advanced beam manipulation, cooling, damping and stability 209
grows with T . To stabilize the beam/light at the focus, a feedback would usually need to be applied, which can be described by the characteristic function of the feedback F(ω). In
this case the variance of the relative misalignment of the output beam/light is a constant, and is given by
+∞
dω dk
σ
2 = P(ω, k) F(ω) G(k)
(10.28)
2π 2π
−∞
Examples of a 2D power spectrum as well as a spectral
response function and characteristic of feedback for a particular beamline (final focus system beamline) are shown in
Fig. 10.30.
A particularly useful and insightful approximation for a
2D spectrum is the one related to the ATL law, which was first
suggested 9 to describe space–time motion of ground (earth),
on which the focusing elements of a beamline are installed.
For this “ATL law,” the variance of the misalignment for
elements separated by distance L and observed after time T
(described above by Eq. 10.26) is simply
σ
2 (T ,L) = A · T · L
(10.29)
where A is the parameter of the model that could be estimated, for example, from ground motion measurements of
our site (where the elements of beamline are installed) or
perhaps of our optical table (where the laser elements are
mounted).
The 2D power spectrum corresponding to the “ATL law”
is given by
A
P(ω, k) =
(10.30)
ω 2 k 2
The 2D spectrum shown in Fig. 10.30 includes the ATL component; however, it is included in a corrected way, at higher
frequencies and wavenumbers, to ensure integrability of the
equations.
The term “law” in “ATL law” should be taken cautiously
here, as this is certainly a model, an estimate that is applicable in certain ranges of parameters and certain situations.
It is, however, amazing how many different situations and
wide parameter ranges 10 there are in which this approximation can be found valid.
10.4 Beam or pulse addition
The topic of beam or pulse addition is one where there is
an important difference between the area of accelerators —
9 B. A. Baklakov et al., INP Report 91–15, 1991.
10 V. Shiltsev, Phys. Rev. Lett. 104, 238501 (2010).
grows with T . To stabilize the beam/light at the focus, a feedback would usually need to be applied, which can be described by the characteristic function of the feedback F(ω). In
this case the variance of the relative misalignment of the output beam/light is a constant, and is given by
+∞
dω dk
σ
2 = P(ω, k) F(ω) G(k)
(10.28)
2π 2π
−∞
Examples of a 2D power spectrum as well as a spectral
response function and characteristic of feedback for a particular beamline (final focus system beamline) are shown in
Fig. 10.30.
A particularly useful and insightful approximation for a
2D spectrum is the one related to the ATL law, which was first
suggested 9 to describe space–time motion of ground (earth),
on which the focusing elements of a beamline are installed.
For this “ATL law,” the variance of the misalignment for
elements separated by distance L and observed after time T
(described above by Eq. 10.26) is simply
σ
2 (T ,L) = A · T · L
(10.29)
where A is the parameter of the model that could be estimated, for example, from ground motion measurements of
our site (where the elements of beamline are installed) or
perhaps of our optical table (where the laser elements are
mounted).
The 2D power spectrum corresponding to the “ATL law”
is given by
A
P(ω, k) =
(10.30)
ω 2 k 2
The 2D spectrum shown in Fig. 10.30 includes the ATL component; however, it is included in a corrected way, at higher
frequencies and wavenumbers, to ensure integrability of the
equations.
The term “law” in “ATL law” should be taken cautiously
here, as this is certainly a model, an estimate that is applicable in certain ranges of parameters and certain situations.
It is, however, amazing how many different situations and
wide parameter ranges 10 there are in which this approximation can be found valid.
10.4 Beam or pulse addition
The topic of beam or pulse addition is one where there is
an important difference between the area of accelerators —
9 B. A. Baklakov et al., INP Report 91–15, 1991.
10 V. Shiltsev, Phys. Rev. Lett. 104, 238501 (2010).