Page 90 - Modern Optical Engineering The Design of Optical Systems

The Primary Aberrations 73

intersects the axis in object space must also intersect it in image space.
Every ray passing through the same axial point in object space and
also passing through the same annular zone in the aperture (i.e.,
with the same value of s) must pass through the same axial point in
image space. A ray in front of the meridional (y, z) plane has a mirror-
image ray behind the meridional plane which is identical except for
the (reversed) signs of x′ and . Similarly, rays originating from h
in the object and passing through corresponding upper and lower
aperture points must have identical x′ intersections and oppositely
signed y′ values. With this sort of logic one can derive equations such
as the following:

y′ A s cos A h
1 2
2
3
2
B s cos B s h(2 cos 2 ) (3B B )sh cos B h 3
1 2 3 4 5
3 2
5
2
4
C s cos (C C cos 2 )s h (C C cos )s h cos
1 2 3 4 6
5
7
4
2 3
(C C cos 2 )s h C sh cos C h D s cos . . . (5.1)
7 8 10 12 1
x′ A s sin
1
2
3
2
B s sin B s h sin 2 (B B )sh sin
1 2 3 4
3
2
4
5
2
C s sin C s h sin 2 (C C cos )s h sin
1 3 5 6
7
3
2
4
C s h sin 2 C sh sin D s sin . . . (5.2)
9 11 1
where A , B , etc., are constants, and h, s, and have been defined
N N
above and in Fig. 5.1.
Notice that in the A terms, the exponents of s and h are unity. In the
3
3
2
B terms the exponents total 3, as in s , s h, sh , and h . In the C terms
2
the exponents total 5, and in the D terms, 7. These are referred to as
the first-order, third-order, and fifth-order terms, etc. There are 2 first-
order terms, 5 third-order, 9 fifth-order, 14 seventh-order, 20 ninth-
order, and
(n 3) (n 5)
1
8
nth-order terms. In an axially symmetrical system there are no even-
order terms; only odd-order terms may exist (unless we depart from
symmetry as, for example, by tilting a surface or introducing a toroidal
or other nonsymmetrical surface).
It is apparent that the A terms relate to the paraxial (or first-order)
imagery discussed in the preceding chapters. A is simply the magnifica-
2
tion (h′/h), and A 1 is a transverse measure of the distance from the par-
axial focus to our “image plane.” All the other terms in Eqs. 5.1 and 5.2