.

In the process of filtering the radiation in a practicable way several optical elements must thus be introduced. Assuming that the refocusing is done by Kirkpatrick-Baez pairs (figure 6.8) four mirrors in addition to the spherical grating are used. For high energy resolution the slits can typically not be larger than a few tens of microns, putting great demands on the quality of the mirrors, both regarding figure errors and surface roughness. Additional optical elements improve resolution and versatility but notoriously reduce the throughput. To find the best compromises and design the optics arrangement for the specific experiment for which it is intended is the art of beamline building.

There are several alternative optical schemes. The SX-700 [10] monochromator principle introduced by Helmuth Petersen is frequently used in the soft x-ray range. It does not use an input slit and takes advantage of the low emittance of modern sources.

The simple grating equation (6.27) presupposes perfectly parallel incoming and outgoing beams, i.e. that the source is infinitely far away from the grating. For finite source distance one can show that a plane reflection grating also has focusing properties. If the distance to the source is r there will be a virtual mononchromatic image at

where the negative sign implies that the monochromatic image is situated in front of the grating. The SX-700 concept uses this image produced by the plane grating as the source for an elliptical mirror which makes a real image at an exit slit.

The energy can now be varied by rotating the grating. This would however, change both the distance and the direction to the virtual image. A rotatable plane mirror in front of the grating is introduced to keep the virtual image in a constant position, i.e. to keep c constant in equation (6.40). In this way the imaging due to the elliptical mirror is well defined, producing a monochromatic real image at the exit slit, with varying energy as both grating and plane pre-mirror are rotated

An advantage of this setup is that the position of the virtual image can be chosen so that the distance between the virtual image and the elliptical mirror is very large compared to the distance to the slit, e.g. at grazing incidence, when , the c value in equation (6.39) becomes very large. This implies large demagnification in the imaging of the elliptical mirror, and high energy resolution. Larger grazing angles imply a smaller c which can be used to suppress higher orders due to the reduced reflectivity for higher energies. To maintain the focusing condition, changing c must be accompanied by a change of the distance between the elliptical mirror and the exit slit.

Today, gratings with varied line spacing (VLS) are often used in monochromators. The line spacing is varied to further exploit the focusing properties of the grating, and it can be used to simplify the scanning mechanisms, and even to reduce the number of optical components. Since plane mirrors can be manufactured with higher precision than concave mirrors VLS grating focusing can be advantageous in high-resolution applications [11].

A deeper analysis of grating properties and monochromators can be found e.g. in [12] and a nice overview is given by Howells in chapter 4.3 of [6].

6.11 Ray tracing

For an understanding of the functioning of a beamline it is prerequisite to be aware of the boundary conditions given by optical properties in the x-ray range, as well as the basic function of the optical components. It is also important to grasp the conceptual idea behind a specific monochromator, both to understand how it is optimized, and how it may be improved. Some grating-monochromator principles have been described above as examples, and similar lines of arguments apply in the realization of most beamlines.

As we have seen, several optical components are needed, and high demands are put on their quality as well as the precision in their positioning. Due to the complexity, the overall performance of a beamline cannot be predicted without advanced computer simulations, i.e. ray tracing.

In ray tracing programs the surfaces of the optical components and their positions are specified, and rays which simulate a synchrotron radiation source are traced numerically through the system. Typically the specifics of undulator radiation, as described in chapter 4, are taken as input. The definition of the surfaces includes estimated slope errors and surface roughness, and the effect of the errors can be analyzed in detail. Ray tracing programs also include deformation due to the heat load, which may be severe for the mirrors close to the primary source.

Consequences of misalignment can also be investigated, giving important information about the required mechanical stability and and reproducibility in the scanning procedures.

Figure 6.17. Typical output from a raytracing calculation, where the effect on the energy resolution of increasing the length of a spherical grating is investigated. The three different colors represent radiation of different energies, and the figure shows where the rays impinge on the detector. A shorter grating (left) separates the energies better than a longer grating (right), as the imaging errors increase. Courtesy of J Forsberg.

Ray tracing simulations are therefore an invaluable aid in guiding beamline design and construction (figure 6.17). There are several shareware programs which are dedicated to synchrotron radiation applications. The program SHADOW [14], originally developed at University of Wisconsin, has been widely used over the years, and is constantly developing. The RAY [15] program is developing at BESSY, Berlin, and the BEAM FOUR [16] program from Stellar Software in Berkeley is also widely used.

References

[1] http://henke.lbl.gov/optical_constants/asf.html

[2] CXRO home page www.cxro.lbl.gov

[3] Morawe Ch X-ray Optics Group, ESRF (France) www.esrf.fr/cms/live/live/en/sites/www/home/Instrumentation/xray-optics/multilayer-lab.html

[4] Balaic D X and Nugent K A 1995 Appl. Opt. 34 7263–72

[5] Snigirev A A 2011 SPIE Newsroom www.spie.org/newsroom/3688-refractive-lenses-allow-compact-precision-focusing-of-x-rays

[6] http://xdb.lbl.gov/

[7] Agåker M, Andersson J, Englund J C, Rausch J, Rubensson J-E and Nordgren J 2011 Spectroscopy in the vacuum-ultraviolet Nature Photonics 5 248

[8] de Oliveira N, Roudjane M, Joyeux D, Phalippou D, Rodier J-C and Nahon L 2011 High-resolution broad-bandwidth Fourier-transform absorption spectroscopy in the VUV range down to 40 nm Nat. Photon. 5 149

[9] Smith R 2000 X-ray optics: making hard light sharper Nature 404 345–7

[10] Petersen H and Baumgärtel H 1980 Bessy SX/700: A monochromator system covering the spectral range 3 eV ≃ ℏ ω ≃700 eV Nucl. Instrum. Meth. 172 191–3

[11] Reeininger 2011 The in-focus variable line spacing plane grating monochromator Nucl. Instrum. Meth. Phys. 10 139–43

[12] Peatman W B 1997 Gratings, Mirrors and Slits: Beamline Design for Soft X-ray Synchrotron Radiation Sources 1st edn (Boca Raton, FL: CRC)

[13] Forsberg J and Strocov V yyyy

[14] Sanchez del Rio M, Canestrari N, Jiang F and Cerrina 2011 SHADOW3: a new version of the synchrotron x-ray optics modelling package J. Synchtrotron Rad. 18 708–16

[15] Schafers F 2008 Modern Developments in X-ray and Neutron Optics, (Springer Series in Optical Science vol 137) (Berlin: Springer) pp 9–14

[16] http://www.stellarsoftware.com/