Overview of Synchrotron Radiation

Abstract

After Schott’s [1] unsuccessful attempt to explain atomic radiation with his electromagnetic theory no further progress was made for some 40 years mainly because of lack of interest. Only in the mid 1940s did the theory of electromagnetic radiation from free electrons become interesting again with the successful development of circular high-energy electron accelerators.

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After Schott’s [1] unsuccessful attempt to explain atomic radiation with his electromagnetic theory no further progress was made for some 40 years mainly because of lack of interest. Only in the mid 1940s did the theory of electromagnetic radiation from free electrons become interesting again with the successful development of circular high-energy electron accelerators. At this time powerful betatrons [2] have been put into operation and it was Ivanenko and Pomeranchouk [3], who first in 1944 pointed out a possible limit to the betatron principle and maximum energy due to energy loss from emission of electromagnetic radiation. This prediction was used by Blewett [4] to calculate the radiation energy loss per turn in a newly constructed 100 MeV betatron at General Electric. In 1946 he measured the shrinkage of the orbit due to radiation losses and the results agreed with predictions. On April 24, 1947 visible radiation was observed for the first time at the 70 MeV synchrotron built at General Electric [5–7] with a transparent glass vacuum chamber. Since then, this radiation is called synchrotron radiation.

The energy loss of particles to synchrotron radiation causes technical and economic limits for circular electron or positron accelerators. As the particle energy is driven higher and higher, more and more rf-power must be supplied to the beam not only to accelerate particles but also to overcome energy losses due to synchrotron radiation. The limit is reached when the radiation power grows to high enough levels exceeding technical cooling capabilities or exceeding the funds available to pay for the high cost of electrical power. To somewhat ameliorate this limit, high-energy electron accelerators have been constructed with ever increasing circumference to allow a more gentle bending of the particle beam. Since the synchrotron radiation power scales like the square of the particle energy (assuming constant magnetic fields) the circumference must scale similar for a constant amount of rf-power. Usually, a compromise is reached by increasing the circumference less and adding more rf-power in spaces along the ring lattice made available by the increased circumference. In general the maximum energy in large circular electron accelerators is limited by the available rf-power while the maximum energy of proton or ion accelerators is more likely limited by the maximum achievable magnetic fields in bending magnets.

What is a nuisance for researchers in one field can provide tremendous opportunities for others. Synchrotron radiation is emitted tangentially from the particle orbit and within a highly collimated angle of ± 1∕γ. The spectrum reaches from the far infrared up to hard x-rays, the radiation is polarized and the intensities greatly exceed other sources specifically in the vacuum ultra violet to x-ray region. With these properties synchrotron radiation was soon recognized to be a powerful research tool for material sciences, crystallography, surface physics, chemistry, biophysics, and medicine to name only a few areas of research. While in the past most of this research was done parasitically on accelerators built and optimized for high-energy physics the usefulness of synchrotron radiation for research has become important in its own right to justify the construction and operation of dedicated synchrotron radiation sources all over the world.

1 Radiation Sources

Deflection of a relativistic particle beam causes the emission of electromagnetic radiation which can be observed in the laboratory system as broadband radiation, highly collimated in the forward direction. The emission is related to the deflection of a charged particle beam and therefore sweeps like a search light across the detection apparatus of the observer. It is this shortness of the observable radiation pulse which implies that the radiation is detected as synchrotron radiation with a broad spectrum. The width of the spectrum is characterized by the critical photon energy (24.49) and depends only on the particle energy and the bending radius of the magnet. Generally, the radiation is produced in bending magnets of a storage ring, where an electron beam is circulating for hours.

In order to adjust the radiation characteristics to special experimental needs, other magnetic devices are being used as synchrotron radiation sources. Such magnets are known as insertion devices since they do not contribute to the overall deflection of the particle beam in the circular accelerator. Their effect is localized and the total deflection in an insertion device is zero. In this chapter, we give a short overview of all radiation sources and their characteristics and postpone more detailed discussions of insertion device radiation to Chap. 26

1.1 Bending Magnet Radiation

The radiation from bending magnets is emitted tangentially from any point along the curved path like that of a searchlight and appears therefore as a swath of radiation around the storage ring as shown in Fig. 24.1. In the vertical, nondeflecting plane, however, the radiation is very much collimated with a typical opening angle of ± 1∕γ.

Fig. 24.1

Radiation swath from bending magnets in an electron storage ring

The temporal structure of synchrotron radiation reflects that of the electron beam. Electrons circulating in the storage ring are concentrated into equidistant bunches equal to an integer multiple (usually equal to unity) of the rf-wavelength (60 cm for 500 MHz) while the bunch length itself is of the order of 1 to 3 cm or 30 to 100 ps depending on beam energy and rf-voltage. As a consequence, the photon beam consists of a series of short 30–100 ps flashes every 2 ns (500 MHz) or integer multiples thereof.

Radiation is emitted in a broad spectrum reaching, in principal, from mircowaves up to the critically photon energy (24.49) and beyond with fast declining intensities. The long wavelength limit of the radiation spectrum is actually limited by the vacuum chamber, which causes the suppression of radiation at wavelength longer than its dimensions. The strength of bending magnets, being a part of the geometry of the storage ring cannot be freely varied to optimize for desired photon beam characteristics. This is specifically limiting in the choice of the critical photon energy. While the lower photon energy spectrum is well covered even for rather low energy storage rings, the x-ray region requires high beam energies and/or high magnetic fields. Often, the requirements for x-rays cannot be met with existing bending magnet and storage ring parameters.

1.2 Superbends

The critical photon energy from bending magnet radiation (24.51) is determined by the magnet field and the particle energy. The combination of both quantities may not be sufficient to extend the synchrotron radiation spectrum into the hard x-ray regime, especially in low energy storage rings. In this case, it is possible to replace some or all original bending magnets by much stronger but shorter magnets, called superbends. To be more specific, conventional bending magnets are replaced by high field, shorter superconducting magnets deflecting the electron beam by the same angle to preserve the storage ring geometry. Since conventional bending magnet fields rarely exceed 1.5 T, but superconducting magnets can be operated at 5–6 T or higher, one can gain a factor of 3 to 4 in the critical photon energy and extend the photon spectrum towards or even into the hard x-ray regime and beyond.

1.3 Wavelength Shifter

The installation of superbends is not always feasible or desirable. To still meet the need for harder x-ray radiation in a low energy storage ring, it is customary to use a wavelength shifter. Such a device may consist of three or five superconducting dipole magnets with alternating magnetic field directions. For this latter reason, a wavelength shifter is a true insertion device. Figure 24.2 shows schematically a three-pole wavelength shifter.

Fig. 24.2

Magnetic field distribution along the beam path for a wave length shifter

The particle beam passing through this wavelength shifter is deflected up and down or left and right in such a way that no net deflection remains. To meet this condition, the longitudinal field distribution of a horizontally deflecting wavelength shifter must obey the condition

$$\displaystyle{ \int _{-\infty }^{\infty }B_{ y}\left (y = 0,z\right )\,\text{d}z = 0\,. }$$

(24.1)

A wavelength shifter with such field properties is neutral on the geometry of the particle beam path through a storage ring and therefore can be made in principle as strong as necessary or technically feasible.

Only the central high field pole is used as the radiation source, while the two side poles compensate the beam deflection from the central pole. In a five-pole wavelength shifter the three central poles would be used as radiators, while both end poles again act as compensators. Mostly, the end poles are longer than the central poles and operate at a lower field. As their name implies, the primary objective in wavelength shifters is to extend the photon spectrum while the enhancement of intensity through radiation accumulation from many poles, while desirable, is of secondary importance. To maximize the desired effect, wavelength shifters are often constructed as high field superconducting magnets to maximize the critical photon energy for the given particle beam energy. Some limitations apply for such devices as well as for any other insertion device. The end fields of magnets can introduce particle focusing and nonlinear field components may introduce aberrations and cause beam instability. Both effects must either be kept below a critical level or be compensated.

1.4 Wiggler Magnet Radiation

The principle of a wavelength shifter is extended in the case of a wiggler-magnet. Such a magnet consists of a series of equal dipole magnets with alternating magnetic field direction. Again, the end poles must be configured to make the total device neutral to the geometry of the particle beam path such that the conditions $\int \boldsymbol{B}\mathrm{d}z = 0$ are met in both planes.

The main advantage of using many magnet poles is to increase the photon flux. Like a single bending magnet, each of the N_pol magnet poles produces a fan of radiation in the forward direction and the total photon flux is N_pol-times larger than that from a single pole. Wiggler-magnets may be constructed as electromagnets with fields up to 2 T to function both as a flux enhancer and as a more modest wavelength shifter compared to the superconducting type. An example of an 8-pole, 1.8 T electromagnetic wiggler-magnet [8] is shown in Fig. 24.3.

Fig. 24.3

Electromagnetic wiggler magnet with eight 1.8 T poles

In this picture, the magnet gap is wide open, to display the flat vacuum chamber running through the magnet between the poles. The pole pieces in the lower row are visible surrounded by water cooled excitation coils. During operation, both rows of wiggler poles are closed to almost touch the flat vacuum chamber. When the magnet is closed, a maximum magnetic field of 1.8 T can be obtained. Strong fields can be obtained from electromagnets, but the space requirement for the excitation coils limits the number of poles that can be installed within a given length.

Progress in the manufacturing of high field permanent magnet material permits the installation of many more poles into the same space compared to an electromagnet. An example of a modern 26 pole, 2.0 T permanent magnet wiggler magnet is shown in Fig. 24.4.¹

Fig. 24.4

Permanent magnet wiggler magnet with 26 poles, a 175 mm period length and a maximum field of 2.0 T

Figure 24.4 shows the wiggler magnet during magnetic measurement with the rail in front of the magnet holding and guiding the Hall probe. The increased number of poles and simplified design compared to the electromagnetic wiggler in Fig. 24.3 are clearly visible.

For short wiggler poles, we express the magnetic field by

$$\displaystyle{ B_{y}\left (x,y = 0,z\right )\, = B_{0}\sin \frac{2\pi z} {\lambda _{\text{p}}} \, }$$

(24.2)

and the maximum beam deflection from the axis is equal to the deflection angle per half pole

$$\displaystyle{ \vartheta = \frac{B_{0}} {B\rho } \int _{0}^{\lambda _{\text{p}}/4}\sin \frac{2\pi z} {\lambda _{\text{p}}} \,\text{d }z = \frac{B_{0}} {B\rho } \frac{\lambda _{\text{p}}} {2\pi }, }$$

(24.3)

where B ρ is the beam rigidity. Multiplying this with the beam energy γ, we define the wiggler strength parameter

$$\displaystyle{ K =\gamma \vartheta = \frac{ecB_{0}} {mc^{2}} \frac{\lambda _{\text{p}}} {2\pi } = 0.934B_{0}\left (\text{T}\right )\lambda _{\text{p}}\left (\text{cm}\right )\,. }$$

(24.4)

For longer magnet poles (24.2) must be replaced by a sum of harmonics. Most wiggler magnets, though, are designed for the lowest harmonic only. This wiggler strength parameter is generally much larger than unity. Conversely, a series of alternating magnet poles is called a wiggler magnet if the strength parameter $K \gg 1$ and condition (24.1) is met. As we will see later, a weak wiggler magnet with K ≪ 1 is called an undulator and produces radiation with significant different characteristics. The magnetic field strength can be varied in both electromagnetic wigglers as well as in permanent magnet wigglers. While this is obvious for electromagnets, the magnetic field strength in permanent magnets depends on the distance between magnet poles or on the gap height g. By varying mechanically the gap height of a permanent magnet wiggler, the magnetic field strength can be varied as well. The field strength also depends on the period length and on the design and magnet materials used. For a wiggler magnet constructed as a hybrid magnet with Vanadium Permendur poles, the field strength along the midplane axis scales approximately like [9]

$$\displaystyle{ B_{y}(\text{T}) \approx 3.33\exp \left [-\frac{g} {\lambda _{\text{p}}}\left (5.47 - 1.8\frac{g} {\lambda _{\text{p}}}\right )\right ]\,,\ \ \ \ \text{for }0.1\,\lambda _{\text{p}} \precapprox g \precapprox 10\,\lambda _{\text{p}}\,, }$$

(24.5)

where g is the gap aperture between magnet poles. This dependency is also shown in Fig. 24.5 and we note immediately that the field strength drops off dramatically for magnet gaps of the order of a period length or greater.

