Fields, Potentials and Equations of Motion
53
TABLE 3.1: A list of multipoles
l Leading Term in V
Name
0 M 0,0 (s) cos (θ 0,0 )
1 M 1,1 (s) cos ( φ + θ 1,1 ) r Dipole
2 M 2,2 (s) cos (2φ + θ 2,2 ) r
2
Quadrupole
3 M 3,3 (s) cos (3φ + θ 3,3 ) r
3
Sextupole/Hexapole
4 M 4,4 (s) cos (4φ + θ 4,4 ) r
4
Octupole
5 M 5,5 (s) cos (5φ + θ 5,5 ) r
5
Decapole
6 M 6,6 (s) cos (6φ + θ 6,6 ) r
6
Duodecapole
3.1.2 Quadrupole Fields
The case k = 2 leads to quadrupoles, and the potential has the form V =
M 2,2 cos (2φ + θ 2,2 ) r
2 . Particularly important in practice will be the sub-cases
θ 2,2 = 0 and θ 2,2 = π/2. In the first case, we have
V = M 2,2 cos (2φ) r
2 = M 2,2
cos
2 φ − sin
2 φ
r
2 = M 2,2
x
2
− y
2
,
and in the second case we have
V = M 2,2 cos
2φ +
π
2
r
2 = −M 2,2 sin (2φ) r
2
= −M 2,2 (2 sin φ cos φ) r
2 = −M 2,2 · 2xy.
All other angles θ 2,2 lead to formulas that are more complicated; they can be
obtained from the ones here by subjecting the x, y coordinates to a suitable
rotation. This again leads to terms of purely second order.
Because the potential is quadratic, the resulting fields
E or
B are linear.
Indeed, the quadrupole is the only s-independent element that leads to
linear motion similar to that in glass optics, and thus has great importance.
In the electric case, one usually chooses θ 2,2 = 0, thus having V = M 2,2 (x
2
−
y
2 ) and resulting in the fields
E x = −2M 2,2 · x, E y = 2M 2,2 · y.
The fields extend throughout the length of the device, and thus provide strong
focusing. Different from the case of glass optics, it turns out that the motion
cannot be rotationally symmetric anymore. If there is focusing in the
x-direction, there is defocusing in the y-direction, and vice versa. This effect,
completely due to Maxwell’s equations, turns out to be perhaps the biggest
nuisance in beam physics; i.e., if one uses piecewise s-independent particle
optical elements, the horizontal and vertical planes are always different
from each other.
To make an electrostatic device that produces a quadrupole field, it is best
to machine the electrodes along the equipotential surfaces, and utilize the fact
that if a sufficient amount of boundary information is specified, the field is
53
TABLE 3.1: A list of multipoles
l Leading Term in V
Name
0 M 0,0 (s) cos (θ 0,0 )
1 M 1,1 (s) cos ( φ + θ 1,1 ) r Dipole
2 M 2,2 (s) cos (2φ + θ 2,2 ) r
2
Quadrupole
3 M 3,3 (s) cos (3φ + θ 3,3 ) r
3
Sextupole/Hexapole
4 M 4,4 (s) cos (4φ + θ 4,4 ) r
4
Octupole
5 M 5,5 (s) cos (5φ + θ 5,5 ) r
5
Decapole
6 M 6,6 (s) cos (6φ + θ 6,6 ) r
6
Duodecapole
3.1.2 Quadrupole Fields
The case k = 2 leads to quadrupoles, and the potential has the form V =
M 2,2 cos (2φ + θ 2,2 ) r
2 . Particularly important in practice will be the sub-cases
θ 2,2 = 0 and θ 2,2 = π/2. In the first case, we have
V = M 2,2 cos (2φ) r
2 = M 2,2
cos
2 φ − sin
2 φ
r
2 = M 2,2
x
2
− y
2
,
and in the second case we have
V = M 2,2 cos
2φ +
π
2
r
2 = −M 2,2 sin (2φ) r
2
= −M 2,2 (2 sin φ cos φ) r
2 = −M 2,2 · 2xy.
All other angles θ 2,2 lead to formulas that are more complicated; they can be
obtained from the ones here by subjecting the x, y coordinates to a suitable
rotation. This again leads to terms of purely second order.
Because the potential is quadratic, the resulting fields
E or
B are linear.
Indeed, the quadrupole is the only s-independent element that leads to
linear motion similar to that in glass optics, and thus has great importance.
In the electric case, one usually chooses θ 2,2 = 0, thus having V = M 2,2 (x
2
−
y
2 ) and resulting in the fields
E x = −2M 2,2 · x, E y = 2M 2,2 · y.
The fields extend throughout the length of the device, and thus provide strong
focusing. Different from the case of glass optics, it turns out that the motion
cannot be rotationally symmetric anymore. If there is focusing in the
x-direction, there is defocusing in the y-direction, and vice versa. This effect,
completely due to Maxwell’s equations, turns out to be perhaps the biggest
nuisance in beam physics; i.e., if one uses piecewise s-independent particle
optical elements, the horizontal and vertical planes are always different
from each other.
To make an electrostatic device that produces a quadrupole field, it is best
to machine the electrodes along the equipotential surfaces, and utilize the fact
that if a sufficient amount of boundary information is specified, the field is