11 Application of Accelerators and Storage Rings
683
Average brilliance values can exceed values B avg ≈ 10 21 mm −2 mrad −2 s −1
(0.1%) −1 .
In case of a perfectly diffraction limited beam in both x/y directions, it is
concluded from Eqs. (11.49), (11.51), and (11.52) that Eq. (11.53) simplifies to:
B = 4 (dn/dt) /
λ
2 dω/ω
.
(11.54)
Coherent Synchrotron Radiation
In sections “Basic Properties of Synchrotron Radiation” through “Wigglers and
Undulators”, emission by a point charge has been considered. In this case, the
intensity scales with the radiating charge q squared. In other words, if this quasipoint charge consists of N e electrons, the total power would be N 2
e larger than the
power P 0 radiated by a single electron.
In general, if radiation of many electrons is considered, we need to add up
first the electric field vectors
− →
E k of all electrons coherently before calculating
the intensity. In the following we restrict ourselves to the one-dimensional case,
where all electrons follow the same path, but just at different times (“pencil beam”).
In most cases, the transverse extension of the bunch does not alter the results
significantly.
In case of a pencil beam, the electric field contributed by any of the electrons
differs from the field of the others only in terms of the phase factor exp{iω(t − R k /c)}
in Eq. (11.23), since, at time t, the electron with index k is located at a retarded
distance R k while others will be located at a different distance. Calling these
individual phases differences ϕ k , the power radiated by the ensemble scales like
P (ω) ∝
E 2
=
− →
E 1 +
− →
E 2 + . . .
− →
E 1 +
− →
E 2 + . . .
∗
=
N e
k,j
− →
E k
− →
E
∗
j ∝
N e
k,j
e
−i(ϕ k −ϕ j )
∝ N e +
N e
k =j
e
i(ϕ k −ϕ j ) .
(11.55)
How much of phase difference is introduced by a difference in longitudinal
position z k depends, of course, on the wavelength λ considered:
ϕ k − ϕ j =
2π
λ
z k − z j
= k
z k − z j
=
ω
c
z k − z j
.
(11.56)
If the longitudinal positions of electrons are random, the last term in Eq. (11.55)
cancels and the resulting power is given by
P inc (ω) = N e P 0 (ω) ,
(11.57)
683
Average brilliance values can exceed values B avg ≈ 10 21 mm −2 mrad −2 s −1
(0.1%) −1 .
In case of a perfectly diffraction limited beam in both x/y directions, it is
concluded from Eqs. (11.49), (11.51), and (11.52) that Eq. (11.53) simplifies to:
B = 4 (dn/dt) /
λ
2 dω/ω
.
(11.54)
Coherent Synchrotron Radiation
In sections “Basic Properties of Synchrotron Radiation” through “Wigglers and
Undulators”, emission by a point charge has been considered. In this case, the
intensity scales with the radiating charge q squared. In other words, if this quasipoint charge consists of N e electrons, the total power would be N 2
e larger than the
power P 0 radiated by a single electron.
In general, if radiation of many electrons is considered, we need to add up
first the electric field vectors
− →
E k of all electrons coherently before calculating
the intensity. In the following we restrict ourselves to the one-dimensional case,
where all electrons follow the same path, but just at different times (“pencil beam”).
In most cases, the transverse extension of the bunch does not alter the results
significantly.
In case of a pencil beam, the electric field contributed by any of the electrons
differs from the field of the others only in terms of the phase factor exp{iω(t − R k /c)}
in Eq. (11.23), since, at time t, the electron with index k is located at a retarded
distance R k while others will be located at a different distance. Calling these
individual phases differences ϕ k , the power radiated by the ensemble scales like
P (ω) ∝
E 2
=
− →
E 1 +
− →
E 2 + . . .
− →
E 1 +
− →
E 2 + . . .
∗
=
N e
k,j
− →
E k
− →
E
∗
j ∝
N e
k,j
e
−i(ϕ k −ϕ j )
∝ N e +
N e
k =j
e
i(ϕ k −ϕ j ) .
(11.55)
How much of phase difference is introduced by a difference in longitudinal
position z k depends, of course, on the wavelength λ considered:
ϕ k − ϕ j =
2π
λ
z k − z j
= k
z k − z j
=
ω
c
z k − z j
.
(11.56)
If the longitudinal positions of electrons are random, the last term in Eq. (11.55)
cancels and the resulting power is given by
P inc (ω) = N e P 0 (ω) ,
(11.57)