Fig. 24.5

On-axis field strength in a vanadium Permendur hybrid wiggler magnet as a function of gap aperture (24.5)

On the other hand, significant field strengths can be obtained for small gap apertures and it is therefore important to install the insertion device at a location, where the beam dimension normal to the deflection plane is small.

The total radiation power can be derived by integrating (24.32) through the wiggler magnet. The result of this integration is

$$\displaystyle{ \left \langle P_{\gamma }\right \rangle = \tfrac{1} {3}r_{\text{c}}mc^{2}\,c\gamma ^{2}K^{2} \frac{4\pi ^{2}} {\lambda _{\text{p}}^{2}}\,, }$$

(24.6)

or in practical units

$$\displaystyle{ \left \langle P_{\gamma }\left (\text{W}\right )\right \rangle = 632.7\,E^{2}B_{ 0}^{2}\,I\,L_{\text{u}}\,, }$$

(24.7)

where I is the circulating beam current, and L_u = N_pλ_p the length of the wiggler magnet.

For a sinusoidal field distribution $B_{0}\sin \frac{2\pi } {\lambda _{\text{p}}}z$, the desired wavelength shifting property of a strong wiggler magnet can be obtained only in the forward direction. Radiation emitted at a finite angle with respect to the wiggler axis is softer because it is generated at a source point where the field is lower. The hardest radiation is emitted in the forward direction from the crest of the magnetic field. For a distance Δ z away from the crest, the emission angle in the deflection plane is $\psi = \frac{1} {\rho _{0}} \frac{\lambda _{\text{p}}} {2\pi } \sin \left ( \frac{2\pi \,} {\lambda _{\text{p}}}\varDelta z\right )$ and the curvature at the source point is $\frac{1} {\rho } = \frac{1} {\rho _{0}} \sqrt{1 - \left ( \frac{\gamma \psi }{K} \right ) ^{2}}$, where we have made use of (24.4). Consequently, the critical photon energy for radiation in the direction ψ with respect to the wiggler axis varies with the emission angle ψ like

$$\displaystyle{ \varepsilon _{\text{c}} =\varepsilon _{\text{c0}}\sqrt{1 - \left ( \frac{\gamma \psi } {K}\right )^{2}}. }$$

(24.8)

At the maximum deflection angle ψ_max = K∕γ the critical photon energy has dropped to zero, reflecting a zero magnetic field at the source point.

This property is undesirable if more than one experimental station is supposed to receive hard radiation from the same wiggler magnet. The strength of the wiggler magnet sweeps the electron beam over a considerable angle, a feature which can be exploited to direct radiation not only to one experimental station along the axis but also to two or more side-stations on either side of the wiggler axis. However, these side beam lines at an angle ψ ≠ 0 receive softer radiation than the main beam line. This can be avoided if the poles of the wiggler magnet are lengthened thus flattening the sinusoidal field crest. As the flat part of the field crest is increased, hard radiation is emitted into an increasing angular cone.

1.5 Undulator Radiation

So far, we discussed insertion devices designed specifically to harden the radiation spectrum or to increase the radiation intensity. Equally common is the implementation of insertion devices to optimize photon beam quality by maximizing its brightness or to provide specific characteristics like elliptically polarized radiation. This is done with the use of undulator magnets, which are constructed similar to wiggler magnets, but are operated at a reduced field strength.

Fundamentally, an undulator magnet causes particles to be only very weakly deflected with an angle of less than ± 1∕γ and consequently the transverse motion of particles is nonrelativistic. In this picture, the electron motion viewed from far away along the beam axis appears as a purely sinusoidal transverse oscillation similar to the electron motion in a linear radio antenna driven by a transmitter and oscillating at the station’s carrier frequency. The radiation emitted is therefore monochromatic with a period equal to the oscillation period.

To be more precise, viewed from far away the particle appears to be at rest or uniform motion as long as the electron has not yet reached the undulator magnet. Upon entering the magnet the electron performs sinusoidal transverse oscillations and returns to its original motion again after it exits the undulator. As a consequence of this motion and in light of earlier discussions, we observe emission of radiation at the frequency of the transverse oscillating beam motion. If N_per is the number of undulator periods, the electric field lines have been perturbed periodically N_per-times and the radiation pulse is composed of N_per oscillations. In the particle rest frame $\mathcal{L}^{{\ast}}$ the undulator period length is Lorentz contracted to $\lambda _{\gamma }^{{\ast}} =\lambda _{\text{p}}/\gamma$ which is the wavelength of the emitted radiation. Because the radiation includes only a finite number of N_per oscillations, the radiation is not quite monochromatic but rather quasi monochromatic with a band width of 1∕N_per as illustrated in Fig. 24.6 (top).

Fig. 24.6

Beam dynamics and radiation lobes in the particle rest system (a) and the laboratory system (b) for a weak undulator (K ≪ 1)

In Fig. 24.6 (bottom) the radiation lobe and spectrum is shown in the laboratory system. The monochromatic nature of the radiation is somewhat lost because radiation emitted at different angles experiences different Doppler shifts. Of course, the radiation is again quasi monochromatic even in the laboratory system when observed through a narrow pin hole along the axis. This monochromatic radiation is called the fundamental undulator radiation and has for K ≪ 1 a wavelength of  

$$\displaystyle{ \lambda _{\gamma } \approx \frac{\lambda _{\text{p}}} {2\gamma ^{2}}. }$$

(24.9)

The situation becomes more complicated as the undulator strength is increased. Two new phenomena appear, an oscillatory forward motion and a transverse relativistic effect. The first phenomenon that we need to discuss is the fact that the transverse motion becomes relativistic. As a consequence of this, the pure sinusoidal transverse motion becomes distorted. There is a periodic Lorentz contraction of the longitudinal coordinate, which is larger when the particle travels almost parallel to the axis in the vicinity of the oscillation crests and is smaller when in between crests. The cusps and valleys of the sinusoidal motion become Lorentz-contracted in the particle system thus perturbing the sinusoidal motion as shown in Fig. 24.7. In addition with increasing undulator strength the transverse motion becomes relativistic and the transverse Lorentz contraction enhances the distortion of the sine-like motion.

Fig. 24.7

Distortion of sinusoidal motion due to relativistic perturbation of transverse motion

This perturbation is symmetric about the cusps and valleys causing the appearance of odd and only odd (3^rd, 5^th, 7^th…) harmonics of the fundamental oscillation period. From an undulator of medium strength ($K \gtrsim 1$) we observe therefore along the axis a line spectrum of odd harmonics in addition to the fundamental undulator radiation.

The second phenomenon to be discussed is the periodic modulation of the longitudinal motion. The longitudinal component of the particle velocity is maximum when the particle travels close to the crest of the oscillations and at a minimum when it is close to the axis crossings. In a reference system which moves uniformly with the average longitudinal particle velocity along the axis, the particle performs periodic longitudinal oscillations in addition to the transverse oscillations. For each transverse period, the particle performs two longitudinal oscillations and its path looks therefore like a figure of eight. This situation is shown in Fig. 24.8.

Fig. 24.8

Beam dynamics (left) and radiation lobes (middle) in the particle rest system together with the harmonics spectrum (right) for a stronger undulator ($K \gtrsim 1$)

We have now two orthogonal accelerations, one transverse and one longitudinal, and two radiation lobes as indicated in Fig. 24.8. Since the longitudinal motion occurs at twice the frequency of the transverse motion, we observe now radiation also at twice the fundamental frequency. Of course, the relativistic perturbation applies here too and we have therefore a line spectrum which includes two series, one with all odd harmonics and one with only even harmonics. Even and odd harmonic radiation is emitted in the particle system in orthogonal directions and therefore we find both radiation lobes in the laboratory system spatially separated as well. The odd harmonics all have their highest intensities along the undulator axis, while the even harmonic radiation is emitted preferentially into an angle 1∕γ with respect to the axis and has zero intensity along the axis.

In another equally valid view of undulator radiation, the static and periodic magnetic undulator field appears in the rest frame of the electron as a Lorentz contracted electromagnetic field or as monochromatic photon of wavelength $\lambda ^{{\ast}} =\lambda _{\text{p}}/\gamma$. The emission of photons can therefore be described as Thomson scattering of virtual photons by free electrons [10] resulting in monochromatic radiation in the direction of the particle path. Viewed from the laboratory system, the radiation is Doppler shifted and applying (1.38) the wavelength of the backscattered photons is

$$\displaystyle{ \lambda _{\text{ph}} = \frac{\lambda _{\text{p}}} {\gamma ^{2}\left (1 +\beta \, n_{z}^{{\ast}}\right )}. }$$

(24.10)

Viewing the radiation parallel to the forward direction ($\vartheta \approx 0)$, (1.39) becomes with $n_{z} =\cos \vartheta ^{{\ast}}\approx 1 -\frac{1} {2}\,\vartheta ^{{\ast}2}$, and $\beta \approx 1$

$$\displaystyle{ 1 +\beta n_{z}^{{\ast}} = \frac{\beta +n_{z}^{{\ast}}} {n_{z}} \approx 2 -\tfrac{1} {2} \frac{\vartheta ^{{\ast}2}} {n_{z}}. }$$

(24.11)

Setting $n_{z} \approx 1$, the fundamental wavelength of the emitted radiation is

$$\displaystyle{ \lambda _{1} = \frac{\lambda _{\text{p}}} {\gamma ^{2}} \frac{1} {2 -\tfrac{1} {2} \frac{\vartheta ^{{\ast}2}} {n_{z}}} \approx \frac{\lambda _{\text{p}}} {2\gamma ^{2}}\left (1 + \tfrac{1} {4}\vartheta ^{{\ast}2}\right ). }$$

(24.12)

