Particle Accelerator Physics pp 895-927 | Cite as
Insertion Device Radiation
Abstract
Synchrotron radiation from bending magnets is characterized by a wide spectrum from microwaves up to soft or hard x-rays as determined by the critical photon energy. To optimally meet the needs of basic research with synchrotron radiation, it is desirable to provide radiation characteristics that cannot be obtained from ring bending magnets but require special magnets. The field strength of bending magnets and the maximum particle beam energy in circular accelerators like a storage ring is fixed leaving no adjustments to optimize the synchrotron radiation spectrum for particular experiments.
Keywords
Storage Ring Elliptical Polarization Harmonic Radiation Insertion Device Undulator Period
In Chap. 24 the properties of wiggler radiation were discussed shortly in an introductory way. Here we concentrate on more detailed and formal derivations of radiation characteristics from relativistic electrons passing through periodic magnets.
There is no fundamental difference between wiggler and undulator radiation. One is the stronger/weaker version of the other. The deflection in an undulator is weak and the transverse particle momentum remains nonrelativistic. The motion is purely sinusoidal in a sinusoidal field, and the emitted radiation is monochromatic at the particle oscillation frequency which is the Lorentz-contracted periodicity of the undulator period. Since the radiation is emitted from a moving source the observer in the laboratory frame of reference then sees a Doppler shifted frequency. We call this monochromatic radiation the fundamental radiation or radiation at the fundamental frequency of the undulator.
As the undulator field is increased, the transverse motion becomes stronger and the transverse momentum starts to become relativistic. As a consequence, the so far purely sinusoidal motion becomes periodically distorted causing the appearance of harmonics of the fundamental monochromatic radiation. These harmonics increase in number and density with further increase of the magnetic field and, at higher frequencies, eventually merge into one broad spectrum characteristic for wiggler or bending magnet radiation. At very low frequencies, the theoretical spectrum is still a line spectrum showing the harmonics of the revolution frequency. Of course, there is a low frequency cut-off at a wavelength comparable or longer than vacuum chamber dimensions which therefore do not show-up as radiation.
An insertion device does not introduce a net deflection of the beam and we may therefore choose any arbitrary field strength which is technically feasible to adjust the radiation spectrum to experimental needs. The radiation intensity from a wiggler magnet also can be made much higher compared to that from a single bending magnet. A wiggler magnet with say ten poles acts like a string of ten bending magnets or radiation sources aligned in a straight line along the photon beam direction. The effective photon source is therefore ten times more intense than the radiation from a single bending magnet with the same field strength.
Wiggler magnets come in a variety of types with the flat wiggler magnet being the most common. In this wiggler type only the component B_{y} is nonzero deflecting the beam in the horizontal plane. To generate circularly or elliptically polarized radiation, a helical wiggler magnet [2] may be used or a combination of several flat wiggler magnets deflecting the beam in orthogonal planes which will be discussed in more detail in Sect. 26.3.2.
26.1 Particle Dynamics in a Periodic Field Magnet
Equations (26.2) describe the coupled motion of a particle in the sinusoidal field of a flat wiggler magnet. This coupling is common to the particle motion in any magnetic field but generally in beam dynamics we set \(\mathrm{d}z/\mathrm{d}t \approx v\) and \(\mathrm{d}x/\mathrm{d}t \approx 0\) because \(\mathrm{d}x/\mathrm{d}t \ll \mathrm{ d}z/\mathrm{d}t\). This approximation is justified in most beam transport applications for relativistic particles, but here we have to be cautious not to neglect effects that might be of relevance on a very short time or small geometric scale comparable to the oscillation period and wavelength of synchrotron radiation.
26.2 Undulator Radiation
The physical process of undulator radiation is not different from the radiation produced from a single bending magnet. However, the radiation received at great distances from the undulator exhibits special features which we will discus in more detail. Basically, we observe an electron performing N_{p} oscillations while passing through an undulator with N_{p} undulator periods. The observed radiation spectrum is the Fourier transform of the electron motion and therefore quasi-monochromatic with a finite line width inversely proportional to the number of oscillations performed.
