56
Chapter 2. Geometrical optics
For many systems it is the case that the radial position of the
ray is roughly constant within the lens field, i.e., R(z) ≈ const.
Such a lens is called a thin lens. It is also often the case that the
electrostatic component of the focusing is weak or nonexistent;
i.e., r
� ≈ Φ
−1/4 R
� . Such a lens is called a weak lens. Using these
approximations, we obtain (2.139)
1 ≈
−∞
+
−
Φ
f
Φ ∞
1/4 R
�
∞ .
(2.143)
R −∞
From (2.140) we obtain
∞
∞
∞
R
� =
R
�� dz =
Q R dz
R
Q dz. (2.144)
∞
−∞
−
−∞
≈ − −∞
−∞
From (2.141, 2.144, 2.145) we obtain
1
Φ
4
�
1/
⎡
�
∞
3 Φ
2
B
2
−∞
1 + Φ
≈
+
dz,
f +
Φ ∞
−∞ 16 Φ
�
⎣
1 + Φ/2 8 Φ (1 + Φ/2)
⎤
(2.145)
⎦
where the first term on the right represents the electrostatic focusing, and the second term represents the magnetic focusing. Similarly, we define a reverse focal length f , where rays enter parallel
−
to the optic axis at radius r , and exit with slope r
�
:
∞
−∞
1
r
�
=
−∞ ,
where
r
� = 0.
(2.146)
f
∞
−
r ∞
The axial positions of principal planes follow directly from f + and
f .
−
The quantity 1/f represents the focal strength of a lens. In the
purely electrostatic case where B = 0, the focal strength is proportional to the charge q, and independent of the mass m, taking
account of the dimensionless units. In the purely magnetic case
where Φ
� = 0, the focal strength is proportional to the ratio of
q/m. Consequently, it is more efficient to use electrostatic lenses
for heavier particles, such as ions, and magnetic lenses for lighter
particles, such as electrons.