Computation and Properties of Maps
123
from
j 1 − j 2 + j 3 − j 4 = ±1,
which is
x f + iy f = (z|z)(x i + iy i ) + (z|w)(a i + ib i ) + (z|¯ z)(x i −iy i ) + (z| ¯
w)(a i −ib i ),
a f + ib f = (w|z)(x i + iy i ) + (w|w)(a i + ib i ) + (w|¯ z)(x i −iy i ) + (w| ¯
w)(a i −ib i ).
Since a quadrupole has midplane symmetry, all the coefficients are real numbers. Thus its linear matrix is
⎛
⎜
⎜
⎝
(z|z) + (z|¯ z)
0
( z|w) + (z| ¯
w)
0
0
( z|z) − (z|¯ z)
0
( z|w) − (z| ¯
w)
(w|z) + (w|¯ z)
0
( w|w) + (w| ¯
w)
0
0
( w|z) − (w|¯ z)
0
( w|w) − (w| ¯
w)
⎞
⎟
⎟
⎠ .
For other multipoles, we have
j 1 − j 2 + j 3 − j 4 = nk + 1 =
⎧
⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
· · ·
−k + 1
1
k + 1
· · ·
k = 3, 4, . . . .
(5.4)
With midplane symmetry, the linear matrix of a 2k-pole is
ˆ
M =
⎛
⎜
⎜
⎝
(z|z) (z|w)
0
0
(w|z) (w|w)
0
0
0
0
(z|z) (z|w)
0
0
(w|z) (w|w)
⎞
⎟
⎟
⎠ .
(5.5)
Since the linear matrix of a 2k-pole (k ≤ 3) is just a drift, it satisfies eq. (5.5).
Eq. (5.4) shows that the geometric aberrations appear only for orders of at
least k − 1. The fact that multipoles do not have dispersion determines that
the chromatic aberrations do not appear until order k. This can be easily seen
from the equations of motion (3.22). Therefore, a 2k-pole is necessarily a drift
up to order k − 2.
A lens with rotational symmetry is frequently called a round lens. The
main examples are magnetic solenoids and electrostatic round lenses. Since
rotational symmetry is the highest degree of symmetry a lens can have, round
lenses have the fewest number of aberrations. As a result, they are widely
used in low energy electron optical devices such as electron microscopes. At
high energy, round lenses are too weak to be effective.
5.1.4 Symplectic Symmetry
Another important symmetry of the motion is due to the fact that the motion is indeed obtained as the solution of Hamiltonian differential equations.
123
from
j 1 − j 2 + j 3 − j 4 = ±1,
which is
x f + iy f = (z|z)(x i + iy i ) + (z|w)(a i + ib i ) + (z|¯ z)(x i −iy i ) + (z| ¯
w)(a i −ib i ),
a f + ib f = (w|z)(x i + iy i ) + (w|w)(a i + ib i ) + (w|¯ z)(x i −iy i ) + (w| ¯
w)(a i −ib i ).
Since a quadrupole has midplane symmetry, all the coefficients are real numbers. Thus its linear matrix is
⎛
⎜
⎜
⎝
(z|z) + (z|¯ z)
0
( z|w) + (z| ¯
w)
0
0
( z|z) − (z|¯ z)
0
( z|w) − (z| ¯
w)
(w|z) + (w|¯ z)
0
( w|w) + (w| ¯
w)
0
0
( w|z) − (w|¯ z)
0
( w|w) − (w| ¯
w)
⎞
⎟
⎟
⎠ .
For other multipoles, we have
j 1 − j 2 + j 3 − j 4 = nk + 1 =
⎧
⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
· · ·
−k + 1
1
k + 1
· · ·
k = 3, 4, . . . .
(5.4)
With midplane symmetry, the linear matrix of a 2k-pole is
ˆ
M =
⎛
⎜
⎜
⎝
(z|z) (z|w)
0
0
(w|z) (w|w)
0
0
0
0
(z|z) (z|w)
0
0
(w|z) (w|w)
⎞
⎟
⎟
⎠ .
(5.5)
Since the linear matrix of a 2k-pole (k ≤ 3) is just a drift, it satisfies eq. (5.5).
Eq. (5.4) shows that the geometric aberrations appear only for orders of at
least k − 1. The fact that multipoles do not have dispersion determines that
the chromatic aberrations do not appear until order k. This can be easily seen
from the equations of motion (3.22). Therefore, a 2k-pole is necessarily a drift
up to order k − 2.
A lens with rotational symmetry is frequently called a round lens. The
main examples are magnetic solenoids and electrostatic round lenses. Since
rotational symmetry is the highest degree of symmetry a lens can have, round
lenses have the fewest number of aberrations. As a result, they are widely
used in low energy electron optical devices such as electron microscopes. At
high energy, round lenses are too weak to be effective.
5.1.4 Symplectic Symmetry
Another important symmetry of the motion is due to the fact that the motion is indeed obtained as the solution of Hamiltonian differential equations.