26.2.1 Fundamental Wavelength
From an infinitely long undulator, the radiation spectrum consists of spectral lines at a wavelength determined by (26.11). In particular, we note that the shortest wavelength is emitted into the forward direction while the radiation at a finite angle \(\vartheta\) appears red shifted by the Doppler effect. For an undulator with a finite number of periods, the spectral lines are widened to a width of about \(1/N_{\text{p}}\) or less as we will discuss in the next section.
26.2.2 Radiation Power
26.2.3 Spatial and Spectral Distribution
The value of this integral reaches a maximum of \(2\pi \frac{N_{\text{p}}} {\omega _{\mathrm{p}}}\) for \(\omega \rightarrow 0\). From (26.34) we note the coefficient of this integral to include the angle \(\vartheta \gtrsim 1/\gamma\) and the whole integral is therefore of the order or less than \(L_{\mathrm{u}}/(c\gamma ),\) where \(L_{\mathrm{u}} = N_{\text{p}}\lambda _{\mathrm{p}}\) is the total length of the wiggler magnet. This value is in general very small compared to the first integral and can therefore be neglected. Actually, this statement is only partially true since the first integral, as we will see, is a fast varying function of the radiation frequency with a distinct line spectrum. Being, however, primarily interested in the peak intensities of the spectrum we may indeed neglect the third integral. Only between the spectral lines does the radiation intensity from the first integral become so small that the third integral would be a relatively significant although absolutely a small contribution.
The spectral width of the undulator radiation is reduced proportional to the number of undulator periods, but reduces also proportional to the harmonic number.
The Bessel functions J_{m}(u) determine mainly the intensity of the line spectrum. For an undulator with \(K \ll 1\), the argument \(u \propto K^{2} \ll 1\) and the contributions of higher order Bessel’s functions are very small. The radiation spectrum consists therefore only of the fundamental line. For stronger undulators with K > 1, higher order Bessel’s functions grow and higher harmonic radiation appears in the line spectrum of the radiation.
where α is the fine structure constant and where we have kept the coordinate unit vectors to keep track of the polarization modes. The vectors x and y are orthogonal unit vectors indicating the directions of the electric field or the polarization of the radiation. Performing the squares does therefore not produce cross terms and the two terms in (26.54) with the expressions (26.53) represent the amplitude factors for both polarization directions, the σ-mode and π-mode respectively.
realizing that the photon spectrum is determined by the \((\sin x/x)^{2}\)-function. For not too few periods, this function is very small for frequencies away from the resonance conditions.
Even for an extremely small pin hole, we would observe a similar spectrum as shown in Fig. 26.5 because of the finite beam divergence of the electron beam. The electrons follow oscillatory trajectories due not only to the undulator field but also due to betatron oscillations. We observe therefore always some radiation at a finite angle given by the particle trajectory with respect to the undulator axis. Figure 26.5 also demonstrates the fact that all experimental circumstances must be included to meet theoretical expectations. The amplitudes of the measured low energy spectrum is significantly suppressed compared to theoretical expectations which is due to a Be-window being used to extract the radiation from the ultra high vacuum chamber of the accelerator. This material absorbs radiation significantly below a photon energy of about 3 keV.
While we observe a line spectrum expressed by the \((\sin x/x)^{2}\)-function, we also notice that this line spectrum is red shifted as we increase the observation angle \(\vartheta\). Only, when we observe the radiation though a very small aperture (pin hole) do we actually see this line spectrum. Viewing the undulator radiation through a large aperture integrates the linespectra over a finite range of angles \(\vartheta\) producing an almost continuous spectrum with small spikes at the locations of the harmonic lines.
We note clearly the strong forward lobe at the fundamental frequency in σ-mode while there is no emission in π-mode along the path of the particle. The second harmonic radiation vanishes in the forward direction, an observation that is true for all even harmonics. By inspection of (26.54), we note that v = 0 for \(\vartheta = 0\) and the square bracket in (26.53b) vanishes for all odd indices or for all even harmonics k. There is therefore no forward radiation for even harmonics of the fundamental undulator frequency.
26.2.4 Line Spectrum
To exhibit other important and desirable features of the radiation spectrum (26.54), we ignore the actual frequency distribution in the vicinity of the harmonics and set \(\varDelta \omega _{k} = 0\) because the spectral lines are narrow for large numbers of wiggler periods N_{p}. Further, we are interested for now only in the forward radiation where \(\vartheta = 0\) keeping in mind that the radiation is mostly emitted into a small angle \(\langle \vartheta \rangle = 1/\gamma\).
