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

Thick Lenses.

Cardinal Points

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.

Equations Governing Thick Lenses

                               ,                           (10.1)

and the focal length f₂ is conveniently expressed in terms of f₁ by

                                                     .                                                               (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.

Matrix Methods

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.

Ray Transfer Matrix

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

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)

Refraction Matrix

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)

Reflection Matrix

Finally, consider reflection at a spherical surface. Since a mirror is so much like a lens, the reflection matrix is very similar to the refraction matrix. To find the reflection matrix, we make the substitution . This leads to the reflection matrix

                                                      .                                                    (10.10)

System Matrix

                                             .                         (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