122
An Introduction to Beam Physics
TABLE 5.2: The second order map of a solenoid (Exponents in the
initial variables x, a, y, b, l, δ)
x f
a f
y f
b f
l f
exponents
0.999662
-0.408336e-3 -0.186647e-1 0.761884e-5 0
100000
0.799815
0.999662
-0.149334e-1 -0.186647e-1 0
010000
0.186647e-1 -0.761884e-5 0.999662
-0.408336e-3 0
001000
0.149334e-1 0.186647e-1 0.799815
0.999662
0
000100
0
0
0
0
1.000000
000010
0
0
0
0
0.163101
000001
0
0
0
0
-0.112007e-3 200000
0
0
0
0
-0.876499e-9 110000
0
0
0
0
-0.219388
020000
0
0
0
0
-0.102394e-1 011000
0
0
0
0
-0.112007e-3 002000
0
0
0
0
0.102394e-1 100100
0
0
0
0
-0.876499e-9 001100
0.370284e-3 0.223860e-3 0.102326e-1 -0.836226e-5 0
100001
-0.438474
0.370284e-3 0.163792e-1 0.102326e-1 0
010001
-0.102326e-1 0.836226e-5 0.370284e-3 0.223860e-3 0
001001
0
0
0
0
-0.219388
000200
-0.163792e-1 -0.102326e-1 -0.438474
0.370284e-3 0
000101
0
0
0
0
-0.134185
000002
complex coordinates, eqs. (5.1) are transformed to
¯
z f
¯
w f
=
F z
F w
(¯ z i , z i , ¯
w i , w i , t i , d i )
⇒
z f
w f
=
¯
F z
¯
F w
(z i , ¯
z i , w i , ¯
w i , t i , d i )
=
j1j2j3j4jtj d
¯
a z
¯
a w
j1j2j3j4jtj d
z
j1 ¯
z
j2 w
j3 ¯
w
j4 t
jt d
j d ,
which shows that all coefficients have to be real numbers in order to preserve
midplane symmetry.
For discrete rotational symmetry, invariance occurs only when φ = 2π/k,
where k is an integer. Hence the nonzero terms satisfy
j 1 − j 2 + j 3 − j 4 − 1 = nk.
In general, a 2k-pole is invariant under rotation of φ = 2π/k. For example,
for a quadrupole, we have k = 2. Hence the nonzero terms satisfy
j 1 − j 2 + j 3 − j 4 = 2n + 1.
Like round lenses, systems with quadrupole symmetry are also free of even
order geometric aberrations. The linear map of a quadrupole can be obtained