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51
2.5. Axial symmetry
therefore refer to these units as natural units. Unless specifically

noted, we will use these units throughout the following section describing the special case with axial symmetry, and drop the tilde.

2.5 Axial symmetry
Systems with a straight optic axis, where the potentials A and
φ are axially symmetric, represent an important special case of
the general curvilinear axis. This includes a large class of useful
instruments, including electron and ion microscopes. It excludes
curved-axis energy analyzers. The reader is referred to Rose [75],
Hawkes and Kasper [43, 44], and Wollnik [93] for further detail
and elaboration, both of axially symmetric and nonsymmetric systems.
2.5.1 Exact equations of motion for axially
symmetric fields
In the absence of space charge, the electrostatic potential φ satisfies Laplace’s equation,
v
2 φ = 0.
(2.118)
In cylindrical coordinates this becomes
∂
2
1 ∂
∂
2
+
+
φ = 0.
(2.119)
∂r 2 r ∂r ∂z 2
We propose a series solution by the method of undetermined coefficients [74]. We assume that φ can be expanded in a series representation given by
φ(r, z) = a 0 (z) + a 2 (z) r
2 + a 4 (z) r
4 + . . . ,
(2.120)