With (1.40) the angle $\vartheta ^{{\ast}}$ of the particle trajectory with respect to the observation is transformed into the laboratory system like $\vartheta ^{{\ast}} = 2\gamma \vartheta$. We distinguish two configurations. One where $\vartheta = K/\gamma =$ const. describing the particle motion in a helical undulator, where the magnetic field, being normal to the undulator axis, rotates about this axis. The other more common case is that of a flat undulator, where the particle motion follows a sinusoidal path in which case $\vartheta =\vartheta _{\text{und}} +\vartheta _{\text{obs}}$. Here $\vartheta _{\text{und}} = \frac{K} {\gamma } \sin k_{\text{p}}z$ is the observation angle due to the periodic motion of the electrons in the undulator and $\vartheta _{\text{obs}}$ is the actual observation angle. With these definitions and taking the average $\left \langle \vartheta _{ \text{und}}^{2}\right \rangle$ we get $\gamma ^{2}\vartheta ^{2} = \frac{1} {2}K^{2} +\gamma ^{2}\vartheta _{ \text{obs}}^{2}$. Depending on the type of undulator, the wavelength of radiation from an undulator with a strength parameter K is

$$\displaystyle{ \lambda _{1} = \left \{\begin{array}{ll} \left. \dfrac{\lambda _{\text{p}}} {2\gamma ^{2}}\left (1 + K^{2} +\gamma ^{2}\vartheta _{\text{obs}}^{2}\right )\right. &\text{ for a helical undulator} \\ \left. \dfrac{\lambda _{\text{p}}} {2\gamma ^{2}}\left (1 + \tfrac{1} {2}K^{2} +\gamma ^{2}\vartheta _{ \text{obs}}^{2}\right )\right.&\text{ for a flat undulator.} \end{array} \right. }$$

(24.13)

From now on only flat undulators will be considered in this text and readers interested in more detail of helical undulators are referred to [11]. No special assumptions have been made here which would prevent us to apply this derivation also to higher harmonic radiation and we get the general expression for the wavelength of the kth harmonic

$$\displaystyle{ \lambda _{k} = \frac{\lambda _{\text{p}}} {2\gamma ^{2}k}\left (1 + \tfrac{1} {2}K^{2} +\gamma ^{2}\vartheta _{ \text{obs}}^{2}\right )\,. }$$

(24.14)

The additional terms $\tfrac{1} {2}K^{2} +\gamma ^{2}\vartheta _{ \text{obs}}^{2}$ compared to (24.9) comes from the correct application of the Doppler effect. Since the particles are deflected periodically in the undulator, we view even the on-axis radiation at a periodically varying angle which accounts for the $\tfrac{1} {2}K^{2}$-term. Of course, observation of the radiation at a finite angle $\vartheta _{ \text{obs}}$ generates an additional red-shift expressed by the term $\gamma ^{2}\vartheta _{ \text{obs}}^{2}$.

In more practical units, the undulator wavelengths for the kth harmonic are expressed from (24.14) by

$$\displaystyle{ \lambda _{k}\left (\mathrm{ \AA }\right ) = 13.056 \frac{\lambda _{\text{p}}\left (\text{cm}\right )} {k\,E^{2}\left (\text{GeV}^{2}\right )}\left (1 + \tfrac{1} {2}\,K^{2} +\gamma ^{2}\vartheta _{ \text{obs}}^{2}\right ) }$$

(24.15)

and the corresponding photon energies are

$$\displaystyle{ \epsilon _{k}\left (\text{eV}\right ) = 950 \frac{k\,E^{2}\left (\text{GeV}^{2}\right )} {\lambda _{\text{p}}\left (\text{cm}\right )\left (1 + \tfrac{1} {2}K^{2} +\gamma ^{2}\vartheta _{ \text{obs}}^{2}\right )}. }$$

(24.16)

Recollecting the discussion of undulator radiation, we found that the first harmonic or fundamental radiation is the only radiation emitted for K ≪ 1. As the undulator parameter increases, however, the oscillatory motion of the particle in the undulator deviates from a pure sinusoidal oscillation. For K > 1 the transverse motion becomes relativistic, causing a deformation of the sinusoidal motion and the creation of higher harmonics. These harmonics appear at integral multiples of the fundamental radiation energy. Only odd harmonics are emitted along the axis ($\vartheta \approx 0$) while even harmonics are emitted into a small angle from the axis. As the undulator strength is further increased more and more harmonics appear, each of them having a finite width due to the finite number of undulator periods, and finally merging into the well-known broad spectrum of bending or wiggler magnet radiation (Fig. 24.9).

Fig. 24.9

Transition from quasi-monochromatic undulator radiation to broad band wiggler radiation

We find no fundamental difference between undulator and wiggler magnets, one being just a stronger/weaker version of the other. From a practical point of view, the radiation characteristics are very different and users of synchrotron radiation make use of this difference to optimize their experimental capabilities. In Chap. 26 we will discuss the features of undulator radiation in much more detail.

The electron motion through an undulator with N_per periods includes that many oscillations and so does the radiation field. Applying a Fourier transformation to the field, we find the spectral width of the radiation to be

$$\displaystyle{ \frac{\varDelta \lambda } {\lambda } = \frac{1} {N_{\text{per}}}. }$$

(24.17)

In reality, this line width is increased due to the finite aperture of the radiation detection elements, and due to a finite energy spread and finite divergence of the electron beam. Typical experimental undulator spectra are shown in Fig. 24.10 for increasing undulator strength K [12].

Fig. 24.10

Measured radiation spectrum from an undulator for different strength parameters K. The intensity at low photon energies are reduced by absorption in a Be-window [12]

Although this radiation was measured through a pin hole and on-axis, we still recognize even harmonic radiation since the pin hole covers a finite solid angle and lets some even harmonic radiation through. Furthermore, the measured intensities of the line spectrum does not reflect the theoretical expectation for the lower harmonics at higher values of K. This is an artifact of the experimental circumstances, where the x-rays have been extracted from the storage ring vacuum chamber through a Be-window. Such a window works very well for hard x-rays but absorbs heavily at photon energies below some 3 keV.

The concentration of all radiation into one or few spectral lines is very desirable for many experiments utilizing monochromatic photon beams since radiation is produced only in the vicinity of the desired wavelength at high brightness. Radiation at other wavelengths creating undesired heating effects on optical elements and samples is greatly eliminated.

1.5.1 Back Scattered Photons

The principle of Thomson backscattering or Compton scattering of the static undulator fields can be expanded to that of photon beams colliding head on with the particle beam. In the electron system of reference the electromagnetic field of this photon beam looks fundamentally no different than the electromagnetic field from the undulator magnet. We may therefore apply similar arguments to determine the wavelength of back scattered photons. The basic difference of both effects is that in the case of back scattered photons the photon beam moves with the velocity of light towards the electron beam and therefore the electron sees twice the Lorentz contracted photon frequency and we expect therefore a back scattered photon beam at twice the Doppler shifted frequency. That extra factor of two does not apply for undulator radiation since the undulator field is static and the relative velocity with respect to the electron beam is c. If λ_L is the wavelength of the incident radiation or incident laser, the wavelength of the backscattered photons is

$$\displaystyle{ \lambda _{\gamma } = \frac{\lambda _{\text{L}}} {4\gamma ^{2}}\left (1 +\gamma ^{2}\vartheta _{ \text{obs}}^{2}\right ), }$$

(24.18)

where $\vartheta _{\text{obs}}$ is the angle between the direction of observation and the particle beam axis. Scattering, for example, a high intensity laser beam from high-energy electrons produces a monochromatic beam of hard x-rays which is highly collimated within an angle of ± 1∕γ. If the laser wavelength is, for example, $\lambda _{\text{L}} = 10\,\upmu$ m and the particle energy is 100 MeV the wavelength of the backscattered x-rays would be 1.3 Å or the photon energy would be 9.5 keV which is well within the hard x-ray regime.

1.5.2 Photon Flux

The intensity of the backscattered photons can be calculated in a simple way utilizing the Thomson scattering cross section [10]

$$\displaystyle{ \sigma _{\text{Th}} = \tfrac{8\pi } {3}r_{\text{c}}^{2} = 6.65 \times 10^{-25}\text{ cm}^{2}. }$$

(24.19)

The total scattering event rate or the number of back scattered photons per unit time is then

$$\displaystyle{ N_{\text{sc}} =\sigma _{\text{Th}}\mathcal{L}\text{,} }$$

(24.20)

where $\mathcal{L}$ is called the luminosity. The value of the luminosity is independent of the nature of the physical reaction and depends only on the intensities and geometrical dimensions of the colliding beams. The definition of the luminosity is the product of the target density of one beam by the “particle”-flux of the other beam onto this target. Therefore the luminosity can be determined by folding the particle density in one beam with the incident “particles” per unit time of the other beam. Obviously, only those parts of the beam cross sections count which overlap with the cross section of the other beam. For simplicity, we assume a Gaussian distribution in both beams and assume that both beam cross sections are the same. In a real setup one would focus the electron beam and the photon beam to the same optimum cross section given by the Rayleigh length (27.59). We further consider the particle beam as the target for the photon beam.

With N_e electrons in each bunch of the particle beam within a cross section of $2\pi \sigma _{x}\sigma _{y}$ the particle density is $N_{\text{e}}/\,2\pi \sigma _{x}\sigma _{y}$. We consider now a photon beam with the same time structure as the electron beam. If this is not the case only that part of the photon beam which actually collides with the particle beam within the collision zone may be considered. For an effective photon flux $\dot{N}_{\text{ph}}$ the luminosity is

$$\displaystyle{ \mathcal{L} =\frac{N_{\text{e}}\dot{N}_{\text{ph}}} {2\pi \sigma _{x}\sigma _{y}}. }$$

(24.21)

Although the Thomson cross-section and therefore the photon yield is very small, this technique can be used to produce photon beams with very specific characteristics. By analyzing the scattering distribution this procedure can also be used to determine the degree of polarization of an electron beam in a storage ring.

So far, it was assumed that the incident and scattered photon energies are much smaller than the particle energy in which case it was appropriate to use the classical case of Thomson scattering. However, we note from (24.18) that the backscattered photon energy increases quadratically with the particle energy and therefore at some energy the photon energy becomes larger than the particle energy which is nonphysical. In case of large photon energies comparable with the particle energy, Compton corrections [13–15] must be included. The Compton cross-section for head-on collision is given by [16]

$$\displaystyle{ \sigma _{\text{C}} = \frac{3\,\sigma _{\text{Th}}} {4x} \left [\left (1 -\frac{4} {x} - \frac{8} {x^{2}}\right )\ln \left (1 + x\right ) + \tfrac{1} {2}\, + \frac{8} {x^{2}} - \frac{1} {2\left (1 + x\right )^{2}}\right ], }$$

(24.22)

where $x = \frac{4\gamma \hslash \omega _{0}} {mc^{2}}$, and $\hslash \omega _{0}$ the incident photon energy. The energy spectrum of the scattered photons is then [16]

$$\displaystyle{ \frac{\text{d }\sigma _{\text{C}}} {\text{d }y} = \frac{3\,\sigma _{\text{Th}}} {4x} \left [1 - y + \frac{1} {1 - y} - \frac{4y} {x\left (1 + y\right )} + \frac{4y^{2}} {x^{2}\left (1 - y\right )^{2}}\right ], }$$

(24.23)

where $y = \hslash \omega /E$ is the scattered photon energy in units of the particle energy.

