- Cardinal Points
- Equations Governing Thick Lenses
- Matrix Methods
- Ray Transfer Matrix
- Refraction Matrix
- Reflection Matrix
- System Matrix

There are six cardinal points on the axis of
a thick lens, from which its imaging properties can be deduced. Planes
normal to the axis at these points are called **cardinal planes**. The
six cardinal points consist of the first and second system **focal points
**(*F _{1}* and

A ray from the first focal point *F _{1}*
is rendered parallel to the axis and a ray parallel to the axis is refracted
by the lens through the second focal point

The positions of all six cardinal points are shown in the figure below:

Distances are *directed*, positive or negative
by a sign convention that makes distances directed to the left negative
and distances to the right positive. Notice that for the thick lens, the
distances *r* and *s* determine the positions of the principle
points relative to the vertices *V _{1}* and

, (10.1)

and the focal length *f _{2}* is
conveniently expressed in terms of

. (10.2)

Notice that the two focal lengths have the same
magnitude if the lens is surrounded by a single refractive medium, so that
*n' = n*. The principal planes can be located next using

. (10.3)

The position of the nodal points are given by

. (10.4)

Notice that when *n = n'*, we get that *r
= v* and *s = w*.

The ray is described at distance *x _{0}*
from the first refracting surface in terms of its height

Let the axial progress of the ray be *L*
such that at point 1 the elevation and direction are given by *y _{1}*
and

and

These equations may be put into matrix notation, where the paraxial approximation has been used,

. (10.5)

This is known as the **ray transfer matrix**,
and it represents the effect of the translation on the ray. It can be written
in standard form by rescaling the vector describing the ray,

, (10.6)

where
is the index of refraction of the * ^{th}*
medium. The transfer matrix then becomes

. (10.7)

Since refraction occurs at a point, there is
no change in elevation, and *y = y'*. The angle *'*,
on the other hand, is given by

Similarly,

Using the paraxial form of Snell's law we get

Writing this and *y = y'* in matrix form,
we get

(10.8)

This is known as the **refraction matrix**.
Notice that we use the same sign convention as earlier. If the surface
is convex, *R* is positive, but if the surface were to be concave,
then *R* is negative. Furthermore, if we allow ,
we get the appropriate refraction matrix for a plane interface. Using
to write this in standard form yields

. (10.9)

. (10.10)

. (10.11)

Notice that the matrices are multiplied together from right to left.

Using these matrices, any arbitrary lens and mirror system can be analyzed. The various elements in the system are multiplied together, with the later matrices being inserted on the left. For example, the general equation describing the lens system given earlier would be

As can be seen, these problems can become very
complicated very rapidly. One way to simplify the problem is to recognize
that all of the matrices which describe a lens system are **unitary**;
that is they have a determinant of 1. The result of multiplying together
unitary matrices is another unitary matrix, so the final matrix should
also have a determinant of one.

Last updated: July 23, 1997

Comments to: D-Suson@tamuk.edu