Amplitudes A_{k}(K) for k = 1, 3, 5, 7, 9, 11
K | A_{1} | A_{3} | A_{5} | A_{7} | A_{9} | A_{11} | |
---|---|---|---|---|---|---|---|
0.1 | 0.010 | 0 | 0 | 0 | 0 | 0 | |
0.2 | 0.038 | 0 | 0 | 0 | 0 | 0 | |
0.4 | 0.132 | 0.004 | 0 | 0 | 0 | 0 | |
0.6 | 0.238 | 0.027 | 0.002 | 0 | 0 | 0 | |
0.8 | 0.322 | 0.087 | 0.015 | 0.002 | 0 | 0 | |
1.0 | 0.368 | 0.179 | 0.055 | 0.015 | 0.004 | 0.001 | |
1.2 | 0.381 | 0.276 | 0.128 | 0.051 | 0.019 | 0.007 | |
1.4 | 0.371 | 0.354 | 0.219 | 0.118 | 0.059 | 0.028 | |
1.8 | 0.320 | 0.423 | 0.371 | 0.286 | 0.206 | 0.142 | |
2.0 | 0.290 | 0.423 | 0.413 | 0.354 | 0.285 | 0.220 | |
5.0 | 0.071 | 0.139 | 0.188 | 0.228 | 0.261 | 0.290 | |
10.0 | 0.019 | 0.037 | 0.051 | 0.064 | 0.075 | 0.085 | |
20.0 | 0.005 | 0.010 | 0.013 | 0.016 | 0.019 | 0.022 |
26.2.5 Spectral Undulator Brightness
Similar to Chap. 27 we define the spectral brightness of undulator radiation as the photon density in six-dimensional phase space. The actual photon brightness is reduced from the diffraction limit due to betatron motion of the particles, transverse beam oscillation in the undulator, apparent source size on axis and under an oblique angle. All of these effects tend to increase the source size and reduce brightness.
Similarly, due to an oblique observation angle \(\vartheta\) with respect to the (y, z)-plane or ψ with respect to the (x, z)-plane we get a further additive contribution \(\frac{1} {6}\vartheta L\) to the apparent beam size. Finally, the apparent source size is widened by the transverse beam wiggle in the periodic undulator field. This oscillation amplitude is from (26.6) \(a =\lambda _{\mathrm{p}}K/(2\pi \gamma )\).
26.3 Elliptical Polarization
During the discussion of bending magnet radiation in Chap. 25 and insertion radiation in this chapter, we noticed the appearance of two orthogonal components of the radiation field which we identified with the σ-mode and π-mode polarization. The π-mode radiation is observable only at a finite angle with the plane defined by the particle trajectory and the acceleration force vector, which is in general the horizontal plane. As we will see, both polarization modes can, under certain circumstances, be out of phase giving rise to elliptical polarization. In this section, we will shortly discuss such conditions.
26.3.1 Elliptical Polarization from Bending Magnet Radiation
The direction of the electric component of the radiation field is parallel to the particle acceleration. Since radiation is the perturbation of electric field lines from the charge at the retarded time to the observer, we must take into account all apparent acceleration. To see this more clear, we assume an electron to travel counter clockwise on an orbit travelling from say a 12-o’clock position to 9-o’clock and then 6-o’clock. Watching the particle in the plane of deflection, the midplane, we notice only a horizontal acceleration which is maximum at 9-o’clock. Radiation observed in the midplane is therefore linearly polarized in the plane of deflection.
Now we observe the same electron at a small angle above the midplane. Apart from the horizontal motion, we notice now also an apparent vertical motion. Since the electron follows pieces of a circle this vertical motion is not uniform but exhibits acceleration. Specifically, at 12-o’clock the particle seems to be accelerated only in the vertical direction (downward), horizontally it is in uniform motion; at 9-o’clock the acceleration is only horizontal (towards 3-o’clock) and the vertical motion is uniform; finally, at 6-o’clock the electron is accelerated only in the vertical plane again (upward). Because light travels faster than the electron, we observe radiation first coming from the 12-o’clock position, then from 9-o’clock and finally from 6-o’clock. The polarization of this radiation pulse changes from downward to horizontal (left-right) to upward which is what we call elliptical polarization where the polarization vector rotates with time. Of course, in reality we do not observe radiation from half the orbit, but only from a very short arc segment of angle ± 1∕γ. However, if we consider Lorentz contraction the 9-o’clock trajectory in the particle system looks very close to a half circle radiation into ± 180 degrees which appears in the laboratory system within ± 1∕γ. Therefore the short piece of arc from which we observe the radiation has all the features just used to explain elliptical polarization in a bending magnet.
