1.6 Thesis Layout
2.1.1 Synchrotron Radiation
Synchrotron radiation is the electromagnetic radiation emitted by relativistic charged particles when they undergo acceleration. Such radiation was first observed from a General Electric synchrotron in 1947 [28]. Synchrotron radiation is produced whenever a relativistic charged particle is bent in a magnetic field, and can cover a range of the electromagnetic spectrum from infrared, through visible light and ultraviolet light to x-rays.
As a consequence of synchrotron radiation, relativistic charged particles in a mag- netic field will lose energy. As accelerator technology has developed, the ability to add energy to charged particles has improved. Synchrotron radiation can be a severe lim- itation for high energy electron accelerators [29]. The power lost through synchrotron radiation varies as:
∆E∝ 1 ρ2m4
0
, (2.1)
wherem0 is the rest mass of the particle, andρis the radius of curvature of the path of the particle in the magnetic field. For a circular machine accelerating electrons, large amounts of energy are radiated and wasted by hitting the beam pipe; this energy needs to be restored to the particles in order to maintain the beam energy. However, with the help of insertion devices such as wigglers and undulators, electron storage rings can be used to produce intense beams of synchrotron radiation that can be used widely in many fields of science, including chemistry and medicine. Since the mass of the proton is much larger than that of the electron, the synchrotron radiation produced by protons is generally negligible.
To understand synchrotron radiation, two key effects are Lorentz contraction and relativistic Doppler shift [30]. For example, consider an electron that travels through an undulator (period λu) at (close to) the speed of light. In the moving frame, the
period of the undulator will be contracted by a factor of γ, and so the electron will emit the radiation with wavelength λu
γ. For the relativistic Doppler shift, in the case of
the particle travelling with the speed of light towards the observer, the frequency will change to:
f =γf′(1 +β), (2.2)
where the observer will see radiation at a frequency f, the source (electron) emits radiation at frequency f′ in its own rest frame, and β = v
c where v is the velocity of
the electron. If we convert the frequency of the radiation to the wavelength, we get:
λ= λ′
γ(1 +β) ≈
λ′
2γ. (2.3)
We see that the wavelength of radiation observed in the rest frame of the undulator isλu/2γ2. Modern accelerators readily achieve energies of a few GeV; for electrons, the
relativistic factorγ can be a few thousands. For an undulator with a period of order 0.1 m, the synchrotron radiation can have a wavelength of a few nanometres. A further important property of synchrotron radiation, is that the radiation is emitted into a narrow cone of opening angle 1/γ around the instantaneous direction of motion of the particle. For an undulator, interference effects lead to a further narrowing of the cone by a factor√N, whereN is the number of periods in the undulator. This means that undulators can be used in high energy accelerators to produce beams of very intense radiation with wavelengths of a few nanometres.
Understanding the properties of undulator radiation is essential for optimising the design of an undulator-based positron source. In the following sections, we will consider the properties of synchrotron radiation from an undulator in more detail. Starting from the motion of a charged particle in a magnetic field, we will use Maxwell’s equations to find the fields produced, and we will then derive equations for the power emitted and the polarisation. The power spectrum and angular distribution of the radiation are also important properties that we will derive.
2.1.2 Electric and Magnetic Fields Around a Relativistic Charged Particle
The electromagnetic potentialsφandA~ around a charged, point-like particle are given by the Li´enard-Wiechert potentials [31]:
φ(t) = e 4πǫ0 " 1 r(1−~n·β~) # ret , (2.4) ~ A(t) = e 4πǫ0c " ~ β r(1−~n·β~) # ret , (2.5)
whereeis the charge on the particle,r is the distance from the position of the particle to the observation point, ~n is a unit vector from the position of the particle to the observation point, and β~ is the velocity of the particle divided by the speed of light. Note that the quantities inside the brackets [·]ret must be evaluated at a timet′, where:
t=t′+r(t′)
c , (2.6)
in order to find the correct values for the potentials at timet. For a relativistic particle moving directly towards the observer (~n ·β~ ≈ 1), there is an enhancement of the electromagnetic potentials; while the potentials are reduced for a particle moving away from the observer.
The electric and magnetic fields may be obtained from the potentials using the usual relations: ~ B = ∇ ×A,~ (2.7) ~ E = −∇φ−∂ ~A ∂t. (2.8)
For a particle on an arbitrary trajectory, application of the derivatives to Eqs. (2.4) and (2.5) is complicated, because r,~n andβ~ are all functions of time; andr and~n are additionally functions of position (of the observer). However, it is possible to perform the derivatives. The result for the electric field is [17]:
~ E = e 4πǫ0 " (1−β2)(~n−β~) r2(1−~n·β~)3 + ~n×((~n−β~)×β~˙) cr(1−~n·β~)3 # ret , (2.9)
and for the magnetic field:
~ B = 1
Note that the second term on the right hand side of Eq. (2.9) depends on the acceleration of the particle β~˙. Also, this term varies with distance from the source as ∼1/r, whereas the first term varies as ∼ r2. This means that for an accelerating particle, at a sufficient distance from the particle, the fields are dominated by the second term in Eq. (2.9). The region where this second term does dominate the fields is known as thefar field orradiation region, and is the region we shall be concerned with.
It turns out to be convenient to work with the frequency spectrum of the fields, rather than the fields expressed as functions of time. The frequency spectrum of the electric field is given by the Fourier transform ofE~(t):
˜ E(ω) = √1 2π Z ∞ −∞ ~ E(t)eiωtdt. (2.11) Working in the radiation region, we can substitute for E~(t) from Eq. (2.9), and at the same time change the variable of integration fromt tot′ using Eq. (2.6). This gives:
˜ E(ω) = e 4π√2πǫ0 Z ∞ −∞ ~n×((~n−β~)×β~˙) cr(1−~n·β~)2 e iω(t′+r c)dt′. (2.12)
Let us assume that the observer is sufficiently far from the particle that~n is constant: this is consistent with working in the radiation region. Then, we can integrate Eq. (2.12) by parts to give: ˜ E(ω) = iωe 4π√2πǫ0cr Z ∞ −∞ (~n×(~n×β~))eiω(t′+rc)dt′. (2.13)