2 Radiation Power

Synchrotron radiation properties can be described in more detail by integrating the Poynting vector (23.7) over a closed surface enclosing the radiating charge. With (23.9) and n^∗d$\mathbf{A}^{{\ast}} = R^{2}\sin \varTheta ^{{\ast}}$ dΘ^∗dΦ^∗ we get the total radiation power from a single electron in its own rest frame

$$\displaystyle{ P^{{\ast}} =\int \boldsymbol{ S}^{{\ast}}\text{d}\boldsymbol{A}^{{\ast}} = \tfrac{2} {3}r_{\text{c}}\frac{mc^{2}} {c^{3}} a^{{\ast}2}, }$$

(24.24)

where we have set $q^{2} = 4\pi \epsilon _{0}r_{\text{c}}mc^{2}$. From the discussion of 4-vectors, we know that the square of the 4-acceleration is invariant to Lorentz transformations and get from (B.21) for the total radiation power in the laboratory system

$$\displaystyle{ P = \tfrac{2} {3}r_{\text{c}}mc\gamma ^{6}\left [\boldsymbol{\dot{\beta }\,}^{2} -\left (\boldsymbol{\beta \,}\times \boldsymbol{\dot{\beta }\,}\right )^{2}\right ]. }$$

(24.25)

Equation (24.25) expresses the radiation power in a simple way and allows us to calculate other radiation characteristics based on beam parameters in the laboratory system. The radiation power is greatly determined by the geometric path of the particle trajectory through the quantities $\boldsymbol{\beta }$ and $\boldsymbol{\dot{\beta }\,}$. Specifically, if this path has strong oscillatory components we expect that motion to be reflected in the synchrotron radiation power spectrum. This aspect will be discussed later in more detail. Here we distinguish only between acceleration parallel $\boldsymbol{\dot{\beta }\,}_{\Vert }$ or perpendicular $\boldsymbol{\dot{\beta }\,}_{\perp }$ to the propagation $\boldsymbol{\beta }$ of the charge and set therefore

$$\displaystyle{ \boldsymbol{\dot{\beta }\,}=\boldsymbol{\dot{\beta }\,} _{\Vert } +\boldsymbol{\dot{\beta }\,} _{\perp }\,. }$$

(24.26)

Insertion into (24.25) shows the total radiation power to be composed of separate contributions from parallel and orthogonal acceleration. Separating both contributions we get the synchrotron radiation power for both parallel and transverse acceleration respectively

$$\displaystyle\begin{array}{rcl} P_{\Vert } = \tfrac{2} {3}r_{\text{c}}mc\gamma ^{6}\boldsymbol{\dot{\beta }\,}_{ \Vert }^{2},& &{}\end{array}$$

(24.27)

$$\displaystyle\begin{array}{rcl} P_{\perp } = \tfrac{2} {3}r_{\text{c}}mc\gamma ^{4}\boldsymbol{\dot{\beta }\,}_{ \perp }^{2}.& &{}\end{array}$$

(24.28)

Expressions have been derived that define the radiation power for parallel acceleration like in a linear accelerator or orthogonal acceleration found in circular accelerators or deflecting systems. We note a similarity for both contributions except for the energy dependence. At highly relativistic energies the same acceleration force leads to much less radiation if the acceleration is parallel to the motion of the particle compared to orthogonal acceleration. Parallel acceleration is related to the accelerating force by $m\boldsymbol{\dot{v}}_{\Vert } = \frac{1} {\gamma ^{3}}$ d$\boldsymbol{p}_{\Vert }/$ dt and after insertion into (24.27) the radiation power due to parallel acceleration becomes

$$\displaystyle{ P_{\Vert } = \frac{2} {3} \frac{r_{\text{c}}} {mc}\left (\frac{\text{d}\boldsymbol{p}_{\Vert }} {\text{d}\,t} \right )^{2}. }$$

(24.29)

The radiation power for acceleration along the propagation of the charged particle is therefore independent of the energy of the particle and depends only on the accelerating force or with d$\boldsymbol{p}_{\Vert }/$ dt = d$\boldsymbol{E}/$ dz on the energy increase per unit length of accelerator. Different from circular electron accelerators we encounter therefore no practical energy limit in a linear accelerator at very high energies. In contrast very different radiation characteristics exist for transverse acceleration as it happens, for example, during the transverse deflection of a charged particle in a magnetic field. The transverse acceleration $\boldsymbol{\dot{v}}_{\perp }$ is expressed by the Lorentz force

$$\displaystyle{ \frac{\text{d}\boldsymbol{p}_{\perp }} {\text{d}\,t} =\gamma m\boldsymbol{\dot{v}}_{\perp } = e\left [\boldsymbol{v}\boldsymbol{ \times }\boldsymbol{ B}\right ] }$$

(24.30)

and after insertion into (24.28) the radiation power from transversely accelerated particles becomes

$$\displaystyle{ P_{\perp } = \tfrac{2} {3} \frac{r_{\text{c}}} {mc}\gamma ^{2}\left (\frac{\text{d}\boldsymbol{p}_{\perp }} {\text{d}\,t} \right )^{2}. }$$

(24.31)

From (24.29), (24.31) we find that the same accelerating force leads to a much higher radiation power by a factor γ² for transverse acceleration compared to longitudinal acceleration. For all practical purposes, technical limitations prevent the occurrence of sufficient longitudinal acceleration to generate noticeable radiation. From here on we will stop considering longitudinal acceleration unless specifically mentioned and eliminate, therefore, the index ⊥ setting for the radiation power P_⊥ = P_γ. We also restrict from now on the discussion to singly charged particles and set q = e ignoring extremely high energies, where multiple charged ions may start to radiate. Replacing the force in (24.31) by the Lorentz force (24.30) we get

$$\displaystyle{ P_{\gamma } = \frac{4\pi } {\mu _{0}} \frac{2\,r_{\text{c}}^{2}c} {3\left (mc^{2}\right )^{2}}B^{2}E^{2} = C_{ B}B^{2}E^{2}, }$$

(24.32)

where $\beta \approx 1$ and

$$\displaystyle{ C_{B} = \frac{4\pi } {\mu _{0}} \frac{2r_{\text{c}}^{2}c} {3\left (mc^{2}\right )^{2}} = 379.35 \frac{1} {\text{T}^{2}\text{GeV}\,\text{s}}. }$$

(24.33)

The synchrotron radiation power scales like the square of the magnetic field and the square of the particle energy. Replacing the deflecting magnetic field B by the bending radius ρ, the instantaneous synchrotron radiation power becomes

$$\displaystyle{ P_{\gamma } = \tfrac{2} {3}r_{\text{c}}mc^{3}\frac{\beta ^{4}\gamma ^{4}} {\rho ^{2}} }$$

(24.34)

or in more practical units,

$$\displaystyle{ P_{\gamma } = \frac{c\,C_{\gamma }} {2\pi } \frac{E^{4}} {\rho ^{2}}, }$$

(24.35)

where

$$\displaystyle{ C_{\gamma } = \frac{4\pi } {3} \frac{r_{\text{c}}} {\left (mc^{2}\right )^{3}} = 8.8463 \times 10^{-5} \frac{\text{m}} {\text{GeV}^{3}}. }$$

(24.36)

The electromagnetic radiation of charged particles in transverse magnetic fields is proportional to the fourth power of the particle momentum β γ and inversely proportional to the square of the bending radius ρ.

The synchrotron radiation power increases very fast for high-energy particles and provides the most severe limitation to the maximum energy achievable in circular accelerators. In storage rings with different magnets and including insertion devices it is important to formulate the average radiation power of an electron during the course of one turn. In this case we calculate the average

$$\displaystyle{ \left \langle P_{\gamma }\right \rangle = \frac{c} {2\pi }C_{\gamma }E^{4}\left \langle \frac{1} {\rho ^{2}} \right \rangle = C_{\gamma }E^{4}\frac{f_{\text{rev}}} {2\pi } \oint \frac{\text{d}z} {\rho ^{2}}. }$$

(24.37)

We note, however, from also a strong dependence on the kind of particles involved in the process of radiation. Because of the much heavier mass of protons compared to the lighter electrons we find appreciable synchrotron radiation only in electron accelerators.

The radiation power of protons actually is smaller compared to that for electrons by the fourth power of the mass ratio or by the factor

$$\displaystyle{ \frac{P_{\text{e}}} {P_{\text{p}}} = 1836^{4} = 1.1367 \times 10^{13}. }$$

(24.38)

In spite of this enormous difference measurable synchrotron radiation has been predicted by Coisson [17] and was indeed detected at the 400 GeV proton synchrotron, SPS (Super Proton Synchrotron), at CERN in Geneva [18, 19]. Substantial synchrotron radiation is expected in multi-TeV proton colliders like the LHC (Large Hadron Collider) at CERN [20].

Knowledge of the synchrotron radiation power allows us now to calculate the energy loss per turn of a particle in a circular accelerator by integrating the radiation power along the circumference of the circular accelerator

$$\displaystyle{ U_{0} =\oint P_{\gamma }\text{d}t = \tfrac{2} {3}r_{\text{c}}mc^{2}\beta ^{3}\gamma ^{4}\oint \frac{\text{d}z} {\rho ^{2}}. }$$

(24.39)

In an isomagnetic lattice, where the bending radius is the same for all bending magnets ρ = const., the integration around a circular accelerator can be performed and the energy loss per turn due to synchrotron radiation is

$$\displaystyle{ U_{0} = P_{\gamma }\frac{2\pi \rho } {\beta c} = \tfrac{4\pi } {3}r_{\text{c}}mc^{2}\beta ^{3}\frac{\gamma ^{4}} {\rho }. }$$

(24.40)

In more practical units, the energy loss of relativistic electrons per revolution in a circular accelerator with an isomagnetic lattice and a bending radius ρ is given by

$$\displaystyle{ U_{0,\text{iso}}\left (\text{GeV}\right ) = C_{\gamma }\frac{E^{4}(\text{GeV}^{4})} {\rho (m)}. }$$

(24.41)

For a beam of N_e particles or a circulating beam current I = ef_revN_e the total average radiation power is

$$\displaystyle{ \left \langle P_{\text{s}}\right \rangle = U_{0}\frac{I} {e}, }$$

(24.42)

or in more practical units

$$\displaystyle{ \left \langle P_{\text{s}}\left (\text{MW}\right )\right \rangle _{\text{iso}} = 0.088463\frac{E^{4}\left (\text{GeV}\right )} {\rho \left (\text{m}\right )} I\left (\text{A}\right ). }$$

(24.43)

The total synchrotron radiation power scales like the fourth power of the particle energy and is inversely proportional to the bending radius. The strong dependence of the radiation on the particle energy causes severe practical limitations on the maximum achievable energy in a circular accelerator.

3 Spectrum

Synchrotron radiation from relativistic charged particles is emitted over a wide spectrum of photon energies. The basic characteristics of this spectrum can be derived from simple principles as suggested in [21]. For an observer synchrotron light has the appearance similar to the light coming from a lighthouse. Although the light is emitted continuously an observer sees only a periodic flash of light as the aperture mechanism rotates in the lighthouse. Similarly, synchrotron light emitted from relativistic particles will appear to an observer as a single flash if it comes from a bending magnet in a transport line passed through by a particle only once or as a series of equidistant light flashes as bunches of particles orbit in a circular accelerator.

