STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA

which excites it. The BBU instability is generated by dipole HOMs which couple
to the beam through the dipole moment of the current. Thus the net effect on the
beam, imparted by the transverse deflecting voltage, is given by the integral [67]
V (t) =
t
−∞
W1(t − t ′ )I(t ′ − Tr)r2(t ′ )dt ′
94
(4.31)
where W1(t) is the delta functional dipole wake function and I(t)r2(t) is the dipole
moment of the beam bunch when it passes the cavity at time t. Equation (4.31)
represents the starting point for an alternate derivation of the BBU threshold current
using a wake function formalism.
For a single dipole higher-order mode with angular frequency ω and loaded
quality factor QL, the long-range wake function can be expressed analytically as
W1(t − t ′ ) =
Rd
Qo
ωk
2
e− ω
2Q L (t−t ′ ) sin(ω(t − t ′ )) (4.32)
where k is the wavenumber. The bunch displacement at the cavity on the second
pass can be written in terms of the voltage kick due to the accumulated wake excited
by all preceding bunches and in terms of the appropriate matrix elements describing
a single recirculation from the cavity back to itself as
r2(t ′ ) = V (t′ − Tr)
(M12 cos 2 α + (M14 + M32) sin α cos α + M34 sin 2 α) (4.33)
Vb
where α is the mode polarization and Vb = pb(c/q) is the beam momentum at the
cavity. The beam bunches are approximated by delta functions
I(t ′
− Tr) = Ioto δ(t ′ − Tr − nto) (4.34)
n
where to is the bunching period. Plugging in Eqs. (4.32), (4.33) and (4.34) into
Eq. (4.31) leads to an integral equation. Assuming a normal mode solution of the
form