Fundamentals of Synchrotron Radiation Emission
Fundamentals of Synchrotron Radiation Emission
Abstract and Keywords
This chapter derives the general solution for the electric and magnetic fields emitted by an accelerated relativistic charged particle using Maxwell’s equations as the starting point. This general solution is then applied to the case of a bending magnet and the photon flux and brightness levels found. Finally, it is shown that by using a special magnet called a wavelength shifter, it is possible to alter the output spectrum significantly to enhance the output at X-ray wavelengths.
Keywords: electric field, bending magnet, Maxwell’s equations, flux, brightness, wavelength shifter, X-ray
Although this book is primarily concerned with the subject of Insertion Devices (IDs) it is important that we cover the basics first before rushing into the exact output characteristics from undulators and wigglers.
The concept involved in the production of Synchrotron Radiation (SR) is essentially straightforward. The radiation is produced when a relativistic charged particle is accelerated, usually in a magnetic field. Any charged particle will produce SR but the electron is the particle most usually associated with the phenomenon since, as will become apparent, the low rest mass of the electron means that it emits by far the most radiation for a given particle energy (˜ 10^{13} times more than a proton). It is only in the very latest generation of proton accelerators (such as the Large Hadron Collider at CERN) that the emission of SR from protons is of any consequence. The vast majority of synchrotron light sources around the world use electrons as their emitter of SR. The remainder rely on the positron which, although it does have some subtle accelerator physics advantages over the electron, produces exactly the same SR output. Throughout this book I have consistently referred to the electron as the emitting particle but I could equally well have used a positron instead without any loss of accuracy.
This chapter covers the very basics of SR from first principles. An arbitrary electron trajectory is studied to give us some fundamental relationships linking the emitting particle to the observer. The general electric field due to the electron at the observer is then derived and this is applied to the scenario of an electron travelling on a circular path through a uniform magnetic field. This is the extremely important case of bending magnet or dipole radiation. Finally, equations are derived for the actual number of photons arriving at the observer at particular photon wavelengths and also the exact angular distribution of these photons.
2.1 Emission and Observation
First we consider an electron moving with relativistic velocity (i.e., close to the velocity of light) on an arbitrary path (Fig. 2.1). If the electron emits a photon in the direction of an observer then by the time it arrives at the observer the electron will no longer be in the same position. So, if we want to consider the light that is arriving at the observer now we must appreciate that this was emitted by the moving electron some time ago.
The electron emits a photon at a time t′, which arrives at the observer at the later time of t. The time t′ generally goes by the peculiar name of the retarded time but perhaps a better, more intuitive, description is the emission time. The time t will be referred to as the observation time. The photon travels at the speed of light, c, towards the observer along the path R(t′) (note that vector quantities are shown in bold font) and so the observation time is related to the emission time by
Recall the standard result that by differentiating the dot product a(t) · a(t) with respect to time we get
2.2 Electric Field at the Observer
A widely used approach to reducing the number of unknowns in Maxwell's equations is to introduce auxiliary variables, which are known as vector and scalar potentials [12]. From the standard vector analysis result [13] ∇ · (∇ × a) = 0 it is clear that for Maxwell's first equation, ∇ · B = 0, to hold, it is sufficient to have
The first term in eqn (2.8) is the gradient of Φ, (p.11)
The second term in (2.8) contains the differential of the magnetic vector potential with respect to the observer time, t, (p.13)
where we have used the result of (2.6). Differentiating the magnetic vector potential with respect to the emission time, t′, we get
Now that we have derived the electric and magnetic field experienced by the observer we can make several remarks. First, the magnetic field, B, must be perpendicular to both the electric field, E, and the unit vector pointing towards the observer, n since by definition the result a × b is perpendicular to both a and b. Second, if the electron is stationary (β = βad; = 0), then the electric field is given by
2.3 Fourier Transform of the Electric Field
Although we now have a means of calculating the electric field as a function of time experienced by an observer due to an electron on an arbitrary path, it is not altogether clear how to interpret this result. It would be much nicer if we could analyse the electric field to determine what frequency content, and so what electromagnetic radiation, it represented. Of course we can convert from time to frequency with a Fourier Transform and this is the next step. We will assume the far field case is valid as this simplifies things somewhat. The Fourier Transform is given by
Again, the magnetic field is related to the electric field by
2.4 Synchrotron Radiation in a Bending Magnet
We will use the equations that we have just derived to look at the special case of a single relativistic electron moving on a circular orbit. Passing electrons through a uniform magnetic field (commonly referred to as a bending magnet or more correctly, dipole magnet) is the most basic method of generating synchrotron radiation in particle accelerators.
