86
An Introduction to Beam Physics
FIGURE 4.7: An electric deflector with cylindrical plates.
FIGURE 4.8: An electric deflector with spherical plates.
and the determinant is unity.
So far, no assumptions have been made about the vertical shapes of the
electrodes, and in fact a variety of choices exist. Two common situations are
the cylindrical plates and the spherical plates, as shown in Fig. 4.7 and Fig.
4.8, respectively.
In the case of the cylindrical field, we have from Gauss’ law that E ∝ 1/R,
which implies the expansion
E x = −E x0
R 0
R 0 + x
= 1 −E x0
1 −
x
R 0
,
and hence corresponds to n = 1. The transfer matrix is
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
x f
a f
y f
b f
l f
δ f
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
cos(
√
2φ)
(R 0 /
√
2) sin(
√
2φ) 0
0 0 (x|δ)
−(
√
2/R 0 ) sin(
√
2φ)
cos(
√
2φ)
0
0 0 (a|δ)
0
0
1R 0 φ 0 0
0
0
0
1
0 0
(l|x)
( l|a)
0
0 1 ( l|δ)
0
0
0
0 0 1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
x i
a i
y i
b i
l i
δ i
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
where
(x|δ) = −(l|a) =
R 0
2
1 − cos
√
2φ
,
(a|δ) = −(l|x) =
1
√
2
sin
√
2φ
,
(l|δ) = −R 0
1
4
φ −
1
2
√
2
sin
√
2φ
.