The elliptical polarization is left or right handed depending on whether we observe the radiation from above or below the horizontal mid plane. Furthermore, the helicity depends on the direction of deflection in the bending magnet or the sign of the curvature sign(1∕ρ). By changing the sign of the bending magnet field the helicity of the elliptical polarization can be reversed. This is of no importance for radiation from a bending magnet since we cannot change the field without loss of the particle beam but is of specific importance for elliptical polarization state of radiation from wiggler and undulator magnets.
26.3.2 Elliptical Polarization from Periodic Insertion Devices
Asymmetric Wiggler Magnet
The degree of polarization from an asymmetric wiggler depends on the desired photon energy. The critical photon energy is high for radiation from the high field pole \(\left (\epsilon _{\mathrm{c}}^{+}\right )\) and lower for radiation from the low field pole \(\left (\epsilon _{\mathrm{c}}^{-}\right )\). For high photon energies \(\left (\epsilon _{\mathrm{ph}} \approx \epsilon _{\mathrm{c}}^{+}\right )\) the radiation from the low field poles is negligible and the radiation is essentially the same as from a series of bending magnets with its particular polarization characteristics. For lower photon energies \(\left (\epsilon _{\mathrm{c}}^{-} <\epsilon _{\mathrm{ph}} <\epsilon _{ \mathrm{c}}^{+}\right )\) the radiation intensity from high and low field pole become similar and cancellation of the elliptical polarization occurs. At low photon energies \(\left (\epsilon _{\mathrm{ph}} <\epsilon _{ \mathrm{c}}^{-}\right )\) the intensity from the low field poles exceeds that from the high field poles and we observe again elliptical polarization although with reversed helicity.
Elliptically Polarizing Undulator
26.4 Problems
26.1 (S). Consider an undulator magnet with a period length of λ_{p} = 5 cm in a 7 GeV storage ring. The strength parameter be K = 1. What is the maximum oscillation amplitude of an electron passing through this undulator? What is the maximum longitudinal oscillation amplitude with respect to the reference system moving with velocity \(\bar{\beta }\)?
26.2 (S). An undulator with 50 poles, a period length of λ_{p} = 5 cm and a strength parameter of K = 1 is to be installed into a 1 GeV storage ring. Calculate the focal length of the undulator magnet. Does the installation of this undulator require compensation of its focusing properties? How about a wiggler magnet with K = 5?
26.3 (S). Consider the expression (26.67) for the photon flux into the forward cone. We also know that the band width of undulator radiation scales like \(\varDelta \omega /\omega \propto 1/N_{\text{p}}\). With this, the photon flux (26.67) becomes independent of the number of undulator periods!? Explain in words, why this expression for the photon flux is indeed a correct scaling law.
26.4 (S). A hybrid undulator is to be installed into a 7 GeV storage ring to produce undulator radiation in a photon energy range of 4 keV to 15 keV. The maximum undulator field shall not exceed a value of B_{0} ≤ 2 T at a gap aperture of 10 mm. The available photon flux in the forward cone shall be at least 10 % of the maximum flux within the whole spectral range. Specify the undulator parameters and show that the required photon energy range can be covered by changing the magnet gap only.
26.5 (S). Consider an electron colliding head-on with a laser beam. What is the wavelength of the laser as seen from the electron system. Derive from this the wavelength of the “undulator“ radiation in the laboratory system.
26.6 (S). An electron of energy 2 GeV performs transverse oscillations in a wiggler magnet of strength K = 1. 5 and period length λ_{p} = 7. 5 cm. Calculate the maximum transverse oscillation amplitude. What is the maximum transverse velocity in units of c during those oscillations. Define and calculate a transverse relativistic factor γ_{ ⊥ }. Note, that for \(K \gtrsim 1\) the transverse relativistic effect becomes significant in the generation of harmonic radiation.