The angular velocity, ω _{0}, of the electron is given by
Introducing our results from (2.21) and (2.23) into the electric field as a function of frequency (2.19) we obtain
Similarly for the electric field in the vertical plane
We can present (2.24) and (2.25) in a more useful form but first we need to make a brief diversion into Airy functions and modified Bessel functions. A standard result from math texts for the Airy function, Ai(x), is [17]
Some example graphs of E _{x}(ω) and E _{y}(ω) are given in Figs. 2.4 and 2.5. The result for E _{x}(ω) looks very much like the classic bending magnet spectrum, as we shall see later in this section. Note that the lower frequencies become more and more important in both the planes as the vertical angle (γψ) increases. This is also a standard bending magnet behaviour where long wavelength radiation extends much further out in vertical angle than shorter wavelength radiation. It is also important to note that the vertical electric field vanishes when ψ = 0, or in other words, the radiation is polarized entirely in the horizontal plane when observed on-axis.
Whether one chooses to work with either the Airy functions or the Bessel functions is purely a matter of personal preference. In the literature, the Bessel function form appears more often though some find the Airy function easier to work with analytically. I have found the Bessel function to be more readily available in commercial software such as spreadsheets and programming libraries and so I generally prefer this form. Table 2.1 gives the value of K _{1/3}(u) and K _{2/3}(u) for a wide range of u, as well as two other useful Bessel functions that we will meet in the following section.
We will need the expressions for the electric fields in the next part where we will look at how the synchrotron radiation power emitted by the electron is distributed in angle.
2.4.1 Angular Power Distribution
A standard result in electromagnetic theory is that the Poynting vector, S, gives the energy flow per unit area per unit time [18].
Table 2.1 Numerical values for modified Bessel functions that are encountered in synchrotron radiation calculations
u |
K _{1/3}(u) |
K _{2/3}(u) |
K _{5/3}(u) |
${\int}_{u}^{\infty}{K}_{5/3}\left(u\right)}du$ |
---|---|---|---|---|
0.0001 |
36.28 |
498.9 |
6.651E+06 |
995.9 |
0.0002 |
28.76 |
314.3 |
2.095E+06 |
626.7 |
0.0004 |
22.79 |
198.0 |
6.599E+05 |
394.1 |
0.0005 |
21.13 |
170.6 |
4.550E+05 |
339.4 |
0.0007 |
18.86 |
136.3 |
2.597E+05 |
270.8 |
0.0008 |
18.03 |
124.7 |
2.079E+05 |
247.6 |
0.001 |
16.72 |
107.5 |
1.433E+05 |
213.1 |
0.002 |
13.19 |
67.69 |
4.514E+04 |
133.6 |
0.004 |
10.38 |
42.62 |
1.422E+04 |
83.49 |
0.005 |
9.594 |
36.72 |
9.802E+03 |
71.70 |
0.007 |
8.514 |
29.33 |
5.594E+03 |
56.93 |
0.008 |
8.116 |
26.82 |
4.478E+03 |
51.93 |
0.01 |
7.486 |
23.10 |
3.087E+03 |
44.50 |
0.02 |
5.781 |
14.50 |
972.3 |
27.36 |
0.04 |
4.386 |
9.052 |
306.1 |
16.57 |
0.05 |
3.991 |
7.762 |
211.0 |
14.03 |
0.07 |
3.437 |
6.136 |
120.3 |
10.85 |
0.08 |
3.231 |
5.581 |
96.25 |
9.777 |
0.1 |
2.900 |
4.753 |
66.27 |
8.182 |
0.2 |
1.979 |
2.802 |
20.66 |
4.517 |
0.4 |
1.206 |
1.517 |
6.263 |
2.255 |
0.5 |
0.9890 |
1.206 |
4.205 |
1.742 |
0.7 |
0.6965 |
0.8148 |
2.248 |
1.126 |
0.8 |
0.5932 |
0.6839 |
1.733 |
0.9280 |
1 |
0.4384 |
0.4945 |
1.098 |
0.6514 |
2 |
0.1165 |
0.1248 |
0.1998 |
0.1508 |
4 |
0.01130 |
0.01173 |
0.01521 |
0.01321 |
5 |
0.003729 |
0.003844 |
0.004754 |
0.004250 |
7 |
4.280E-04 |
4.376E-04 |
5.113E-04 |
4.725E-04 |
8 |
1.474E-04 |
1.504E-04 |
1.725E-04 |
1.611E-04 |
10 |
1.787E-05 |
1.816E-05 |
2.030E-05 |
1.922E-05 |
The functions S, S _{x}, and S _{y} are plotted in Fig. 2.6 and values for the integral of K _{5/3} are given in Table 2.1. It is clear from this graph that the majority of the power is horizontally polarized. We can integrate dP/dω over frequency to find some remarkable results. First if we consider all frequencies then
