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 (F1 and F2), the first and second principal points (H1 and H2), and the first and second nodal points (N1 and N2).
A ray from the first focal point F1 is rendered parallel to the axis and a ray parallel to the axis is refracted by the lens through the second focal point F2. The extensions of the incident and resultant rays in each case intersect, by definition, in the principal planes, and these cross the axis at the principle points H1 and H2. Once the principle planes are known, accurate ray diagrams can be drawn. The usual rays, determined by the focal points, bend at their intersections with the principal planes. The third ray usually drawn for thin-lens diagrams is one through the lens center, undeviated and negligibly displaced. The nodal points of a thick lens, or of any optical system, permit the correction to this ray. Any ray directed toward the first nodal point N1 emerges from the optical system parallel to the incident ray, but displaced so that it appears to come from the second nodal point on the axis N2.
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 V1 and V2, while f1 and f2 determine focal point positions relative to the principle points. It is important to note that these focal points are not measured from the vertices of the lens.
and the focal length f2 is conveniently expressed in terms of f1 by
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
The position of the nodal points are given by
Notice that when n = n', we get that r = v and s = w.
The ray is described at distance x0 from the first refracting surface in terms of its height y0 and slope angle 0 relative to the optical axis. Changes in angle occur at each refraction, such as at points 1 through 5, and at each reflection, such as point 6. The height of the ray changes during translations between these points. We want a procedure that will allow us to calculate the height and slope angle of the ray at any point in the optical system, for example at point 7, which is a distance x7 from the mirror.
Let the axial progress of the ray be L such that at point 1 the elevation and direction are given by y1 and 1 respectively. Then we have that
These equations may be put into matrix notation, where the paraxial approximation has been used,
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,
where is the index of refraction of the th medium. The transfer matrix then becomes
Since refraction occurs at a point, there is no change in elevation, and y = y'. The angle ', on the other hand, is given by
Using the paraxial form of Snell's law we get
Writing this and y = y' in matrix form, we get
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
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, 1997Comments to: D-Suson@tamuk.edu