11 Application of Accelerators and Storage Rings
681
Gaussian optical mode, the minimum-emittance is achieved, given by [4, 7]
ε γ = σ
γ σ γ =
λ
4π
.
(11.49)
with the wavelength of radiation λ.
As the emittance of the photon beam can never be smaller than indicated by Eq.
(11.49), the very narrow opening angle of undulator radiation, Eq. (11.46), means
that photons radiated from an undulator have a rather large apparent source size.
If a radiating electron beam has an emittance ε e smaller than given by Eq.
(11.49), the resulting radiation will be indistinguishable from radiation of a point
source, if the opening angle σ
e of the electron beam is matched to the opening
angle of the radiation. A crude estimate for σ
e is σ
e ≈
√
ε e /β, where β is the
magnet optics beta function. Thus, a comparison with Eq. (11.46) would result in
an estimated matched beta function ≈L u /4π. However, such a small value cannot be
kept constant within the undulator length if there are no further magnetic focusing
elements. A more appropriate value is thus somewhat larger [4]:
β match ≈
L u
2
.
(11.50)
It should be noted that these considerations apply for both transverse directions
x/y independently.
An electron storage ring providing an electron emittance
ε e ≤
λ
4π
(11.51)
in both x and y is said to work at the diffraction limit. Designing electron storage
rings operating at (or close to) the diffraction limit is a difficult task, in particular
if the wavelengths of interest are very short, e.g. in the hard X-ray regime. In
tendency, this requires large ring diameters and sophisticated electron beam optics
arrangements. Quite some efforts in this direction are under way at several big
laboratories at the time of writing the present article.
In practice, Eq. (11.51) can often not be fulfilled. In this case, since both the
photon and electron distributions are Gaussian in good approximation, the effective,
combined distribution is given by a convolution and results in
2
≈ σ
2
γ + σ
2
e .
(11.52)
This holds for both, transverse dimensions and opening angles, and applies again
for both transverse directions x/y independently.
Many experiments using synchrotron (or undulator) radiation rely on the possibility to focus a small spectral fraction Δω/ω of the photon beam, selected e.g. by
a monochromator, onto a small spot at the experiment. The figure of merit for such
681
Gaussian optical mode, the minimum-emittance is achieved, given by [4, 7]
ε γ = σ
γ σ γ =
λ
4π
.
(11.49)
with the wavelength of radiation λ.
As the emittance of the photon beam can never be smaller than indicated by Eq.
(11.49), the very narrow opening angle of undulator radiation, Eq. (11.46), means
that photons radiated from an undulator have a rather large apparent source size.
If a radiating electron beam has an emittance ε e smaller than given by Eq.
(11.49), the resulting radiation will be indistinguishable from radiation of a point
source, if the opening angle σ
e of the electron beam is matched to the opening
angle of the radiation. A crude estimate for σ
e is σ
e ≈
√
ε e /β, where β is the
magnet optics beta function. Thus, a comparison with Eq. (11.46) would result in
an estimated matched beta function ≈L u /4π. However, such a small value cannot be
kept constant within the undulator length if there are no further magnetic focusing
elements. A more appropriate value is thus somewhat larger [4]:
β match ≈
L u
2
.
(11.50)
It should be noted that these considerations apply for both transverse directions
x/y independently.
An electron storage ring providing an electron emittance
ε e ≤
λ
4π
(11.51)
in both x and y is said to work at the diffraction limit. Designing electron storage
rings operating at (or close to) the diffraction limit is a difficult task, in particular
if the wavelengths of interest are very short, e.g. in the hard X-ray regime. In
tendency, this requires large ring diameters and sophisticated electron beam optics
arrangements. Quite some efforts in this direction are under way at several big
laboratories at the time of writing the present article.
In practice, Eq. (11.51) can often not be fulfilled. In this case, since both the
photon and electron distributions are Gaussian in good approximation, the effective,
combined distribution is given by a convolution and results in
2
≈ σ
2
γ + σ
2
e .
(11.52)
This holds for both, transverse dimensions and opening angles, and applies again
for both transverse directions x/y independently.
Many experiments using synchrotron (or undulator) radiation rely on the possibility to focus a small spectral fraction Δω/ω of the photon beam, selected e.g. by
a monochromator, onto a small spot at the experiment. The figure of merit for such