Having integrated (2.29) over all angles to examine the power emitted as a function of frequency, we will now do the reverse and integrate over frequency to determine how the power varies with angle [14].
The first and second terms in the square brackets again correspond to the horizontal and vertical polarizations, respectively. This function is plotted in (p.26)
2.4.2 Photon Flux
It is quite straightforward to convert the power results from the previous section into a more commonly used result in terms of the number of photons. The energy (p.27)
Table 2.2 Example values for the critical photon energy and angular frequency for three light sources
Ring |
Energy |
γ |
B |
ρ |
ω _{c} |
є _{c} |
λ_{c} |
---|---|---|---|---|---|---|---|
(GeV) |
(T) |
(m) |
(×10^{18}s^{−1}) |
(keV) |
(nm) |
||
SRS |
2 |
3914 |
1.2 |
5.56 |
4.9 |
3.19 |
0.39 |
DIAMOND |
3 |
5871 |
1.4 |
7.15 |
13 |
8.38 |
0.15 |
ESRF |
6 |
11742 |
0.8 |
25.0 |
29 |
19.2 |
0.06 |
If the number of photons emitted per second with energy є is N˙ then the power emitted at that photon energy is simply N˙є. The number of photons emitted per second per solid angle by one electron into a relative photon energy bandwidth Δє/є is
Three examples for the spectral intensity are given in Fig. 2.8, each for a different photon angular frequency. Note that at lower frequency and longer wavelength, the radiation extends further out in the vertical angle and also that the vertical polarization contribution becomes quite significant. Since the vertical electric field is 90° out of phase with the horizontal component (by inspection of (2.26) and (2.27)), the result is circular polarization. We can also see that at frequencies close to ω _{c} the approximation that the radiation is emitted with a (p.29)
We can use the same approach to look at the number of photons emitted per electron per second into all angles (p.30)
Again, multiplying this result by the number of electrons will give us the number of photons emitted per second into all angles for a beam current, I _{b}, into a relative energy bandwidth Δє/є
The parameter N˙ is referred to as the spectral photon flux or the vertically integrated spectral flux, this latter name conveys the message that the photon emission has been summed over all angles (as the electron travels on a circle of 2π radians it is automatically integrated horizontally). Again, in practical units this reduces to
A plot of N˙ versus є/є_{c} is known as the universal curve (Fig. 2.9). Absolute flux levels for a particular electron energy and beam current can be quickly scaled off the curve for any photon energy once the critical photon energy is calculated. It should be clear from this that all bending magnet sources have the same characteristic spectrum. The spectral flux always increases slowly from the low photon energies, peaking at approximately є/є_{c} = 0.25. The spectrum then falls off sharply, with a typical consideration being that the flux is useful up to about є/є_{c} ≍ 5. The low photon energies extend down until the emitted radiation reaches a wavelength of the order of the vacuum chamber dimensions (p.31)
In summary then, the spectral range covered by a bending magnet is fixed by the critical photon energy, which is a function of the electron energy and the bending radius (2.37). The bending radius itself depends upon the electron (p.32) energy and the magnetic field, B
2.4.3 Vertical Opening Angle
We have already seen that different photon energies emit SR over quite different vertical angular distributions. It is useful to be able to estimate what a ‘typical’ angle might be for each particular photon energy. This ‘typical’ angle is the so-called vertical opening angle of the radiation, denoted as σ_{r′}. The most common method for estimating this opening angle is to assume that the angular distribution follows a Gaussian or Normal distribution. This is not a particularly good assumption, especially when є « є_{c} where the vertical polarization component can become quite significant (see Fig. 2.8, for example) so the results should always be treated with some caution.
