11 Application of Accelerators and Storage Rings
677
It should be noted that, in order this periodic trajectory to happen, the electron
beam must enter the device on an orbit representing the appropriate initial conditions. In addition, the undulator field must begin and end with a quarter-period
undulator section.
From Eq. (11.37) the maximum deflection angle can be calculated:
ϑ max ≈
dx
dz
max
= K·
1
γβ
≈
K
γ
.
(11.38)
The maximum orbit excursion of the electron is
x max ∼ =
K
γ k u
.
(11.39)
Under many typical conditions, this results in only a few micrometers and is thus
much smaller than the typical electron beam diameter.
In Eq. (11.37) the longitudinal motion z(t)was described only to first order, not
taking into account its coupling to the transverse motion. A more precise, second
order calculation results in
z(t) = v z t −
K 2
8γ 2 k u
sin (2ω u t) ,
(11.40)
where the abbreviation ω u = βck u and the average longitudinal velocity
v z ≈ c
1 −
1
2γ 2
1 +
K 2
2
≡ βc
(11.41)
have been introduced. According to Eq. (11.40), the oscillatory motion in the
transverse plane translates into a small modulation of the longitudinal velocity
around v z . This average longitudinal velocity differs, as described by Eq. (11.41),
from c due to two effects: the factor 1 − 1/(2γ 2 ) ≈ β describes by how much
the electron’s total speed differs from c. The factor 1 + K 2 /2 describes the mean
additional longitudinal retardation due to the transverse velocity components.
Radiation Properties
Calculation of the radiation spectrum starts again from Eqs. (11.22) and (11.23),
however, now the oscillatory motion in the undulator field must be considered when
evaluating Eq. (11.23). Qualitatively, it should be clear from Figs. 11.4 and 11.9 that
the time profile of the wiggler radiation field (K > 1) should resemble the standard
dipole field case with the only difference that there will be a series of radiation
spikes, see Fig. 11.10. The radiation spectrum differs thus not very much from the
synchrotron radiation case but will be N times more intense with N being the number
of wiggler periods (see Fig. 11.11).
On the contrary, in the undulator case K ≤ 1, an observer will only see radiation
field within a small angle of approximately ±1/γ with respect to the forward
677
It should be noted that, in order this periodic trajectory to happen, the electron
beam must enter the device on an orbit representing the appropriate initial conditions. In addition, the undulator field must begin and end with a quarter-period
undulator section.
From Eq. (11.37) the maximum deflection angle can be calculated:
ϑ max ≈
dx
dz
max
= K·
1
γβ
≈
K
γ
.
(11.38)
The maximum orbit excursion of the electron is
x max ∼ =
K
γ k u
.
(11.39)
Under many typical conditions, this results in only a few micrometers and is thus
much smaller than the typical electron beam diameter.
In Eq. (11.37) the longitudinal motion z(t)was described only to first order, not
taking into account its coupling to the transverse motion. A more precise, second
order calculation results in
z(t) = v z t −
K 2
8γ 2 k u
sin (2ω u t) ,
(11.40)
where the abbreviation ω u = βck u and the average longitudinal velocity
v z ≈ c
1 −
1
2γ 2
1 +
K 2
2
≡ βc
(11.41)
have been introduced. According to Eq. (11.40), the oscillatory motion in the
transverse plane translates into a small modulation of the longitudinal velocity
around v z . This average longitudinal velocity differs, as described by Eq. (11.41),
from c due to two effects: the factor 1 − 1/(2γ 2 ) ≈ β describes by how much
the electron’s total speed differs from c. The factor 1 + K 2 /2 describes the mean
additional longitudinal retardation due to the transverse velocity components.
Radiation Properties
Calculation of the radiation spectrum starts again from Eqs. (11.22) and (11.23),
however, now the oscillatory motion in the undulator field must be considered when
evaluating Eq. (11.23). Qualitatively, it should be clear from Figs. 11.4 and 11.9 that
the time profile of the wiggler radiation field (K > 1) should resemble the standard
dipole field case with the only difference that there will be a series of radiation
spikes, see Fig. 11.10. The radiation spectrum differs thus not very much from the
synchrotron radiation case but will be N times more intense with N being the number
of wiggler periods (see Fig. 11.11).
On the contrary, in the undulator case K ≤ 1, an observer will only see radiation
field within a small angle of approximately ±1/γ with respect to the forward