STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA

consider an HOM polarized at an arbitrary angle, α, with respect to the x-axis.
The unit vector describing the HOM polarization is êHOM = (sin α, cos α) and the
bunch displacement is given by
r = r · êHOM = x cos α + y sin α (4.8)
On the first pass through the cavity the bunch deposits an energy to the HOM
given by inserting Eq. (4.8) into Eq. (4.7)
85
∆U1 = −q Va
a (x1 cos α + y1 sin α) cos φ (4.9)
The bunch’s contribution to the energy of the HOM on the second pass through the
cavity can be written in a similar manner
∆U2 = −q Va
a (x2 cos α + y2 sin α) cos(φ + ωTr) (4.10)
where x2 and y2 denote the second pass displacements, ω is the HOM angular
frequency and Tr is the recirculation time of the machine. The additional phase
term includes the effects of the beam recirculation. Note that this derivation assumes
that the variation of the HOM voltage on the time scale of the recirculation time is
negligible.
The horizontal displacement on the second pass can be rewritten in terms of
the first pass coordinates and the HOM imparted angular kick as
x2 = M11x1 + M12x ′
1 + M13y1 + M14y ′
1 − qVa
and likewise for the vertical plane
y2 = M31x1 + M32x ′
1 + M33y1 + M34y ′
1 − qVa
(M12 cos α + M14 sin α) sin φ (4.11)
ωapb
(M32 cos α + M34 sin α) sin φ (4.12)
ωapb