E =
z I
[ 4 L g
3 h + 2 M g g
� (g h
� + g
� h) + 4 N g
� 3 h
� ] dz
z O
F =
z I
[ 4 L g h
3 + 2 M h h
� (g h
� + g
� h) + 4 N g
� h
� 3 ] dz
z O
e = 2 k
z I
p
−1 (P g
2 + Q g
� 2 ) dz
z O
c = 4 k
z I
p
−1 (P g h + Q g
� h
� ) dz
z O
f = 2 k
z I
p
−1 (P h
2 + Q h
� 2 ) dz,
(2.214)
z O
remembering that g(z) and h(z) are the two linearly independent
solutions to the paraxial ray equation (2.162) satisfying boundary
conditions (2.166). From (2.213) we have
∂
z I
∂
∂
∂
m 4 dz = A
(R
2 ) + B
(ρ
2 ) + C
(χ
2 ) + . . . .
∂x Aj z O
∂x Aj
∂x Aj
∂x Aj
(2.215)
The primary aberration δx I in the Gaussian image plane is thus
given in the rotated coordinate system by (2.203, 2.215)
2
2
k M
−1 δx I =
B [ 4 x A (x A + y A ) ]
+ C [ 2 x O (x O x A + y O y A ) ]
2
2
+ D [ 2 x A (x O + y O ) ]
2
2
+ E [ x O (x O + y O ) ]
2
2
+ F [ x O (x A + y A ) + 2 x A (x O x A + y O y A ) ]
2
2
+ e [ −y O (x O + y O ) ]
+ c [ y A (x
2 − y
2 ) − 2 x O y O x A ]
O
O
+ f [ −y O (x A
2 + y
2 ) + 2 x A (x O y A − y O x A ) ],
A
(2.216)
remembering the definition (2.165) of the constant k, and where
M is the magnification. Making use of the axial symmetry, the
y-coordinate of the aberration is obtained by making the substitutions x → y, and y → −x for all occurrences in (2.216).
73
2.5. Axial symmetry