26.7 (S). Calculate for a 3 GeV electron beam the fundamental photon energy for a 100 period-undulator with K = 1 and a period length of λ_{p} = 5 cm. What is the maximum angular acceptance angle \(\vartheta\) (as determined by adjustable slits) of the beam line, if the radiation spectrum is to be restricted to a bandwidth of 10 %?
26.8 (S). Strong mechanical forces exist between the magnetic poles of an undulator when energized. Are these forces attracting or repelling the poles? Why? Consider a \(\ell=\) 1 m long undulator with a pole width w = 0. 1 m, 15 periods each λ_{p} = 7 cm long and a maximum field of B_{0} = 1. 5 T. Estimate the total force between the two magnet poles?
26.9 (S). In Chap. 23 we mentioned undulator radiation as a result of Compton scattering of the undulator field by electrons. Derive the fundamental undulator wavelength from the process of Compton scattering.
26.10 (S). The undulator radiation intensity is a function of the strength parameter K. Find the strength parameter K for which the fundamental radiation intensity is a maximum. Determine the range of K-values for which the intensity of the fundamental radiation is within 10 % of the maximum.
26.11 (S). Show from (26.54) that along the axis \(\left (\vartheta = 0\right )\) radiation is emitted only in odd harmonics.
26.12 (S). Show from (26.51) that undulator radiation does not produce elliptically polarized radiation in the forward direction \(\left (\vartheta = 0\right )\).
26.13 (S). Try to design a hybrid undulator for a 3 GeV storage ring to produce 4 keV to 15 keV photon radiation. Is it possible? Why not? Optimize the undulator parameters such that this photon energy range can be covered with the highest flux possible and utilizing lower order harmonics (order 7 or less). Plot the radiation spectrum that can be covered by changing the gap height of the undulator.
26.14 (S). An undulator is constructed from hybrid permanent magnet material with a period length of \(\lambda _{\text{,p}} = 5.0\) cm. What is the fundamental wavelength range in a 800 MeV storage ring and in a 7 GeV storage ring if the undulator gap is to be at least 10 mm?
26.15 (S). Determine the tuning range for a hybrid magnet undulator in a 2.5 GeV storage ring with an adjustable gap g ≥ 10 mm. Plot the fundamental wavelength as a function of magnet gap for two different period lengths, λ_{,p} = 15 mm and λ_{,p} = 75 mm. Why are the tuning ranges so different?
26.16. Consider a 26-pole wiggler magnet with a field \(B_{y}\left (\text{T}\right ) = 1.5\sin \left ( \frac{2\pi } {\lambda _{\text{,p}}}z\right )\) and a period length of λ_{,p} = 15 cm as the radiation source for a straight through photon beam line and two side stations at an angle \(\vartheta = 4\) mr and \(\vartheta = 8\) mr in a storage ring with a beam energy of 2.0 GeV. What is the critical photon energy of the photon beam in the straight ahead beam line and in the two side stations?
26.17. Verify the relative intensities of σ-mode and π-mode radiation in Fig.26.12 for two quantitatively different pairs of observation angles \(\vartheta\) and photon energies \(\varepsilon /\varepsilon _{\text{c}}\).
26.18. Design an asymmetric wiggler magnet assuming hard edge fields and optimized for the production of elliptical polarized radiation at a photon energy of your choice. Calculate and plot the photon flux of polarized radiation in the vicinity of the optimum photon energy.
26.19. Calculate the total undulator \(\left (N_{\text{p}} = 50,\lambda _{\text{p}} = 4.5\text{cm, }K = 1.0\right )\) radiation power from a 200 mA, 6 GeV electron beam. Pessimistically, assume all radiation to come from a point source and be contained within the central cone. This is a safe assumption for the design of the vacuum chamber or mask absorbers. Determine the power density at a distance of 15 m from the source. Compare this power density with the maximum acceptable of 10 W/mm^{2}. How can you reduce the power density, on say a mask, to the acceptable value or below?
26.20. Use the beam and undulator from problem 26.19 and estimate the total radiation power into the forward cone alone. What percentage of all radiation falls within the forward cone? [hint: make reasonable approximations to simplify the math but keep the result reasonably close to the correct answer].
26.21. Derive an expression for the average velocity component \(\bar{\beta }=\bar{ v}/c\) of a particle traveling through an undulator magnet of strength K.
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