Chapter 3

The Basic Matrix Transformations of Geometrical Optics

In this chapter we shall first define the variables y and alpha 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

Figure 1

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 alpha 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 alpha represents the angle the ray makes with the x axis and since the slope (dy/dx) = tan(alpha) = alpha for small angles, we may take alpha 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 alpha is determined from the sign of the slope and is positive in Fig. 1.

2. The Translation Matrix

Figure 2

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, alpha, at PP and y ', alpha ', at QQ. From Fig. 2 we may write

$$ y = y' -D \alpha', {\rm (for\  small\  angles)}$$

(for small angles) and since alpha ' equals alpha we may write another linear equation as

$$  \alpha = 0 y'+ \alpha'.$$

These linear transformation equations can be written as
$$\bmatrix{y\cr \alpha} = \bmatrix{1&-D\cr 0 &1}
\bmatrix{y'\cr \alpha'}.\eqno(15)$$ (15)

The matrix,
$$  \bmatrix{1&-D\cr 0 &1}$$
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 alpha '' and height y ''. In matrix form we could write

$$\bmatrix{y\cr \alpha} = \bmatrix{1&-D\cr 0 &1}
\bmatrix{1&-T\cr 0 &1}
\bmatrix{y''\cr \alpha''}=
\bmatrix{1&-(D+T)\cr 0 &1}
\bmatrix{y''\cr \alpha''}.$$

That is,
$$  y =  y'' - (T + D) \alpha'' ,$$


which can be verified by examining Fig. 3.
Figure 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 matrix.

Figure 4.

In Fig. 4, VV represents the spherical refracting surface separating 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 alpha may be considered to refer to the plane through O perpendicular to the optic axis.

At the surface the angle alpha changes from alpha to alpha ' 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
$$y=y'+0\alpha'. \eqno(16)$$ (16)

We may now derive the transformation equation for alpha" at this surface using Snell's law. From Fig. 4,

$$  n \theta = n'\theta' {\rm(approximately)}$$

$$  n (\delta+\alpha) = n' (\delta+\alpha') $$

$$n'\alpha' = n \alpha+(n-n')\delta$$
$$\delta = y'/R.$$

Therefore we obtain
$$n \alpha = n' \alpha' + (n-n') y'/R.\eqno(17)$$ (17)

Thus the transformation equations can be written for the plane through O using equation (16) and rearranging (17) as,
$$y=y'+0\alpha', \eqno(18)$$

$$\alpha = ({n'-n\over nR})y'+({n'\over n}) \alpha', \eqno(19)$$ (19)

or in matrix form
$$\bmatrix{y \cr \alpha}\bmatrix{1&0\cr
{{\displaystyle n'-n}\over {\displaystyle nR}}&
{{\displaystyle n'}\over {\displaystyle n}} }
\bmatrix{y' \cr \alpha'}\eqno(19)$$

and the matrix,
{{\displaystyle n'-n}\over {\displaystyle nR}}&
{{\displaystyle n'}\over {\displaystyle n}} }$$

is defined as the refraction matrix for the plane through the vertex O of the refracting surface.