The Basic Matrix Transformations of Geometrical Optics
In this chapter we shall first define
the variables y and that were
used in the preceding pages. Then we shall derive two different linear
transformations, one corresponding to a translation,
the other to a refraction
operation. They will be characterized by a translation
matrix and a refraction matrix respectively.
With these two matrices we shall be able to treat any
optical system according to the first order theory.
1. The Variables
Let us consider a ray of light EF as in Fig. 1 which is traversing a
uniform portion of a transparent
medium which forms part of an optical system.
The ray makes a small angle with respect to the x
direction which we will
designate as the optic axis of the system.
The centres of curvature of all
the spherical refracting surfaces lie
on the optic axis. We shall choose a
plane PP which is perpendicular to the optic axis.
The meaning of the variables y and
can be seen
from Fig. 1. For a
given plane PP the variable y represents the height
above the x axis that the
ray strikes the plane PP. The variable
the angle the ray makes
with the x axis and since the slope
(dy/dx) = tan()
small angles, we may
as representing the slope of the ray as it passes
the plane PP. The
distance y above the plane is positive, below the plane
negative. The sign
of the angle
is determined from the sign of the slope
and is positive in Fig. 1.
2. The Translation Matrix
In Fig. 2 the ray EF is progressing (without change in direction)
through the uniform medium as described for Fig. 1.
We shall always draw
the ray as progressing from left to right in the x
direction. We may
designate two planes, one PP and the other QQ,
a distance D from PP; the
ray is characterised by the variables y,
at PP and y ',
From Fig. 2 we may write
(for small angles) and since
we may write another linear equation as
These linear transformation equations can be written as
is known as the translation matrix.
A series of translations may be represented according to the rules of
matrix multiplication already discussed. For example,
suppose that the ray
strikes a further plane WW at a distance T from QQ at
angle '' and height y ''.
In matrix form we could write
which can be verified by examining Fig. 3.
The distances T and D are always taken as positive quantities as the
transformation is defined in terms of the ray direction.
With the convention
chosen here for expressing the unprimed coordinates as
functions of the primed
coordinates as in equation 15, the order of the matrices
is in the direction of
the ongoing ray, i.e. from left to right.
3. The Refraction Matrix
At each surface separating two transparent media
within an optical system,
the direction of the ray may be changed according
to Snell's law, which we take
in the approximate form for small angles. We may
express this refraction at
the surface in terms of two linear equations which
gives rise to the refraction
In Fig. 4, VV represents the spherical refracting
transparent media of indices n and
n ' respectively.
The intersection of
the surface and the optic axis is the vertex O.
The radius of curvature is R
and is considered positive (as in the figure) if
the direction OC is the
positive x direction where C is the centre of curvature
of the surface. Since
we have confined the discussion to small angles and
rays close to the axis,
the variables y and
may be considered to refer to
the plane through O
perpendicular to the optic axis.
At the surface the angle changes from
' while the
variable y is
simply the height above the x axis of the point at which
the ray strikes the
surface and refraction takes place. Thus we can write
either y or y ' for this
height and the transformation equation may be written as
We may now derive the transformation equation for
Snell's law. From Fig. 4,
Therefore we obtain
Thus the transformation equations can be written for the
plane through O using
equation (16) and rearranging (17) as,
or in matrix form
and the matrix,
is defined as the refraction matrix for the plane through
the vertex O of the