STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA

The projection of the beam displacement on the HOM is given by the dot product
of Eq. (4.49) with Eq. (4.50)
104
V ∝ r · êHOM = xo cos α + yo sin α (4.51)
As described above, the method of measuring the polarization requires varying the
horizontal displacement of the electron beam while measuring the response of the
induced HOM voltage
∆Vx ∝ ∆x cos α + yo sin α (4.52)
and then varying the vertical beam displacement while measuring the response
∆Vy ∝ xo cos α + ∆y sin α (4.53)
Because the beam is initially on-axis, (xo, yo), and cannot couple to the dipole HOM,
by taking the ratio of Eq. (4.53) with Eq. (4.52), the polarization can be calculated
using the following relation
α = tan −1
∆Vy
∆Vx
where ∆Vy and ∆Vx are extracted from fits of the measured data.
(4.54)
An example of measured data is displayed in Fig. 4.8 which shows the results
for a mode in cavity 7 with a frequency of 2106.007 MHz. From just an inspection of
the HOM response, it is clear that this mode is polarized nearly vertically. A more
thorough analysis where ∆Vy and ∆Vx were extracted and Eq. (4.54) used, yielded
a polarization of (88 ± 2) ◦ . A summary of the measurements for all the HOMs is
given in Table 4.1. Within each dipole HOM, the two polarizations are separated in
frequency by a few hundred kHz, making it possible to excite each independently.
In addition, one polarization typically has a loaded Q an order of magnitude larger