Since the duration of the light flashes is very short the observer notes a broad spectrum of frequencies as his eyes or instruments Fourier analyze the pulse of electromagnetic energy. The spectrum of synchrotron light from a circular accelerator is composed of a large number of harmonics of the particle revolution frequency. These harmonics reach a cutoff, where the period of the radiation becomes comparable to the duration of the light pulse. Even though the aperture of the observers eyes or instruments are assumed to be infinitely narrow we still note a finite duration of the light flash.

This is a consequence of the finite opening angle of the radiation as illustrated in Fig. 24.11. Synchrotron light emitted by a particle travelling along the orbit cannot reach the observer before it has reached the point P₀ when those photons emitted on one edge of the radiation cone at an angle − 1∕γ aim directly toward the observer. Similarly, the last photons to reach the observer are emitted from point P₁ at an angle of + 1∕γ. Between point P₀ and point P₁ we have therefore a deflection angle of 2∕γ. The duration of the light flash for the observer is not the time it takes the particle to travel from point P₀ to point P₁ but must be corrected for the finite time of flight for the photon emitted at P₀. If particle and photon would travel toward the observer with exactly the same velocity the light pulse would be infinitely short. However, particles move slower following a slight detour and therefore the duration of the light pulse equals the time difference between the first photons from point P₀arriving at the observer and the last photons being emitted by the particles at point P₁. Although the particle reaches point P₀ at time t = 0 the first photon can be observed at point P₁only after a time

$$\displaystyle{ t_{\gamma } = \frac{2\rho \sin \frac{1} {\gamma } } {c}. }$$

(24.44)

The last photon to reach the observer is emitted when the particle arrives at point P₁ at the time

$$\displaystyle{ t_{\text{e}} = \frac{2\rho } {\beta c\gamma }. }$$

(24.45)

The duration of the light pulse δ t is therefore given by the difference of both travel times (24.44), (24.45)

$$\displaystyle{ \delta t = t_{\text{e}} - t_{\gamma } = \frac{2\rho } {\beta c\gamma } -\frac{2\rho \sin \frac{1} {\gamma } } {c}. }$$

(24.46)

Fig. 24.11

Temporal pulse formation of synchrotron radiation

The sine-function can be expanded for small angles keeping linear and third order terms only and the duration of the light pulse at the location of the observer is after some manipulation

$$\displaystyle{ \delta t = \frac{4\rho } {3c\gamma ^{3}}. }$$

(24.47)

The total duration of the electromagnetic pulse is very short scaling inversely proportional to the third power of γ. This short pulse translates into a broad spectrum. Using only half the pulse length for the effective pulse duration the spectrum reaches up to a maximum frequency of about

$$\displaystyle{ \omega _{\text{c}} \approx \frac{1} {\frac{1} {2}\delta t} \approx \tfrac{3} {2}c\frac{\gamma ^{3}} {\rho }, }$$

(24.48)

which is called the critical photon frequency of synchrotron radiation. The critical photon energy$\varepsilon _{\text{c}} = \hslash \omega _{\text{c}}$ is then given by

$$\displaystyle{ \varepsilon _{\text{c}} = C_{\text{c}}\frac{E^{3}} {\rho }, }$$

(24.49)

with

$$\displaystyle{ C_{\text{c}} = \frac{3\hslash c} {2\left (mc^{2}\right )^{3}}. }$$

(24.50)

For electrons numerical expressions are

$$\displaystyle{ \varepsilon _{\text{c}}\left (\text{keV}\right ) = 2.2183\frac{E^{3}\left (\text{GeV}^{3}\right )} {\rho \left (\text{m}\right )} = 0.66503\,E^{2}\left (\text{GeV}^{2}\right )B\left (\text{T}\right ). }$$

(24.51)

The synchrotron radiation spectrum from relativistic particles in a circular accelerator is made up of harmonics of the particle revolution frequency ω₀ with values up to and beyond the critical frequency (24.51). Generally, a real synchrotron radiation beam from say a storage ring will not display this harmonic structure. The distance between harmonics is extremely small compared to the extracted photon frequencies in the VUV and x-ray regime while the line width is finite due to the energy spread and beam emittance.

For a single pass of particles through a bending magnet in a beam transport line we observe the same spectrum. Specifically, the maximum frequency is the same assuming similar parameters. Synchrotron radiation is emitted in a particular spatial and spectral distribution, both of which will be derived in Chap. 25, and we will use here only some of these results. A useful parameter to characterize the photon intensity is the photon flux per unit solid angle into a frequency bin Δ ω∕ω and from a circulating beam current I defined by

$$\displaystyle{ \frac{\text{d}^{2}\dot{N}_{\text{ph}}} {\text{d}\theta \text{d}\psi } = C_{\varOmega }E^{2}I \frac{\varDelta \omega } {\omega }\left ( \frac{\omega } {\omega _{\text{c}}}\right )^{2}K_{ 2/3}^{2}\left (\xi \right )\,F\left (\xi,\theta \right ), }$$

(24.52)

where ψ is the angle in the deflecting plane and θ the angle normal to the deflecting plane,

$$\displaystyle{ C_{\varOmega } = \frac{3\alpha } {4\pi ^{2}e\left (mc^{2}\right )^{2}} = 1.3255 \times 10^{16} \frac{\text{photons}} {\text{s mrad}^{2}\text{GeV}^{2}\mathrm{ A\ 100\%BW}}, }$$

(24.53)

α the fine structure constant and

$$\displaystyle{ F\left (\xi,\theta \right ) = \left (1 +\gamma ^{2}\theta ^{2}\right )^{2}\left (1 + \frac{\gamma ^{2}\theta ^{2}} {1 +\gamma ^{2}\theta ^{2}} \frac{K_{1/3}^{2}\left (\xi \right )} {K_{2/3}^{2}\left (\xi \right )}\right )\,. }$$

(24.54)

The functions $K_{1/3}\left (\xi \right )$ and $K_{2/3}\left (\xi \right )$, displayed in Fig. 24.12, are modified Bessel’s functions with the argument

$$\displaystyle{ \xi = \tfrac{1} {2} \frac{\omega } {\omega _{\text{c}}}\left (1 +\gamma ^{2}\theta ^{2}\right )^{3/2}. }$$

(24.55)

Fig. 24.12

Modified Bessel’s functions $K_{1/3}\left (\xi \right )$ and $K_{2/3}\left (\xi \right )$

Synchrotron radiation is highly polarized in the plane normal (σ-mode), and parallel (π-mode), to the deflecting magnetic field. The relative flux in both polarization directions is given by the two components in the second bracket of function $F\left (\xi,\theta \right )$ in (24.54). The first component is equal to unity and determines the photon flux for the polarization normal to the magnetic field or σ-mode, while the second term relates to the polarization parallel to the magnetic field which is also called the π-mode. Equation (24.52) expresses both the spectral and spatial photon flux for both the σ-mode radiation in the forward direction within an angle of about ± 1∕γ and for the π-mode off axis.

For highly relativistic particles the synchrotron radiation is collimated very much in the forward direction and we may assume that all radiation in the nondeflecting plane is accepted by the experimental beam line. In this case we are interested in the photon flux integrated over all angles θ. This integration will be performed in Chap. 26 with the result (25.140)

$$\displaystyle{ \frac{\text{d}\dot{N}_{\text{ph}}} {\text{d}\psi } = \frac{4\alpha } {9}\gamma \frac{I} {e} \frac{\varDelta \omega } {\omega }S\left ( \frac{\omega } {\omega _{\text{c}}}\right ), }$$

(24.56)

where ψ is the deflection angle in the bending magnet, α the fine structure constant and the function $S\left (x\right )$ is defined by

$$\displaystyle{ S\left ( \frac{\omega } {\omega _{\text{c}}}\right ) = \frac{9\sqrt{3}} {8\pi } \frac{\omega } {\omega _{\text{c}}}\int \limits _{\omega /\omega _{\text{c}}}^{\infty }K_{ 5/3}\left (\bar{x}\right )\,\text{d}\bar{x} }$$

(24.57)

with K_5∕3(x) a modified Bessel’s function. The function S(ω∕ω_c) is known as the universal function of synchrotron radiation and is shown in Fig. 24.13. In practical units, the angle integrated photon flux is

$$\displaystyle{ \frac{\text{d}\dot{N}_{\text{ph}}} {\text{d}\psi } = C_{\psi }E\,I \frac{\varDelta \omega } {\omega }S\left ( \frac{\omega } {\omega _{\text{c}}}\right ) }$$

(24.58)

with C_ψ defined by

$$\displaystyle{ C_{\psi } = \frac{4\alpha } {9e\,mc^{2}} = 3.9614 \times 10^{19} \frac{\text{photons}} {\text{s rad A GeV}}\,. }$$

(24.59)

Fig. 24.13

Universal function of the synchrotron radiation spectrum, S(ω∕ω_c)

The spectral distribution depends only on the particle energy, the critical frequency ω_c and a purely mathematical function. This result has been derived originally by Ivanenko and Sokolov [22] and independently by Schwinger [23]. Specifically it should be noted that the spectral distribution, if normalized to the critical frequency, does not depend on the particle energy and can therefore be represented by a universal distribution shown in Fig. 24.13.

The energy dependence is contained in the cubic dependence of the critical frequency acting as a scaling factor for the actual spectral distribution. The synchrotron radiation spectrum in Fig. 24.13 is rather uniform up to the critical frequency beyond which the intensity falls off rapidly. This synchrotron radiation spectrum has been verified experimentally soon after such radiation sources became available [24, 25].

Equation (24.56) is not well suited for quick calculation of the radiation intensity at a particular frequency. We may, however, express (24.56) in much simpler form for very low and very large frequencies making use of limiting expressions of Bessel’s functions for large and small arguments. For small arguments $x = \frac{\omega } {\omega _{\text{c}}} \ll 1$ an asymptotic approximation [26] for the modified Bessel’s function may be used to give instead of (24.58) with AS(9.6.9) [26]

$$\displaystyle{ \frac{\text{d}\dot{N}_{\text{ph}}} {\text{d}\psi } \approx C_{\psi }EI \frac{\varDelta \omega } {\omega }1.333\left ( \frac{\omega } {\omega _{\text{c}}}\right )^{1/3}. }$$

(24.60)

Similarly, for high photon frequencies $x = \frac{\omega } {\omega _{\text{c}}} \gg 1$ we get with AS(9.7.2) [26]

$$\displaystyle{ \frac{\text{d}\dot{N}_{\text{ph}}} {\text{d}\psi } \approx C_{\psi }EI \frac{\varDelta \omega } {\omega }0.8460\frac{\sqrt{x}} {e^{x}}, }$$

(24.61)

where $x = \frac{\omega } {\omega _{\text{c}}}$. Both approximations are included in Fig. 24.13 and display actually a rather good representation of the real spectral radiation distribution over all but the central portion of the spectrum where S(x) ≈ 0. 4. Specifically, we note the slow increase in the radiation intensity at low frequencies and the exponential drop off above the critical frequency.

