Chapter 5
Image Formation
The Positive Nodal Points
The positive nodal points, N and
N ', are two
points lying on the optic
axis which have the property that a ray
incident on N emerges from N ' in a
direction parallel to the incident ray.
Thus the angular magnification
' = 1 for these two
points.
To determine the position of the
nodal points, we again use (36) but with
the conditions now that for y = y ' = 0,
'
= .
Equation (36) states that
for y ' = 0,
=
(z '/f) '
and therefore

(51) 
Also (36) states that for
y ' = y = 0,
0 = (zz '/f  f ')
' =
which combined with equation (51) gives

(52) 
Figure 20.
Thus for a positive system the first nodal
point is a distance
f ' to the right
of F and the second nodal point is a distance
f to the left of F '. The use
of the nodal points in a ray diagram is
illustrated in Fig. 20. The distances
NN ' and HH ' are obviously equal. If
f
= f ',
N and H are coincident, as are
N ' and H '.
Equations (51) and (52) together with (36) give the
transformation from one nodal plane to the other as

(53) 
Again the two planes through N and
N '
are recognized to be objectimage
planes as there is a zero in the upper right hand corner of the matrix.
The linear magnification for an object at the first
nodal plane of an image
at the second nodal plane is seen to be
m = y '/y = f/f ' =
n/n '.
It is left to the reader to show that the equations for
image formation
may be expressed in terms of the object distance v from N
and the image distance
v ' from N ' as

(54) 
and

(55) 
The points N and N '
may lie outside the optical system OO ' and also the
rays may not in fact pass through the nodal points.
Example 4
Compute the nodal points of the lens of example 2.
Figure 21.
Solution:
We had O 'F ' = 2.167 cm and OF = 4.167 cm.
The second nodal point is 3.333
cm to the left of F ' and the first
nodal point 4.333 cm to the right of
F as illustrated in Fig. 21. A ray
incident towards N emerges parallel to the
incident ray as if it came from N '.
The two negative nodal points, N()
and N '(), are two points on the optic
axis such that a ray incident on the
first point at an angle,
,
say, emerges
from the second point at an angle
of .
The angular magnification for
these points is then
'/
= 1.
With this definition it is left to the
reader to show that

(56) 

(57) 
and that

(58) 
It is further left as an exercise to show
that the equations for image
formation may be expressed in terms of the
object distance, w, from N() and
the image distance,
w ', from N '() as

(59) 
and

(60) 
Figure 22.
The ray diagram of Fig. 22 illustrates the
use of the negative nodal
points. The parallel rays through F and
N() are brought to a focus in the
second focal plane while the ray through
F is rendered parallel to the optic
axis.
Exercise 2
Compute the positions of all the cardinal
points of the lens in the example of
chapter 4.
Exercise 3
Compute the positions of all the cardinal
points of the lens in exercise 4.