Particle Accelerator Physics pp 815-855 | Cite as

# 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.

## Keywords

Synchrotron Radiation Radiation Power Storage Ring Photon Beam Beam EmittanceAfter 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, 6, 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.

## 24.1 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

### 24.1.1 Bending Magnet Radiation

*γ*.

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.

### 24.1.2 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.

### 24.1.3 Wavelength Shifter

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.

### 24.1.4 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.

*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.

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.

^{1}

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.

*B ρ*is the beam rigidity. Multiplying this with the beam energy

*γ*, we define the wiggler strength parameter

*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]

*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.

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.

*I*is the circulating beam current, and

*L*

_{u}=

*N*

_{p}

*λ*

_{p}the length of the wiggler magnet.

*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

*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.

### 24.1.5 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.

*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).

*K*≪ 1 a wavelength of

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.

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.

*K*is

*k*th harmonic

*k*th harmonic are expressed from (24.14) by

*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).

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.

*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

*K*[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.

#### Back Scattered Photons

*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

*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.

#### Photon Flux

*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

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.

## 24.2 Radiation Power

**n**

^{∗}d\(\mathbf{A}^{{\ast}} = R^{2}\sin \varTheta ^{{\ast}}\) d

*Θ*

^{∗}d

*Φ*

^{∗}we get the total radiation power from a single electron in its own rest frame

*t*and after insertion into (24.27) the radiation power due to parallel acceleration becomes

*t*= d\(\boldsymbol{E}/\) d

*z*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

*γ*

^{2}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

*B*by the bending radius

*ρ*, the instantaneous synchrotron radiation power becomes

*β γ*and inversely proportional to the square of the bending radius

*ρ*.

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].

*ρ*= const., the integration around a circular accelerator can be performed and the energy loss per turn due to synchrotron radiation is

*ρ*is given by

*N*

_{e}particles or a circulating beam current

*I*=

*ef*

_{rev}

*N*

_{e}the total average radiation power is

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.

## 24.3 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.

*P*

_{0}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

*P*

_{1}at an angle of + 1∕

*γ*. Between point

*P*

_{0}and point

*P*

_{1}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

*P*

_{0}to point

*P*

_{1}but must be corrected for the finite time of flight for the photon emitted at

*P*

_{0}. 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

*P*

_{0}arriving at the observer and the last photons being emitted by the particles at point

*P*

_{1}. Although the particle reaches point

*P*

_{0}at time

*t*= 0 the first photon can be observed at point

*P*

_{1}only after a time

*P*

_{1}at the time

*δ t*is therefore given by the difference of both travel times (24.44), (24.45)

*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

*ω*

_{0}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.

*Δ ω*∕

*ω*and from a circulating beam current

*I*defined by

*ψ*is the angle in the deflecting plane and

*θ*the angle normal to the deflecting plane,

*α*the fine structure constant and

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.

*θ*. This integration will be performed in Chap. 26 with the result ( 25.140)

*ψ*is the deflection angle in the bending magnet,

*α*the fine structure constant and the function \(S\left (x\right )\) is defined by

*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

*C*

_{ψ}defined by

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].

*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.

## 24.4 Spatial Photon Distribution

*σ*-mode is

*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.

*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.

## 24.5 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.

*a*. The radiation field at point

*P*in the image plane is then determined by the Fraunhofer diffraction integral [27]

*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.

*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

^{−ixcosα}d

*α*, (24.65) can be expressed by the integral

*y*=

*xJ*

_{1}(

*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*

*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)

*w*with a standard width of \(\sigma _{r^{{\prime}}}^{2} = \left \langle w^{2}\right \rangle\) or

## 24.6 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 10^{8} − 10^{11} 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.

^{−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)

*D*from both equations gives the diffraction limited photon beam divergence

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

## 24.7 Temporal Coherence

*ω*from a single electron is

*j*th electron with respect to the bunch center. With

*z*

_{j}the distance from the bunch center, the phase is

*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

*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)

*I*

_{1}and

*I*

_{2}are defined by

*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

*ω*= 2

*π c*∕

*λ*

*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

*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.

## 24.8 Spectral Brightness

^{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.

*σ*

_{b}refers to the respective particle beam parameters.

### 24.8.1 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.

*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.

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.

*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

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* = 2*ρ*∕*γ*.

## 24.9 Photon Source Parameters

*σ*

_{β, x, y}and the energy phase space

*σ*

_{η, x, y}and is

*α*

_{x, y}= −\(\frac{1} {2}\)\(\beta _{x,y}^{^{{\prime}} }\). Similarly, we get for the beam divergence

*λ*, the diffraction limited radial photon source parameters are

^{3}

*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* = *ρ*2*θ*_{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 10^{6} 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 10^{9} electrons per pulse at 50 GeV due to longitudinal acceleration. Compare with the radiation power if this bunch of 10^{9} 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\(\dot{N}/\) d*ψ* 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 CO_{2}-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.

## Footnotes

- 1.
The author would like to thank T. Rabedau, Stanford for this picture.

- 2.
Sometimes the term brilliance is used. Since there is no common definition for brilliance and the dictionary does not connect brilliance with say luminescence of a source we do not use this term in this book.

- 3.
Many authors use a different definition \(\sigma _{\text{r}} =\sigma _{r}/\sqrt{2}\). The difference is mainly that the subscript

_{r}refers to radiation and the related beam parameters are already projected to the*x*or*y*-plane. In this text, we use the subscript_{r}from the radial coordinate since we derive the diffraction effects from a round beam.

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