where ϕ is the difference in phase between y- and x-electric field components. From
the above, we see that pure circular polarization (degree of circular polarization
P 3 ¼ 1) occurs when b
E x0 ¼ b
E y0 : (hence b
B x0 = b
B x0 ) and and ϕ ¼ π/2. In the
convention of Born and Wolf, this is called right circular polarization. A nice
example of the variable polarization from an elliptical undulator was reported by
Sasaki and coworkers [78], using the low-energy JSR ring operating at 138 MeV
which put the radiation in the visible region of the spectrum (Fig. 3.17).
3.9 Helical Undulators
A helical undulator is simply the special case of an elliptical undulator with equal b
B x0
and b
B y0 that are 90
out of phase. The net amplitude of the field stays constant, while
the direction rotates around a circle along the axis of the device. Consequently, the
electrons spiral in a circle as they travel down the device, and the resulting synchrotron radiation has pure circular polarization. Several special properties are worth
pointing out.
3.9.1 Fundamental Energy
The on-axis fundamental energy for a helical undulator can be obtained from
Eq. 3.62 by setting K x ¼ K y and the observation angle θ ¼ 0:
E 1 keV
½
¼ 0:950
E
2
e GeV
½
1 þ K
2
À
Á λ u cm
½
ð3:64Þ
3.9.2 Harmonics
The helical undulator possesses the interesting and useful property that there are no
higher harmonics produced on-axis in the forward direction.
3.9.3 Power
For a given magnetic field B 0 , the helical undulator produces twice the power
radiated by a planar undulator. This is easy to understand by referring back to
Eq. 3.30 and noting that the magnitude of the magnetic field does not change in
this device, only the direction.
3.9 Helical Undulators
65
the above, we see that pure circular polarization (degree of circular polarization
P 3 ¼ 1) occurs when b
E x0 ¼ b
E y0 : (hence b
B x0 = b
B x0 ) and and ϕ ¼ π/2. In the
convention of Born and Wolf, this is called right circular polarization. A nice
example of the variable polarization from an elliptical undulator was reported by
Sasaki and coworkers [78], using the low-energy JSR ring operating at 138 MeV
which put the radiation in the visible region of the spectrum (Fig. 3.17).
3.9 Helical Undulators
A helical undulator is simply the special case of an elliptical undulator with equal b
B x0
and b
B y0 that are 90
out of phase. The net amplitude of the field stays constant, while
the direction rotates around a circle along the axis of the device. Consequently, the
electrons spiral in a circle as they travel down the device, and the resulting synchrotron radiation has pure circular polarization. Several special properties are worth
pointing out.
3.9.1 Fundamental Energy
The on-axis fundamental energy for a helical undulator can be obtained from
Eq. 3.62 by setting K x ¼ K y and the observation angle θ ¼ 0:
E 1 keV
½
¼ 0:950
E
2
e GeV
½
1 þ K
2
À
Á λ u cm
½
ð3:64Þ
3.9.2 Harmonics
The helical undulator possesses the interesting and useful property that there are no
higher harmonics produced on-axis in the forward direction.
3.9.3 Power
For a given magnetic field B 0 , the helical undulator produces twice the power
radiated by a planar undulator. This is easy to understand by referring back to
Eq. 3.30 and noting that the magnitude of the magnetic field does not change in
this device, only the direction.
3.9 Helical Undulators
65