STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA

to a condition where M ∗ sin(ωTr) < 0 and the system is pseudo-stable. Fourth,
the machine lattice can be changed, which amounts to modifying the quantity M ∗
defined in Eq. (4.19). The use of beam optical suppression techniques is the topic
of this chapter and methods for modifying the properties of the mode are covered
more fully in Chapter 7.
Methods to manipulate the transverse beam optics in order to suppress BBU
were first presented in 1980 [84]. The strategy of beam optical control techniques
is to modify the machine lattice in such a way that the beam cannot couple as
effectively to the dangerous dipole mode. This can be achieved with point-to-point
focusing, reflecting the betatron planes about an axis that is at 45 ◦ between the
vertical and horizontal axes and a 90 ◦ rotation. While the ability of point-to-point
focusing to increase the threshold current was demonstrated at the SCA [11] and
at MUSL-2 [27], the latter two methods, which require introducing strong betatron
coupling into the system, had never before been tested experimentally. In 2005,
the ability to raise the threshold current by each of these methods was successfully
demonstrated in the FEL Driver, and these methods are described in the following
sections.
6.2 Point-to-Point Focusing
Because it does not involve complicated transverse coupling schemes, point-to-
point focusing was the first beam optical suppression technique employed to combat
the effects of BBU in the SCA and at MUSL-2. With a judicious change in the
betatron phase advance, point-to-point focusing can be achieved (M12 or M34 = 0)
at the location of the cavity containing a dangerous mode so that an HOM-induced
kick on the first pass results in a zero displacement on the second pass. In this
139
way the beam cannot transfer energy to the mode by coupling to the electric field