We now turn to more complex optical systems, including systems which have lenses that are thick enough that we must consider the refractions at the leading and trailing surfaces separately. To start, consider a spherical **thick lens**, that is a lens whose thickness along its optical axis cannot be ignored without leading to serious errors in analysis. Exactly when a lens moves from the thin to thick category depends on the accuracy required. We can treat the thick lens exactly the way we described the thin lens; viewing it as a glass medium bounded by two spherical refracting surfaces. The image of a given object, formed by refraction at the first surface, becomes the object for refraction at the second surface. The object distance for the second surface takes into account the thickness of the lens. The image formed by the second surface is then the final image due to the action of the composite thick lens.

## Cardinal Points

The thick lens can also be described in a way that allows graphical determination of images corresponding to arbitrary objects, much like the ray rules for a thin lens. This description, in terms of the **cardinal points **of the lens, is useful because it can be applied to more complex optical systems.

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 **(*F _{1}* and

*F*), the first and second

_{2}**principal points**(

*H*and

_{1}*H*), and the first and second

_{2}**nodal points**(

*N*and

_{1}*N*).

_{2}A ray from the first focal point *F _{1}* is rendered parallel to the axis and a ray parallel to the axis is refracted by the lens through the second focal point

*F*. The extensions of the incident and resultant rays in each case intersect, by definition, in the

_{2}**principal planes**, and these cross the axis at the principle points

*H*and

_{1}*H*. 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

_{2}*N*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

_{1}*N*.

_{2}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 *V _{1}* and

*V*, while

_{2}*f*and

_{1}*f*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.

_{2}## Equations Governing Thick Lenses

We can summarize the basic equations for the thick lens without proof. Utilizing the symbols defined above, the focal length *f _{1}* is given by

and the focal length *f _{2}* is conveniently expressed in terms of

*f*by

_{1}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*.

## Matrix Methods

When the optical system consists of several elements, we need a systematic approach that facilitates analysis. As long as we restrict our analysis to paraxial rays, this systematic approach is well handled by the matrix method. The figure below shows the progress of a single ray through an arbitrary optical system.

The ray is described at distance *x _{0}* from the first refracting surface in terms of its height

*y*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*

_{0}*x*from the mirror.

_{7}## Ray Transfer Matrix

Consider a simple translation of the ray in a homogeneous medium.

Let the axial progress of the ray be *L* such that at point 1 the elevation and direction are given by *y _{1}* and

*respectively. Then we have that*

_{1}and

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

## Refraction Matrix

Consider next the refraction of a ray at a spherical surface separating media of refractive indices *n* and *n'*.

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

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

## 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**

## System Matrix

The matrix description of a lens can be simplified by realizing that when a ray passes through the lens, refraction occurs *twice*, once at each surface, with a translation between the surfaces. Therefore, in order to describe the lens, we can combine the three relevant matrices together to form the **system matrix**

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 r_{7} = **T**_{7}**MT**_{6}**S**_{5}**S**_{4}**T**_{3}**S**_{2}**T**_{1} r_{0}.

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

Source: D-Suson@tamuk.edu