Chapter 4
The General Optical System
The General Matrix
Figure 10.
In the preceding discussion we were able to describe the passage of a
ray from one point to another in a simple optical system by means of
two translation matrices and one refraction matrix. A more complicated
optical system may contain many refracting surfaces of different indices
of refraction. Let us for example consider the system of
Fig. 10.
A ray passes the plane PP
at height y, angle .
These nine matrices may be multiplied together to give
one two by two matrix
relating y '
'
and y
at the planes QQ and PP respectively.
If the plane PP
is moved to the vertex O and QQ to O ' then we have the
matrix,
which represents the effect on a ray in going from the plane
through the first vertex O of the system through to the
plane through the final vertex O ' of the system.
It is useful in describing the general optical system to base our
discussion on this matrix which we will designate as
which in this example equals
Obtaining the elements of the matrix may involve a lengthy series of
matrix multiplications as in our example above, involving 7 matrices.
However, all the necessary information about the optical system is
included in this resultant two by two matrix and from it the cardinal
points and planes of an optical system are readily obtainable.
From these cardinal points we may completely
describe the imageforming characteristics of the system.
Figure 11.
In general then, we can assume
the situation of Fig. 11. We know (or can
compute) the matrix relating the rays going from the plane through
O to
the plane through O ' namely
The matrix equation relating the emergent
ray (y '
') at plane QQ a distance
D ' from O ' to the incident ray
(y ) at the
plane PP a distance D to the left
of O is
or,

(22) 
In Fig. 10 we had D ' = 2 cm and D = 3 cm.
We are now in a position to discuss the first two cardinal points (and
planes), the first and second focal points (and planes).