�
x
x Aj =
Ij .
(2.226)
h
�
I
From (2.169) we have
1
p I M
=
,
(2.227)
h
�
I
k
in which case
p I M �
x Aj =
x Ij .
(2.228)
k
We now define radial quantities r A , r I
� , δr I as
2
2
2
r = x A + y A
A
� 2
� 2
� 2
r
= x + y
I
I
I
δr I
2 = δx
2
I + δy I
2 ,
(2.229)
from which, due to the axial symmetry, it follows that
4 B M
4 B M
4 p
3
3
I � 3
δr I =
r =
r .
(2.230)
A
k 4
I
k
We define α I as the angle in radians which the ray makes with the
optic axis at the Gaussian image plane. Assuming α « 1,
r I
� = tan α I ≈ α I .
(2.231)
78
Chapter 2. Geometrical optics
come to a focus at the Gaussian image plane, which is shown by
the rightmost vertical line in the figure. Rays at the edge are more
strongly focused, and intersect the axis in front of the Gaussian
image plane. This plane is shown by the leftmost vertical line in
the figure. A disk of least confusion is formed by the entire bundle
at an intermediate plane. This represents the optimum focus.
Spherical aberration is easily expressed as a function of the ray
slope x Ij
� in the Gaussian image plane. This slope is directly measurable by defocusing, whereas the aperture coordinate x Aj is
not easily measured. This necessitates transforming from the set
(x Oj , x Aj ) to the set (x Oj , x Ij
� ). Evaluating (2.168) in the Gaussian
image plane, and setting x Oj = 0, we find