32
E. Wilson and B. J. Holzer
the beam any vertical dimensions. Vertical emittance, and the magnet gap needed to
accept it, is directly proportional to coupling.
2.2.1 Coupling Fields
There are two principal configurations of field which excite coupling. The first we
shall consider is a skew quadrupole, i.e. a quadrupole whose poles lie symmetrically
in the horizontal and vertical planes (Fig. 2.12).
A particle with horizontal position x, experiences not a B z as would be the case
in a normal quadrupole and which would change its x , but a B x which together with
the paraxial velocity deflects vertically in the direction of v × B. Of course once the
particle has acquired a vertical displacement z after a number of turns it experiences
a vertical field, for in a skew quadrupole the field is:
B x / (Bρ) = kx,
B z / (Bρ) = −kz.
(2.40)
Thus a horizontal displacement couples into the vertical plane leading to a vertical divergence and displacement. The vertical displacement goes on to couple back
into the horizontal plane modifying the horizontal displacement and divergence—
and so it proceeds transferring transverse momentum back and forth from one plane
to the other.
A solenoid is the other field configuration that can couple the two planes but this
kind of coupling is less important in synchrotrons and we leave it to the reader to
consult a more exhaustive treatment of coupling in [9].
Fig. 2.12 The magnetic field and force in a skew quadrupole
E. Wilson and B. J. Holzer
the beam any vertical dimensions. Vertical emittance, and the magnet gap needed to
accept it, is directly proportional to coupling.
2.2.1 Coupling Fields
There are two principal configurations of field which excite coupling. The first we
shall consider is a skew quadrupole, i.e. a quadrupole whose poles lie symmetrically
in the horizontal and vertical planes (Fig. 2.12).
A particle with horizontal position x, experiences not a B z as would be the case
in a normal quadrupole and which would change its x , but a B x which together with
the paraxial velocity deflects vertically in the direction of v × B. Of course once the
particle has acquired a vertical displacement z after a number of turns it experiences
a vertical field, for in a skew quadrupole the field is:
B x / (Bρ) = kx,
B z / (Bρ) = −kz.
(2.40)
Thus a horizontal displacement couples into the vertical plane leading to a vertical divergence and displacement. The vertical displacement goes on to couple back
into the horizontal plane modifying the horizontal displacement and divergence—
and so it proceeds transferring transverse momentum back and forth from one plane
to the other.
A solenoid is the other field configuration that can couple the two planes but this
kind of coupling is less important in synchrotrons and we leave it to the reader to
consult a more exhaustive treatment of coupling in [9].
Fig. 2.12 The magnetic field and force in a skew quadrupole