Let's just remind ourselves of the main features of a Gaussian distribution. First it has a functional form
Now, we are assuming that the vertical angular distribution is of Gaussian form and symmetric about ψ = 0, so
A plot of the output from the above result for σ _{r′} is given in Fig. 2.11 for a 3 GeV electron beam. Note that the vertical opening angle changes by two orders of magnitude between low and high photon energies. The examples that were used for Fig. 2.8 are plotted again in Fig. 2.12 but this time with the superimposed ideal Gaussian distribution using the calculated value of σ _{r′} as well. It is clear from this that the Gaussian assumption is only correct at the highest photon energies, the approximation is already struggling at є = є_{c}. Since this treatment is only an approximation, if the exact vertical angular distribution is of particular interest the distribution dN˙/dΩ should be examined directly.
2.4.4 Bending Magnet Brightness
If we take a perpendicular slice through the SR travelling towards the observer then we will intercept millions of photons. Each of these photons in this slice will have a particular position and angular direction. Some will have a position close to the axis but a relatively large divergence and others will be far from the axis but have a very shallow trajectory. This concept of particles having a position and an angle that evolves with time as they travel towards the observer is known as the phase space. It is an important concept that also crops up in many other areas of physics. In particular, it is often used in accelerator physics to describe the distribution of the charged particles travelling around an accelerator.
The brightness of a source is the phase space density of the photon flux (i.e. the photons per unit solid angle per unit solid area) and it is a figure of merit that takes into account not only the number of photons emitted but also their concentration. It is often encountered in geometric optics where it is widely used because it is a quantity which is invariant in an ideal optical beam transport system (a result from thermodynamics known as Liouville's theorem [20]), unlike angular flux density, for instance.
To calculate the bending magnet brightness we need to consider the effective phase space area from which the photon flux is being emitted taking account of both the finite electron and photon beam sizes and divergences. First, there is no need to consider any horizontal angle effects as the light is emitted smoothly over the full horizontal 2π radians. The effective vertical angle, ∑_{y′}, will be a combination of the electron vertical beam divergence, σ _{y′}, and the photon beam opening angle, σ _{r′}. Since these are both from Gaussian distributions they are added in quadrature
In general, σ _{r} « σ _{x,y} and the brightness equation can be simplified to (p.36)
2.5 Power and Power Density from a Bending Magnet
A considerable amount of power can be generated in the form of synchrotron radiation in a storage ring. This is not only important for the users of the radiation but also those who have to design and operate the accelerator. The power and power density levels are often high enough in a synchrotron light source to cause damage to the accelerator itself. A number of accelerators around the world have melted components inside the vacuum chamber and the vacuum chamber itself in some cases! For this reason all light sources have to water cool many items inside the vacuum system that the synchrotron radiation impinges upon. In some accelerators sophisticated monitoring of the electron beam position is necessary to ensure that it is operating safely and if the beam moves outside of certain prescribed limits it is quickly dumped to prevent any possible thermal damage.