4 Spatial Photon Distribution

The expressions for the photon fluxes (24.52), (24.56) provide the opportunity to calculate the spectral distribution of the photon beam divergence. Photons are emitted into a narrow angle and we may represent this narrow angular distribution by a Gaussian distribution. The effective width of a Gaussian distribution is $\sqrt{ 2\pi }\sigma _{\theta }$ and we set

$$\displaystyle{ \frac{\text{d}\dot{N}_{\text{ph}}} {\text{d}\psi } \approx \frac{\text{d}^{2}\dot{N}_{\text{ph}}} {\text{d}\theta \,\text{d}\psi } \sqrt{2\pi }\sigma _{\theta }. }$$

(24.62)

With (24.52), (24.58) the angular divergence of the forward lobe of the photon beam or for a beam polarized in the σ-mode is

$$\displaystyle{ \sigma _{\theta }\left (\text{mrad}\right ) = \frac{C_{\psi }} {\sqrt{2\pi }C_{\varOmega }} \frac{1} {E} \frac{S\left (x\right )} {x^{2}K_{2/3}^{2}\left (\frac{1} {2}x\right )} = \frac{f\,\left (x\right )} {E\left (\text{GeV}\right )}, }$$

(24.63)

where x = ω∕ω_c. For the forward direction θ ≈ 0 the function $f\,(x) =\sigma _{\theta }\left (\text{mrad}\right )E\left (\text{GeV}\right )$ is shown in Fig. 24.14 for easy numerical calculations.

Fig. 24.14

Scaling function f(x) = σ_θ(mrad) E(GeV) for the photon beam divergence in (24.63)

For wavelengths $\omega \ll \omega _{\text{c}},\left (\mbox{ 24.63}\right )$ can be greatly simplified to become in more practical units

$$\displaystyle{ \sigma _{\theta }\left (\text{mrad}\right ) \approx \frac{0.54626} {E\left (\text{GeV}\right )}\left ( \frac{\omega } {\omega _{\text{c}}}\right )^{1/3} = \frac{7.124} {\left [\rho \left (\text{m}\right )\,\epsilon _{\text{ph}}\left (\text{eV}\right )\right ]^{1/3}}, }$$

(24.64)

where ρ is the bending radius and ε_ph the photon energy. The photon beam divergence for low photon energies compared to the critical photon energy is independent of the particle energy and scales inversely proportional to the third root of the bending radius and photon energy.

5 Fraunhofer Diffraction

Synchrotron radiation is emitted from a rather small area equal to the cross section of the electron beam. In the extreme and depending on the photon wavelength the radiation may be spatially coherent because the beam cross section in phase space is smaller than the wavelength. This possibility to create spatially coherent radiation is important for many experiments specifically for holography and we will discuss in more detail the conditions for the particle beam to emit such radiation.

Reducing the particle beam cross section in phase space by diminishing the particle beam emittance reduces also the source size of the photon beam. This process of reducing the beam emittance is, however, effective only to some point. Further reduction of the particle beam emittance would have no effect on the photon beam emittance because of diffraction effects. A point like photon source appears in an optical instrument as a disk with concentric illuminated rings. For synchrotron radiation sources it is of great interest to maximize the photon beam brightness which is the photon density in phase space. On the other hand designing a lattice for a very small beam emittance can cause beam stability problems. It is therefore prudent not to push the particle beam emittance to values much less than the diffraction limited photon beam emittance. In the following we will therefore define diffraction limited photon beam emittance as a guide for low emittance lattice design.

For highly collimated synchrotron radiation it is appropriate to assume Fraunhofer diffraction. Radiation from an extended light source appears diffracted in the image plane with a radiation pattern which is characteristic for the particular source size and radiation distribution as well as for the geometry of the apertures involved. For simplicity, we will use the case of a round aperture being the boundaries of the beam itself although in most cases the beam cross section is more elliptical. In spite of this simplification, however, we will obtain all basic physical properties of diffraction which are of interest to us. We consider a circular light source with diameter 2a. The radiation field at point P in the image plane is then determined by the Fraunhofer diffraction integral [27]

$$\displaystyle{ U(P) = C\,\int _{0}^{a}\int _{ 0}^{2\pi }\text{e}^{-\text{i}k\rho w\cos \left (\varTheta -\psi \right )}\text{d}\varTheta \,\rho \text{d}\rho, }$$

(24.65)

where k is the wave number of the radiation and w is the sine of the angle between the light ray and the optical axis as shown in Fig. 24.15.

Fig. 24.15

Diffraction geometry

With α = Θ −ψ and the definition of the lowest order Bessel’s function $J_{0}\left (x\right ) = \frac{1} {2\pi }\int _{0}^{2\pi }$ e^−ixcosαdα, (24.65) can be expressed by the integral

$$\displaystyle{ U(P) = 2\pi C\,\int _{0}^{a}J_{ 0}\left (k\rho w\right )\rho \text{d}\rho. }$$

(24.66)

This integral can be solved analytically as well with the identity $\int _{0}^{x}J_{0}\left (y\right )\,y\,$ dy = xJ₁(x). The radiation intensity is proportional to the square of the radiation field and we get finally for the radiation intensity in the image plane at the point P

$$\displaystyle{ I(P) = I_{0}\frac{4J_{1}^{\,2}(kaw)} {(kaw)^{2}}, }$$

(24.67)

where $I(P) = \left \vert U(P)\right \vert ^{2}$ and $I_{0} = I(w \rightarrow 0)$ is the radiation intensity at the image center. This result has been derived first by Airy [28]. The radiation intensity from a light source of small circular cross section is distributed in the image plane due to diffraction into a central circle and concentric rings illuminated as shown in Fig. 24.16.

Fig. 24.16

Fraunhofer diffraction for a circular uniform light source

Tacitly, we have assumed that the distribution of emission at the source is uniform which is generally not correct for a particle beam. A Gaussian distribution is more realistic resembling the distribution of independently radiating particles. We must be careful in the choice of the scaling parameter. The relevant quantity for the Fraunhofer integral is not the actual particle beam size at the source point but rather the apparent beam size and distribution. By folding the particle density distribution with the argument of the Fraunhofer diffraction integral we get the radiation field from a round, Gaussian particle beam,

$$\displaystyle{ U_{\text{G}}(P) \propto \int _{0}^{\infty }\exp \left (- \frac{\rho ^{2}} {2\sigma _{r}^{2}}\right )\,J_{0}\left (k\rho w\right )\rho \text{d}\rho, }$$

(24.68)

where σ_r is the apparent standard source radius. Introducing the variable $x =\rho /\sqrt{2}\sigma _{r}$ and replacing $k\rho w = \sqrt{2}xk\,\sigma _{r}w = 2x\sqrt{z}$ we get from (24.68)

$$\displaystyle{ U_{\text{G}}(P) \propto \int _{0}^{\infty }\text{e}^{-x^{2} }x\,J_{0}\left (2x\sqrt{z}\right )\,\text{d}x }$$

(24.69)

and after integration

$$\displaystyle{ U_{\text{G}}(P) \propto \exp \left [-\tfrac{1} {2}\,\left (k\sigma _{r}w\right )^{2}\right ]. }$$

(24.70)

The diffraction pattern from a Gaussian light source (Fig. 24.17) does not exhibit the ring structure of a uniform source. The radiation field assumes rather the form of a Gaussian distribution in the emission angles w with a standard width of $\sigma _{r^{{\prime}}}^{2} = \left \langle w^{2}\right \rangle$ or

$$\displaystyle{ \,\sigma _{r^{{\prime}}} = \frac{1} {k\,\sigma _{r}}\,. }$$

(24.71)

Fig. 24.17

Fraunhofer diffraction for a Gaussian luminescence at the light source

6 Spatial Coherence

Synchrotron radiation is emitted into a broad spectrum with the lowest frequency equal to the revolution frequency and the highest frequency not far above the critical photon energy. Detailed observation of the whole radiation spectrum, however, may reveal significant differences to these theoretical spectra at the low frequency end. At low photon frequencies we may observe an enhancement of the synchrotron radiation beyond intensities predicted by the theory of synchrotron radiation as discussed so far. We note from the definition of the Poynting vector that the radiation power is a quadratic effect with respect to the electric charge. For photon wavelengths equal and longer than the bunch length, we expect therefore all particles within a bunch to radiate coherently and the intensity to be proportional to the square of the number N_e of particles rather than linearly proportional as is the case for high frequencies. This quadratic effect can greatly enhance the radiation since the bunch population can be 10⁸ − 10¹¹ electrons.

Generally such radiation is not emitted from a storage ring beam because radiation with wavelengths longer than the vacuum chamber dimensions are shielded and will not propagate along a metallic beam pipe [29]. This radiation shielding is fortunate for storage ring operation since it eliminates an otherwise significant energy loss mechanism. Actually, since this shielding affects all radiation of sufficient wavelength both the ordinary synchrotron radiation and the coherent radiation is suppressed. New developments in storage ring physics, however, may make it possible to reduce the bunch length by as much as an order of magnitude below presently achieved short bunches of the order of 5–10 mm. Such bunches would then be much shorter than vacuum chamber dimensions and the emission of coherent radiation in some limited frequency range would be possible. Much shorter electron bunches down to a few fs can be produced in linear accelerators [30, 31], and specifically with bunch compression [32] a significant fraction of synchrotron radiation is emitted spontaneously as coherent radiation [33].

In this section we will discuss the physics of spontaneous coherent synchrotron radiation while distinguishing two kinds of coherence in synchrotron radiation, the temporal coherence and the spatial coherence. Temporal coherence occurs when all radiating electrons are located within a short bunch length of the order of the wavelength of the radiation. In this case the radiation from all electrons is emitted with about the same phase. For spatial coherence the electrons may be contained in a long bunch but the transverse beam emittance must be smaller than the radiation wavelength. In either case there is a smooth transition from incoherent radiation to coherent radiation as determined by a formfactor which depends on the bunch length or transverse emittance.

Similar to the particle beam characterization through its emittance we may do the same for the photon beam and doing so for the horizontal or vertical plane we have with $\sigma _{x,y} =\sigma _{r}/\sqrt{2}$ and $\sigma _{x^{{\prime}},y^{{\prime}}} =\sigma _{r^{{\prime}}}/\sqrt{2}$ the photon beam emittance

$$\displaystyle{ \epsilon _{\text{ph,}x,y} = \tfrac{1} {2}\sigma _{r}\sigma _{r^{{\prime}}} = \frac{\lambda } {4\pi }. }$$

(24.72)

This is the diffraction limited photon emittance and reducing the electron beam emittance below this value would not lead to an additional reduction in the photon beam emittance. To produce a spatially coherent or diffraction limited radiation source the particle beam emittance must be less than the diffraction limited photon emittance

$$\displaystyle{ \epsilon _{x,y} \leq \frac{\lambda } {4\pi }. }$$

(24.73)

Obviously, this condition is easier to achieve for long wavelengths. For visible light, for example, the electron beam emittance must be smaller than about 5 × 10⁻⁸ rad-m to be a spatially coherent radiation source. After having determined the diffraction limited photon emittance we may also determine the apparent photon beam size and divergence. The photon source extends over some finite length L along the particle path which could be either the path length required for a deflection angle of 2∕γ or a much longer length in the case of an undulator to be discussed in the next chapter. With $\sigma _{r^{{\prime}}}$ the diffraction limited beam divergence the photons seem to come from a disc with diameter (Fig. 24.18)

$$\displaystyle{ D =\sigma _{r^{{\prime}}}L. }$$

(24.74)

Fig. 24.18

Apparent photon source size

On the other hand, we know from interference theory the correlation

$$\displaystyle{ D\sin \sigma _{r^{{\prime}}} \approx D\sigma _{r^{{\prime}}} =\lambda }$$

(24.75)

and eliminating D from both equations gives the diffraction limited photon beam divergence

$$\displaystyle{ \sigma _{r^{{\prime}}} = \sqrt{ \frac{\lambda } {L}}. }$$

(24.76)

With this we get finally from (24.71) also the diffraction limited source size

$$\displaystyle{ \sigma _{r} = \frac{1} {2\pi }\sqrt{\lambda L}. }$$

(24.77)

The apparent diffraction limited, radial photon beam size and divergence depend both on the photon wavelength of interest and the length of the source.

