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An Introduction to Beam Physics
a sign of (−1)
iy+i b in each monomial. So i y + i b must be odd for y f , b f and
i y + i b must be even for all others. So we obtain
x|x
ix a
ia y
iy b
i b l
i l δ
i δ
= 0 for i y + i b odd,
a|x
ix a
ia y
iy b
i b l
i l δ
i δ
= 0 for i y + i b odd,
y|x
ix a
ia y
iy b
i b l
i l δ
i δ
= 0 for i y + i b even,
b|x
ix a
ia y
iy b
i b l
i l δ
i δ
= 0 for i y + i b even,
l|x
ix a
ia y
iy b
i b l
i l δ
i δ
= 0 for i y + i b odd,
δ|x
ix a
ia y
iy b
i b l
i l δ
i δ
= 0 for i y + i b odd.
For the first order map, the transfer matrix, this leads to the form
ˆ
M =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
(x|x) (x|a)
0
0
(x|l) (x|δ)
(a|x) (a|a)
0
0
(a|l) (a|δ)
0
0
(y|y) (y|b)
0
0
0
0
(b|y) (b|b)
0
0
(l|x) (l|a)
0
0
(l|l) (l|δ)
(δ|x) (δ|a)
0
0
(δ|l) (δ|δ)
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
which is a form seen earlier for electrostatic and magnetic bending elements.
Altogether, the symmetry entails that to any given order, roughly half of all
aberrations vanish.
5.1.2 Double Midplane Symmetry
Several devices have a midplane symmetry not only around the horizontal
plane, but also around a vertical plane. This is the case for all electric cylindrically symmetric devices, as well as quadrupoles, octupoles, and in general
4k poles. In this case, in addition to the requirements we just had, we obtain
a second set in which the roles of x, a and y, b are interchanged. In this case
we obtain
(x| . . .) = 0 for i y + i b odd or i x + i a even,
(a| . . .) = 0 for i y + i b odd or i x + i a even,
(y| . . .) = 0 for i y + i b even or i x + i a odd,
(b| . . .) = 0 for i y + i b even or i x + i a odd,
(l| . . .) = 0 for i y + i b odd or i x + i a even,
(δ| . . .) = 0 for i y + i b odd or i x + i a even,