2.5.1 Total Power
The instantaneous total power emitted by a single relativistic electron is [12]
Table 2.3 Example values for the power, power per horizontal angle and power density on-axis for three light sources
Ring |
Energy (GeV) |
ρ (m) |
I _{b} (mA) |
P _{total} (kW) |
dP/dθ (W/mrad) |
dP/dΩ (W/mrad^{2}) |
---|---|---|---|---|---|---|
SRS |
2 |
5.56 |
200 |
50.9 |
8.1 |
20.8 |
DIAMOND |
3 |
7.15 |
300 |
300.7 |
47.9 |
184.4 |
ESRF |
6 |
25.0 |
200 |
916.5 |
145.9 |
1124.0 |
Another useful number to know is the power per horizontal angle. Again, expressed in practical units this is given by
2.5.2 Power Density
The bending magnet power distribution in the vertical plane has already been derived (2.31) and plotted in Fig. 2.7. We can use this result to find the power density. In particular, the power density on-axis (ψ = 0) is given by
2.6 Wavelength Shifters
Wavelength shifters are a type of insertion device, which essentially produces bending magnet style radiation. The advantage that they have over the storage ring bend magnets is that their magnetic field, and so the critical photon energy, can be tailored to a specific beamline requirement. Generally, wavelength shifters are used to shift the spectrum towards the higher photon energy end. In the SRS, for instance, the 6.0 T magnetic field in one of the wavelength shifters produces a critical photon energy five times higher than the 1.2 T bending magnets [21]. This gives a larger flux at the high photon energy end of the spectrum than is normally available (Fig. 2.13). From this plot it becomes apparent why the term wavelength shifter is used since the spectrum literally shifts along the photon energy x-axis. (p.38)
A wavelength shifter usually consists of one high magnetic field central pole surrounded by two weaker side poles of opposite polarity. A typical magnetic field profile along a wavelength shifter and the electron trajectory through such a magnet is shown in Fig. 2.14. The magnetic field strengths are arranged so that the total integrated field strength along the longitudinal axis is equal to zero. This ensures that the overall angular deflection to the electron is zero and that the electron exits the insertion device on the same axis that it entered on.
Since the magnetic field along the length of the wavelength shifter is not constant, the critical photon energy also varies along its length (see Fig. 2.14). This means that although the SR produced has the same characteristics of bending magnet radiation, the exact characteristics observed depend on which part of the electron trajectory the observer is looking at. Furthermore, the observer may simultaneously also see SR produced by the side poles, which may enhance the flux but also give a light source with more than one source point.
A wavelength shifter will typically deflect the electron beam by the order of 10 mm at the peak of the bump, which is of course the optimum source point (highest magnetic field strength). Clearly if the magnetic field strength of the wavelength shifter is altered the size of this deflection will alter, in turn changing the source point position. It is possible to produce a wavelength shifter, which always has the main source point on-axis by the inclusion of two further side poles. Examples of this are the 7 T wavelength shifters at CAMD [23] and BESSY II [24].
2.7 Extension to Multipole Wigglers
We have just seen that a wavelength shifter, which is simply a single, large bump on the electron trajectory can produce synchrotron radiation that is essentially bending magnet radiation. Imagine putting several identical wavelength shifters one after the other in a straight section of a storage ring. The electron would simply travel through each wavelength shifter in turn emitting synchrotron radiation in the forward direction. Each wavelength shifter is independent of the other and the electron returns back to the beam axis after passing through each one. Although there is no fundamental relationship between each wavelength shifter (p.40)
Of course, putting several wavelength shifters in a straight section is not the most efficient way of creating a multiple source. A better arrangement of alternating magnetic fields is shown in Fig. 2.16. In fact a typical multipole wiggler has a magnetic field, which closely resembles a sinusoidal profile. A detailed derivation of the synchrotron radiation emission from a multipole wiggler is given in Chapter 3 and the design of magnets to create the desired magnetic field is covered in Chapter 7 for permanent magnet based solutions and Chapter 8 for electromagnet based solutions.