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M. Dohlus et al.
with P 0 the power of a single electron. This contribution is called incoherent
radiation.
If, however, the charge distribution is non-random, we describe the longitudinal
charge distribution by a normalized density function ρ(z). This function determines,
when evaluating the double sum of phase factors in Eq. (11.55), how often each
phase difference value appears, so the double sum can be translated into a Fourier
transform of ρ(z). The expectation value of the power radiated by the ensemble is
then
P (ω) = N e P 0 + N e (N e − 1)
F long (ω)
2 P 0 (ω) .
(11.58)
Here,
F long (ω) =
∞
−∞
ρ(z)e
i
ω
c z dz
(11.59)
is the longitudinal form factor of the charge distribution inside the electron bunch.
The contribution
P coh (ω) = N e (N e − 1)
F long (ω)
2 P 0 (ω) .
(11.60)
to the total radiation power is called coherent synchrotron (or undulator, respectively) radiation. Due to the large number N e of electrons in the bunch, P coh can
exceed P inc by many orders of magnitude, even if the longitudinal charge profile
has only a small Fourier content at the wavelength of interest.
At storage rings, where the charge distribution can be described very well by a
Gaussian, with typical rms bunch length values of a few millimetres, P coh becomes
noticeable only at wavelengths in the far infrared. On the other hand, emission of
radiation is suppressed very effectively by shielding due to the vaccum chamber.
This happens, if in the vacuum chamber, which can be regarded as a curved
waveguide, is no propagating mode whose phase velocity matches with the particle
velocity. A typical parameter is the shielding wavelength 2π
h 3 /R that can be
calculated for a flat chamber of height h and curvature radius R [8]. Thus, coherent
synchrotron radiation is not observed at most storage rings, but it has been provoked
by arranging the storage ring parameters to achieve very short bunches [9].
The dramatic increase of radiation power can also be achieved if the longitudinal
charge density is modulated at the wavelength of interest. This is the physical basics
of the free-electron laser (FEL).
At linear accelerators much shorter bunches can be realized than at storage rings,
and longitudinal bunch profiles may exhibit a very rich internal structure at scales
down to the micrometer range, in particular at high-gain FEL facilities, where
bunch lengths in the few micrometer range are needed, see Sect. 11.1.2. At such
accelerators, infrared spectroscopy of coherent synchrotron radiation (or of optical
transition radiation) represents a powerful tool for electron beam diagnostics [10].
M. Dohlus et al.
with P 0 the power of a single electron. This contribution is called incoherent
radiation.
If, however, the charge distribution is non-random, we describe the longitudinal
charge distribution by a normalized density function ρ(z). This function determines,
when evaluating the double sum of phase factors in Eq. (11.55), how often each
phase difference value appears, so the double sum can be translated into a Fourier
transform of ρ(z). The expectation value of the power radiated by the ensemble is
then
P (ω) = N e P 0 + N e (N e − 1)
F long (ω)
2 P 0 (ω) .
(11.58)
Here,
F long (ω) =
∞
−∞
ρ(z)e
i
ω
c z dz
(11.59)
is the longitudinal form factor of the charge distribution inside the electron bunch.
The contribution
P coh (ω) = N e (N e − 1)
F long (ω)
2 P 0 (ω) .
(11.60)
to the total radiation power is called coherent synchrotron (or undulator, respectively) radiation. Due to the large number N e of electrons in the bunch, P coh can
exceed P inc by many orders of magnitude, even if the longitudinal charge profile
has only a small Fourier content at the wavelength of interest.
At storage rings, where the charge distribution can be described very well by a
Gaussian, with typical rms bunch length values of a few millimetres, P coh becomes
noticeable only at wavelengths in the far infrared. On the other hand, emission of
radiation is suppressed very effectively by shielding due to the vaccum chamber.
This happens, if in the vacuum chamber, which can be regarded as a curved
waveguide, is no propagating mode whose phase velocity matches with the particle
velocity. A typical parameter is the shielding wavelength 2π
h 3 /R that can be
calculated for a flat chamber of height h and curvature radius R [8]. Thus, coherent
synchrotron radiation is not observed at most storage rings, but it has been provoked
by arranging the storage ring parameters to achieve very short bunches [9].
The dramatic increase of radiation power can also be achieved if the longitudinal
charge density is modulated at the wavelength of interest. This is the physical basics
of the free-electron laser (FEL).
At linear accelerators much shorter bunches can be realized than at storage rings,
and longitudinal bunch profiles may exhibit a very rich internal structure at scales
down to the micrometer range, in particular at high-gain FEL facilities, where
bunch lengths in the few micrometer range are needed, see Sect. 11.1.2. At such
accelerators, infrared spectroscopy of coherent synchrotron radiation (or of optical
transition radiation) represents a powerful tool for electron beam diagnostics [10].