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

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
represents
the angle the ray makes
with the *x* axis and since the slope
* (dy/dx) = *tan()
= for
small angles, we may
take
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.

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

(for small angles) and since

These linear transformation equations can be written as

(15) |

The 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

That is,

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 matrix.

In Fig. 4, VV represents the spherical refracting
surface separating
transparent media of indices *n* and
*n* '*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
to
*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* '

(16) |

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

or

or

Therefore we obtain

(17) |

Thus the transformation equations can be written for the plane through O using equation (16) and rearranging (17) as,

| (18) |

(19) |

or in matrix form

| (20) |

and the matrix,

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