this in the unperturbed coordinate system v(z). Because the ray
v 1 (z) has energy Φ 1 which differs infinitesimally from Φ, it follows
that the rotation χ(z) differs by an infinitesimal amount δχ between the v- and v 1 -systems. Applying this rotation, we define the
perturbation δv(z I ) in the unperturbed coordinates v(z) according
to
−i δχ
v + δv = v 1 e
≈ (v + δv 1 ) (1 − i v δχ).
(2.244)
Expanding this, the aberration expressed in the Gaussian image
plane z I of the unperturbed system is
δv(z I ) = δv 1 (z I ) − i v(z I ) δχ OI ,
(2.245)
retaining only terms to first order in small quantities. From
(2.157), the perturbation δχ is found to be
δχ OI =
1
2
z I
δ(p
−1 ) B dz = −(δΦ)·
1
2
z I
p
−3 (1+Φ) B dz, (2.246)
z O
z O
where we have made use of (2.124), the definition of the on-axis
kinetic momentum p. The minus sign expresses the fact that the
rotation χ is smaller for higher particle energy, δΦ > 0. Substituting (2.242, 2.243, 2.246) into (2.245), and making use of the
solution (2.167), we find, after collecting terms,
δv I = (δΦ) [ (C 1 + iC 2 ) v O + C 3 v A ],
(2.247)
where we have defined the chromatic aberration coefficients C 1 , C 2 ,
and C 3 in natural units as
M z I
C 1 = −
p
−1 { (2 p
−2 + 1) Φ
� g
� h
k z O
+
1 [ (2 p
−2 + 1) Φ
�� + p
−2 (1 + Φ) B
2 ] g h } dz
2
C 2 =
M z I
p
−3 (1 + Φ) B dz
2 z O
M z I
C 3 = −
p
−1 { (2 p
−2 + 1) Φ
� h h
�
k z O
+
1 [ (2 p
−2 + 1) Φ
�� + p
−2 (1 + Φ) B
2 ] h
2 } dz.
2
(2.248)
87
2.5. Axial symmetry