7 Temporal Coherence

To discuss the appearance of temporal coherent synchrotron radiation, we consider the radiation emitted from each particle within a bunch. The radiation field at a frequency ω from a single electron is

$$\displaystyle{ \mathcal{E}_{j} \propto \text{e}^{\text{i}\left (\omega t+\varphi _{j}\right )}, }$$

(24.78)

where $\varphi _{j}$ describes the position of the jth electron with respect to the bunch center. With z_j the distance from the bunch center, the phase is

$$\displaystyle{ \varphi _{j} = \frac{2\pi } {\lambda } z_{j}\,. }$$

(24.79)

Here we assume that the cross section of the particle beam is small compared to the distance to the observer such that the path length differences from any point of the beam cross section to observer are small compared to the shortest wavelength involved. The radiation power is proportional to the square of the radiation field and summing over all electrons we get

$$\displaystyle\begin{array}{rcl} P\left (\omega \right )& \propto & \sum _{j,l}^{N_{\text{e}}}\mathcal{E}_{ j}\mathcal{E}_{l}^{{\ast}}\propto \sum _{ j,l}^{N_{\text{e}}}\text{e}^{\text{i}\left (\omega t+\varphi _{j}\right )}\text{e}^{-\text{i}\left (\omega t+\varphi _{l}\right )} \\ & =& \sum _{j,l}^{N_{\text{e}}}\exp \mbox{ i(}\varphi _{ j} -\varphi _{l}) = N_{\text{e}} +\sum _{ j\neq l}^{N_{\text{e}}}\exp \text{i}\left (\varphi _{ j} -\varphi _{l}\right )\,.{}\end{array}$$

(24.80)

The first term N_e on the r.h.s. of (24.80) represents the ordinary incoherent synchrotron radiation with a power proportional to the number of radiating particles. The second term describes the coherent power averaging to zero for all but long wavelengths. The actual coherent radiation power spectrum depends on the particular particle distribution in the bunch. For a storage ring bunch it is safe to assume a Gaussian particle distribution and we use therefore the density distribution

$$\displaystyle{ \varPsi _{\text{G}}\left (z\right ) = \frac{N_{\text{e}}} {\sqrt{2\pi }\sigma }\exp \left (-\frac{z^{2}} {2\sigma ^{2}}\right ), }$$

(24.81)

where σ is the standard value of the Gaussian bunch length. Instead of summing over all electrons we integrate over all phases and folding the density distribution (24.81) with the radiation power (24.80) we get with (24.79)

$$\displaystyle{ P\left (\omega \right ) \propto N_{\text{e}} + N_{\text{e}}\frac{N_{\text{e}} - 1} {2\pi \sigma ^{2}} \,I_{1}I_{2}\,, }$$

(24.82)

where the integrals I₁ and I₂ are defined by

$$\displaystyle\begin{array}{rcl} I_{1}& =\int _{ -\infty }^{+\infty }\exp \left (-\frac{z^{2}} {2\sigma ^{2}} + \text{i }2\pi \frac{z} {\lambda } \right )\text{d }z\,,&{}\end{array}$$

(24.83a)

$$\displaystyle\begin{array}{rcl} I_{2}& =\int _{ -\infty }^{+\infty }\exp \left (-\frac{w^{2}} {2\sigma ^{2}} + \text{i }2\pi \frac{w} {\lambda } \right )\text{d}w\,,&{}\end{array}$$

(24.83b)

and $z = \frac{1} {2}\,\pi \lambda \varphi _{j}$ and $w = \frac{1} {2}\pi \lambda \varphi _{l}$. The factor N_e − 1 reflects the fact that we integrate only over different particles. Both integrals are equal to the Fourier transform for a Gaussian particle distribution. With

$$\displaystyle{ \int _{-\infty }^{+\infty }\exp \left (-\frac{z^{2}} {2\sigma ^{2}} + \text{i }2\pi \frac{z} {\lambda } \right )\text{d}\,z = \sqrt{2\pi }\sigma \exp \left [-2\pi ^{2}\left (\frac{\sigma } {\lambda }\right )^{2}\right ] }$$

(24.84)

we get from (24.82) for the total radiation power at the frequency ω = 2π c∕λ

$$\displaystyle{ P\left (\omega \right ) = p\left (\omega \right )\,N_{\text{e}}\left [1 + \left (N_{\text{e}} - 1\right )g^{2}\left (\sigma,\lambda \right )\right ]\,, }$$

(24.85)

where $p\left (\omega \right )$ is the radiation power from one electron and the Fourier transform

$$\displaystyle{ g^{2}\left (\sigma,\lambda \right ) =\exp \left [-2\pi ^{2}\left (\frac{\sigma } {\lambda }\right )^{2}\right ] }$$

(24.86)

is called the formfactor. With the effective bunch length

$$\displaystyle{ \ell=\sqrt{ 2\pi }\sigma }$$

(24.87)

this formfactor becomes finally

$$\displaystyle{ g^{2}\left (\ell,\lambda \right ) =\exp \left [-\pi \frac{\ell^{2}} {\lambda ^{2}}\right ]\,. }$$

(24.88)

The coherent radiation power falls off rapidly for wavelengths as short or even shorter than the effective bunch length ℓ. In Fig. 24.19 the relative coherent radiation power is shown as a function of the effective bunch length in units of the radiation wavelength. The fast drop off is evident and for an effective bunch length of about $\ell\approx 0.6$ λ the radiation power is reduced to only about 10 % of the maximum power for very short bunches. Particle beams from a linear accelerator have often a more compressed particle distribution of a form between a Gaussian and a rectangular distribution. If we take the extreme of a rectangular distribution

Fig. 24.19

Formfactor g²(ℓ, λ) for a Gaussian and rectangular particle distribution

$$\displaystyle{ \varPsi _{\text{r}}\left (z\right ) = \left \{\begin{array}{ll} 1&\qquad \text{for } -\tfrac{1} {2}\ell <z <\frac{1} {2}\ell \\ 0&\qquad \text{otherwise} \end{array} \right.\,, }$$

(24.89)

we expect to extend the radiation spectrum since the corners and sharp changes of the particle density require a broader spectrum in the Fourier transform. Following the procedure for the Gaussian beam we get for a rectangular particle distribution the Fourier transform

$$\displaystyle{ g\left (\ell\right ) = \frac{\sin x} {x}\,, }$$

(24.90)

where x = π ℓ∕λ. Figure 24.19 also shows the relative coherent radiation power for this distribution and we note a significant but scalloping extension to higher radiation frequencies. Experiments have been performed with picosecond electron bunches from linear accelerators both at Tohoku University [30] and at Cornell University [31] which confirm the appearance of this coherent part of synchrotron radiation.

8 Spectral Brightness

The optical quality of a photon beam is characterized by the spectral brightness defined as the six-dimensional volume occupied by the photon beam in phase space

$$\displaystyle{ \mathcal{B} = \frac{\dot{N}_{\text{ph}}} {4\pi ^{2}\sigma _{x}\sigma _{x^{{\prime}}\,}\sigma _{y}\sigma _{y^{{\prime}}\,}\frac{\text{d}\omega } {\omega } }\,, }$$

(24.91)

where $\dot{N}_{\text{ph}}$ is the photon flux defined in (24.58). In the laser community, this quantity is called the radiance while the term spectral brightness is common in the synchrotron radiation community.²

For bending magnet radiation there is a uniform angular distribution in the deflecting plane and we must therefore replace the Gaussian divergence $\sigma _{x^{{\prime}}}$ by the total acceptance angle Δ ψ of the photon beam line or experiment. The particle beam emittance must be minimized to achieve maximum spectral photon beam brightness. However, unlimited reduction of the particle beam emittance will, at some point, seize to further increase the brightness. Because of diffraction effects the electron beam emittance need not be reduced significantly below the limit (24.72) discussed in the previous section.

For a negligible particle beam emittance the maximum diffraction limited spectral brightness is from (24.72), (24.91)

$$\displaystyle{ \mathcal{B}_{\text{max}}= \frac{4} {\lambda ^{2}\frac{\text{d}\omega } {\omega } }\dot{N}_{\text{ph}}. }$$

(24.92)

For a realistic synchrotron light source the finite beam emittance of the particle beam must be taken into account as well which is often even the dominant emittance being larger than the diffraction limited photon beam emittance. We may add both contributions in quadrature and have for the total source parameters

$$\displaystyle\begin{array}{rcl} \sigma _{\text{tot,}x}& = \sqrt{\sigma _{\text{b,} x }^{2 } + \tfrac{1} {2}\sigma _{r}^{2}}\,,\qquad \quad \sigma _{\text{tot,} x^{{\prime}}} = \sqrt{\sigma _{\text{b,} x^{{\prime} } }^{2 } + \tfrac{1} {2}\sigma _{r^{{\prime}}}^{2}}\,,&{}\end{array}$$

(24.93)

$$\displaystyle\begin{array}{rcl} \sigma _{\text{tot,}y}& = \sqrt{\sigma _{\mbox{ b},y }^{2 } + \tfrac{1} {2}\sigma _{r}^{2}}\,,\qquad \quad \sigma _{\text{tot,} y^{{\prime}}} = \sqrt{\sigma _{\text{b,} y^{{\prime} } }^{2 } + \tfrac{1} {2}\sigma _{r^{{\prime}}}^{2}}\,,&{}\end{array}$$

(24.94)

where σ_b refers to the respective particle beam parameters.

8.1 Matching

A finite particle beam emittance does reduce the photon beam brightness from it’s ideal maximum. The amount of reduction, however, depends on the matching to the photon beam. The photon beam size and divergence are the result of folding the diffraction limited source emittance with the electron beam size and divergence. In cases where the electron beam emittance becomes comparable to the diffraction limited emittance the effective photon beam brightness can be greatly affected by the mutual orientation of both emittances. Matching both orientations will maximize the photon beam brightness.

