11 Application of Accelerators and Storage Rings
679
Fig. 11.11 Undulator spectrum in forward direction for γ = 1000 and N = 10 undulator periods.
The undulator parameter is K = 0.5 (top) and K = 2 (bottom). The undulator period is λ u = 2 cm.
The vertical scales are in arbitrary units. The broken line in the lower diagram shows the spectrum
of synchrotron radiation for a magnetic dipole field equal to the peak field of the undulator, with
the vertical scale matched to the undulator spectrum arbitrarily
The sinc-function with argument πN(ω − kω 1 )/ω 1 describes the spectral distribution of the individual undulator harmonics. The total line width of harmonics k
is
Δω
ω k
≈
1
kN
.
(11.44)
The photon number spectrum radiated into the frequency interval k in
forward direction is given by
dN γ
dd = αN 2 γ 2 Δω
ω k
k 2 K 2
1+
K 2
2
2
J k−1
2
kK 2
4
1+
K 2
2
− J k+1
2
kK 2
4
1+
K 2
2
2
.
(11.45)
α is again the fine structure constant.
As compared to the synchrotron radiation spectrum Eq. (11.24), undulator radiation provides a spectral density in forward direction larger by some factor N 2 . One
factor N results from having a number of N undulator periods. The other factor N
stems from the fact that, due to interference, the radiation spectrum is concentrated
into narrow resonance lines. Thus, in forward direction, the field amplitudes add
up coherently, not the intensities. The total radiation energy increases with N as
expected.
679
Fig. 11.11 Undulator spectrum in forward direction for γ = 1000 and N = 10 undulator periods.
The undulator parameter is K = 0.5 (top) and K = 2 (bottom). The undulator period is λ u = 2 cm.
The vertical scales are in arbitrary units. The broken line in the lower diagram shows the spectrum
of synchrotron radiation for a magnetic dipole field equal to the peak field of the undulator, with
the vertical scale matched to the undulator spectrum arbitrarily
The sinc-function with argument πN(ω − kω 1 )/ω 1 describes the spectral distribution of the individual undulator harmonics. The total line width of harmonics k
is
Δω
ω k
≈
1
kN
.
(11.44)
The photon number spectrum radiated into the frequency interval k in
forward direction is given by
dN γ
dd = αN 2 γ 2 Δω
ω k
k 2 K 2
1+
K 2
2
2
J k−1
2
kK 2
4
1+
K 2
2
− J k+1
2
kK 2
4
1+
K 2
2
2
.
(11.45)
α is again the fine structure constant.
As compared to the synchrotron radiation spectrum Eq. (11.24), undulator radiation provides a spectral density in forward direction larger by some factor N 2 . One
factor N results from having a number of N undulator periods. The other factor N
stems from the fact that, due to interference, the radiation spectrum is concentrated
into narrow resonance lines. Thus, in forward direction, the field amplitudes add
up coherently, not the intensities. The total radiation energy increases with N as
expected.