18
P. R. Willmott
0
10
20
30
40
Photon energy [keV]
10
16
10
17
10
18
10
19
10
20
10
21
10
22
Brilliance [ph/s/0.1% BW/mm
2
/mrad
2
]
Fig. 1.13 Comparison of brilliances at a 3 GeV DLSR running at 400 mA between a U14 undulator
with K = 1.6 (blue), a bending magnet with B = 1.41 T (red), a superbend with B = 4 T (yellow),
and a wiggler with the same field strength as the bending magnet and 100 periods (green). Note
that the peak brilliance of synchrotron sources can be calculated from the average brilliance shown
in this figure by multiplying by the ratio of the pulse separation ∼ 5 ns to the pulse width
∼ 50 ps, that is, approximately a factor 100. The peak brilliance of XFELs is of the order of
10 34 ph s −1 mm −2 mrad −2 0.1% BW −1 , some ten orders of magnitude greater than that produced by
DLSR undulators. Reproduced from [3] with permission (Copyright 2019, John Wiley and Sons)
the order of 20. The spectrum from a wiggler has the same form as that from a bending
magnet—it is broadband and thus produces a large amount of integrated radiative
power, of the order of several kW. Thermal management of optical components
is thus critical. Wigglers are therefore becoming fairly uncommon, particularly in
fourth-generation DLSRs.
1.3.2 Undulators
In undulators, the angular deviation of the electrons away from the central axis is of
the order of 1/γ ; the radiation cones emitted by the electrons thus overlap as they
execute the slalom motion. Consequently, radiation from the dipoles interferes with
one another. As such, the field amplitudes are added vectorially (i.e. including the
phase difference from each contribution) and the sum of this is squared to produce
the intensity, which peaks at those wavelengths where interference is constructive.
Undulators therefore differ fundamentally from bending magnets and wigglers in
that their spectral flux reflects this interference phenomenon and is hence concentrated in evenly separated, narrow bands of radiation (Fig. 1.13). The first practical
undulator device to operate in the X-ray regime was constructed by Klaus Halbach
and co-workers at the Lawrence–Berkeley National Laboratory and tested at the
SSRL synchrotron at Stanford in 1981. This breakthrough was thanks on the one
hand to the development of novel magnetic alloys such as SmCo 5 [6], allowing the
P. R. Willmott
0
10
20
30
40
Photon energy [keV]
10
16
10
17
10
18
10
19
10
20
10
21
10
22
Brilliance [ph/s/0.1% BW/mm
2
/mrad
2
]
Fig. 1.13 Comparison of brilliances at a 3 GeV DLSR running at 400 mA between a U14 undulator
with K = 1.6 (blue), a bending magnet with B = 1.41 T (red), a superbend with B = 4 T (yellow),
and a wiggler with the same field strength as the bending magnet and 100 periods (green). Note
that the peak brilliance of synchrotron sources can be calculated from the average brilliance shown
in this figure by multiplying by the ratio of the pulse separation ∼ 5 ns to the pulse width
∼ 50 ps, that is, approximately a factor 100. The peak brilliance of XFELs is of the order of
10 34 ph s −1 mm −2 mrad −2 0.1% BW −1 , some ten orders of magnitude greater than that produced by
DLSR undulators. Reproduced from [3] with permission (Copyright 2019, John Wiley and Sons)
the order of 20. The spectrum from a wiggler has the same form as that from a bending
magnet—it is broadband and thus produces a large amount of integrated radiative
power, of the order of several kW. Thermal management of optical components
is thus critical. Wigglers are therefore becoming fairly uncommon, particularly in
fourth-generation DLSRs.
1.3.2 Undulators
In undulators, the angular deviation of the electrons away from the central axis is of
the order of 1/γ ; the radiation cones emitted by the electrons thus overlap as they
execute the slalom motion. Consequently, radiation from the dipoles interferes with
one another. As such, the field amplitudes are added vectorially (i.e. including the
phase difference from each contribution) and the sum of this is squared to produce
the intensity, which peaks at those wavelengths where interference is constructive.
Undulators therefore differ fundamentally from bending magnets and wigglers in
that their spectral flux reflects this interference phenomenon and is hence concentrated in evenly separated, narrow bands of radiation (Fig. 1.13). The first practical
undulator device to operate in the X-ray regime was constructed by Klaus Halbach
and co-workers at the Lawrence–Berkeley National Laboratory and tested at the
SSRL synchrotron at Stanford in 1981. This breakthrough was thanks on the one
hand to the development of novel magnetic alloys such as SmCo 5 [6], allowing the