This matching process is demonstrated in Fig. 24.20. The left side shows a situation of poor matching in 2-dimensional x − x^′-phase space. In this case the electron beam width is much larger than the diffraction limited source size while its divergence is small compared to the diffraction limit. The effective photon beam distribution in phase space is the folding of both electron beam parameters and diffraction limit and is much larger than either one of its components. The photon beam width is dominated by the electron beam width and the photon beam divergence is dominated by the diffraction limit. Consequently, the effective photon density in phase space and photon beam brightness is reduced.

Fig. 24.20

Matching of the electron beam emittance to the diffraction limited emittance to gain maximum photon beam brightness

To improve the situation one would focus the electron beam to a smaller beam size at the source point at the expense of beam divergence. The reduction of the electron beam width increases directly the photon beam brightness while the related increase of the electron beam divergence is ineffective because the diffraction limit is the dominant term. Applying more focusing may give a situation shown on the right side of Fig. 24.20 where the folded photon phase space distribution is reduced and the brightness correspondingly increased. Of course, if the electron beam is focused too much we have the opposite situation as discussed. There is an optimum focusing for optimum matching.

To find this optimum we use the particle beam parameters

$$\displaystyle{ \sigma _{\mathrm{b},x,y}^{2} =\epsilon _{ x,y}\beta _{x,y}\qquad \mathrm{and}\qquad \sigma _{\mathrm{b},x^{{\prime}},y^{{\prime}}}^{2} = \frac{\epsilon _{x,y}} {\beta _{x,y}}\,, }$$

(24.95)

where β_x, y are the betatron functions at the photon source location and ε_x, y the beam emittances, in the horizontal and vertical plane respectively. Including diffraction limits, the product

$$\displaystyle{ \sigma _{\mathrm{tot},x}\sigma _{\mathrm{tot},x^{{\prime}}} = \sqrt{\epsilon _{x } \beta _{x } + \tfrac{1} {2}\sigma _{r}^{2}}\sqrt{\frac{\epsilon _{x } } {\beta _{x}} + \tfrac{1} {2}\sigma _{r^{{\prime}}}^{2}} }$$

(24.96)

has a minimum ($\frac{\mathrm{d}} {\mathrm{d}\beta _{x}}\sigma _{\mathrm{tot},x}\sigma _{\mathrm{tot},x^{{\prime}}} = 0$) for

$$\displaystyle{ \beta _{x} = \frac{\sigma _{r}} {\sigma _{r^{{\prime}}}} = \frac{L} {2\pi }. }$$

(24.97)

A similar optimum occurs for the vertical betatron function at the source point. The optimum value of the betatron functions at the source point depends only on the length of the undulator.

The values of the horizontal and vertical betatron functions should be adjusted according to (24.97) for optimum photon beam brightness. In case the particle beam emittance is much larger than the diffraction limited photon beam emittance, this minimum is very shallow and almost nonexistent in which case the importance of matching becomes irrelevant. As useful as matching may appear to be, it is not always possible to reach perfect matching because of limitations in the storage ring focusing system. Furthermore it is practically impossible to get a perfect matching for bending magnet radiation since the effective source length L is very small, L = 2ρ∕γ.

9 Photon Source Parameters

In the previous paragraph, we have assumed that there is no dispersion at the source point. This is not always true and we have to modify our beam sizes to take the effect of energy spread and dispersion into account. Still simplifying, we use only the horizontal dispersion. Where this is not acceptable, the vertical dispersion effects have to be added in quadrature. The beam width or height is defined by the contribution of the betatron phase space σ_β, x, y and the energy phase space σ_η, x, y and is

$$\displaystyle{ \sigma _{\text{b},x,y} =\, \sqrt{\sigma _{\beta,x,y }^{2 } +\sigma _{ \eta }^{2}} =\, \sqrt{\epsilon _{x,y } \beta, x, y + \left (\eta \frac{\sigma _{\varepsilon }} {E_{0}}\right )^{2}} }$$

(24.98)

with $\sigma _{\beta,x,y}^{2} =\epsilon _{x,y}\beta _{x,y}$ and $\sigma _{\eta } =\eta \frac{\sigma _{\varepsilon }} {E_{ 0}}$, $\gamma _{x,y} = \frac{1+\alpha _{x,y}^{2}} {\beta _{x,y}}$ and α_x, y = −$\frac{1} {2}$$\beta _{x,y}^{^{{\prime}} }$. Similarly, we get for the beam divergence

$$\displaystyle{ \sigma _{\mathrm{b},x^{{\prime}},y^{{\prime}}} = \sqrt{\sigma _{\beta,x^{{\prime} },y^{{\prime} } }^{2 } +\sigma _{ \eta ^{{\prime} } }^{2}} =\, \sqrt{\epsilon _{x,y } \gamma _{x,y } + \left (\eta ^{{\prime} } \frac{\sigma _{\varepsilon }} {E_{0}}\right )^{2}}. }$$

(24.99)

These beam parameters resemble in general the source parameters of the photon beam. Deviations occur when the beam emittance becomes very small, comparable to the photon wavelength of interest. First the matching conditions should be checked and modified if necessary. Second, the photon source parameters may be modified by diffraction effects which limit the apparent source size and divergence to some minimum values even if the electron beam cross section and divergence should be very small. For radiation at a wavelength λ, the diffraction limited radial photon source parameters are³

$$\displaystyle{ \sigma _{r} = \frac{1} {2\pi }\sqrt{\lambda L}\qquad \mathrm{and}\qquad \sigma _{r^{{\prime}}} = \sqrt{ \frac{\lambda } {L}}. }$$

(24.100)

Projection onto the horizontal or vertical plane gives $\sigma _{x,y} =\sigma _{r}/\sqrt{2}$ etc. Due to diffraction, it is not useful to push the electron beam emittance to values much smaller than

$$\displaystyle{ \epsilon _{x,y} = \frac{\lambda } {4\pi }. }$$

(24.101)

For an arbitrary electron beam cross section the photon source parameters are the quadratic sums of both contributions

$$\displaystyle\begin{array}{rcl} \sigma _{\mathrm{ph},x,y}^{2}& =\sigma _{ \mathrm{ b},x,y}^{2} + \tfrac{1} {2}\sigma _{r}^{2}\,,&{}\end{array}$$

(24.102)

$$\displaystyle\begin{array}{rcl} \sigma _{\mathrm{ph},x^{{\prime}},y^{{\prime}}}^{2}& = \sigma _{\mathrm{ b},x^{{\prime}},y^{{\prime}}}^{2} + \tfrac{1} {2}\sigma _{r^{{\prime}}}^{2}\,.&{}\end{array}$$

(24.103)

The contribution from diffraction can be ignored if

$$\displaystyle{ \epsilon _{x,y} \gg \frac{\lambda } {4\pi }, }$$

(24.104)

which is generally true in the x-direction but not in the y-direction because of the small coupling in a storage ring.

Problems

24.1 (S). Bending magnet radiation (ρ = 2 m) from a 800 MeV, 500 mA storage ring includes a high intensity component of infrared radiation. Calculate the photon beam brightness for $\lambda = 10\,\upmu \mathrm{m}$ radiation at the experimental station which is 5 m away from the source. The electron beam cross section is $\sigma _{\text{b,}x} \times \sigma _{\text{b,}y} = 1.1 \times 0.11$ mm and its divergence $\sigma _{\text{b,}x^{{\prime}}}\times \sigma _{\text{b,}y^{{\prime}}} = 0.11 \times 0.011$ mrad. What is the corresponding brightness for infrared radiation from a black body radiator at 2,000 K with a source size of x × y = 10 × 2 mm? (Hint: the source length L = ρ2θ_rad where ±θ_rad is the vertical opening angle of the radiation.)

24.2 (S). What is the probability for a 6 GeV electron to emit a photon with an energy of $\varepsilon =\sigma _{\varepsilon }$ per unit time travelling on a circle with radius ρ = 25 m. How likely is it that this particle emits another such photon within a damping time? In evaluating quantum excitation and equilibrium emittances, do we need to consider multiple photon emissions? (use isomagnetic ring)

24.3 (S). Derive a formula for the average number of photons emitted by an electron of energy E per turn. How many are these for E = 3 GeV and ρ = 10 m.

24.4 (S). In a 7 GeV electron ring the circulating beam current is 200 mA and the bending radius ρ = 20 m. Your experiment requires a photon flux of 10⁶ photons/sec at a photon energy of 8 keV, within a band width of 10⁻⁴ onto a sample with a cross section of $10 \times 10\,\upmu \mathrm{m}^{2}$ and your experiment is 15 m away from the source point. Can you do your experiment on a bending magnet beam line of this ring?

24.5 (S). How well is the electron beam phase space of exercise 24.1 at the source matched to the photon beam? Show the phase space ellipses of both the electron and the photon beam in phase space and in x and y.

24.6 (S). Derive an expression for the total synchrotron radiation power from a wiggler magnet.

24.7. Verify the numerical validity of Eqs. (24.4), (24.43), (24.51), (24.53), (24.59)

24.8 (S). In the SLAC linear accelerator operating at 100 Hz electrons can be accelerated to 50 GeV at a rate of 17 MeV/m. Calculate to total radiation power from 10⁹ electrons per pulse at 50 GeV due to longitudinal acceleration. Compare with the radiation power if this bunch of 10⁹ electrons is deflected at the same energy by 1 mrad in a 0.6 T bending magnet.

24.9 (S). Consider an electron storage ring at an energy of 1 GeV, a circulating current of 200 mA and a bending radius of ρ = 2. 22 m. Calculate the energy loss per turn, the critical energy and the total synchrotron radiation power. At what frequency in units of the critical frequency has the intensity dropped to 1 % of the maximum? Plot the radiation spectrum and determine the frequency range available for experimentation.

24.10. What beam energy would be required to produce x-rays from the storage ring of problem 24.9 at a critical photon energy of 10 keV? Is that energy feasible from a conventional magnet point of view or would the ring have to be larger? What would the new beam energy and bending radius have to be?

24.11. Consider a storage ring with an energy of 1 GeV and a bending radius of ρ = 2. 5 m. Calculate the angular photon flux density d$\dot{N}/$ dψ for a high photon energy $\hat{\varepsilon }$ where the intensity is still 1 % of the maximum spectral intensity. What is this maximum photon energy? Installing a wavelength shifter with a field of B = 6 T allows the spectrum to be greatly extended. By how much does the spectral intensity increase at the photon energy $\hat{\varepsilon }$ and what is the new photon energy limit for the wavelength shifter?

24.12. Consider an electromagnetic wavelength shifter in a 1 GeV storage ring with a central pole length of 30 cm and a maximum field of 6 T. The side poles are 60 cm long and for simplicity assume that the field in all poles has a sinusoidal distribution along the axis. Determine the focal length due to edge focusing for the total wavelength shifter. To be negligible, the focal length should typically be longer than about 30 m. Is this the case for this wavelength shifter?

24.13. Collide a 25 MeV electron beam with a 1 kW CO₂-laser beam ($\lambda = 10\,\upmu \mathrm{m}$). What is the energy of the backscattered photons? Assume a diffraction limited interaction length of twice the Rayleigh length and an electron beam cross section matching the photon beam. Calculate the x-ray photon flux for an electron beam from a 3 GHz linear accelerator with a pulse length of $1\upmu \mathrm{s}$, a repetition rate of 10 Hz and a pulse current of 100 mA.