Chapter 4

The General Optical System

The General Matrix
Figure 10.

In the preceding discussion we were able to describe the passage of a ray from one point to another in a simple optical system by means of two translation matrices and one refraction matrix. A more complicated optical system may contain many refracting surfaces of different indices of refraction. Let us for example consider the system of Fig. 10. A ray passes the plane PP at height y, angle alpha.

$$\bmatrix{y\cr \alpha}=\bmatrix{1&-3\cr 0 & 1}\bmatrix{1&0\cr .1 & 1.5}\bmatrix{1&-7\cr 0 & 1}\bmatrix{1&0\cr .0667 & .667} \bmatrix{1&-4\cr 0 & 1}\bmatrix{1& 0\cr -.15 & 1.6}\bmatrix{1&-3\cr 0 & 1}\bmatrix{1&0\cr 0 & .625}\bmatrix{1&-2\cr 0 & 1} \bmatrix{y'\cr \alpha'}$$

These nine matrices may be multiplied together to give one two by two matrix relating y ' alpha ' and y alpha at the planes QQ and PP respectively. If the plane PP is moved to the vertex O and QQ to O ' then we have the matrix,
$$\bmatrix{1&0\cr .1 & 1.5} \bmatrix{1&-7\cr 0 & 1}\bmatrix{1&0\cr .0667 & .667} \bmatrix{1&-4\cr 0 & 1}\bmatrix{1& 0\cr -.15 & 1.6} \bmatrix{1&-3\cr 0 & 1}\bmatrix{1&0\cr 0 & .625},$$

which represents the effect on a ray in going from the plane through the first vertex O of the system through to the plane through the final vertex O ' of the system. It is useful in describing the general optical system to base our discussion on this matrix which we will designate as
$$\bmatrix{ a_{11} & a_{12}\cr a_{21} & a_{22} }_{OO'},$$

which in this example equals
$$\bmatrix{1.554 & -9.713 \cr 0.1654 & -0.390 }.$$

Obtaining the elements of the matrix may involve a lengthy series of matrix multiplications as in our example above, involving 7 matrices. However, all the necessary information about the optical system is included in this resultant two by two matrix and from it the cardinal points and planes of an optical system are readily obtainable. From these cardinal points we may completely describe the image-forming characteristics of the system.

Figure 11.

In general then, we can assume the situation of Fig. 11. We know (or can compute) the matrix relating the rays going from the plane through O to the plane through O ' namely

$$[{\bf A}]_{OO'}=\bmatrix{a_{11}&a_{12}\cr a_{21}&a_{22}}_{OO'}.$$

The matrix equation relating the emergent ray (y ' alpha ') at plane QQ a distance D ' from O ' to the incident ray (y alpha) at the plane PP a distance D to the left of O is

$$\bmatrix{y\cr \alpha}_P=\bmatrix{1&-D\cr 0 & 1} \bmatrix{a_{11}&a_{12}\cr a_{21}&a_{22}}_{OO'}\bmatrix{1&-D'\cr 0 & 1} \bmatrix{y'\cr \alpha'}_Q$$

or,
$$\bmatrix{y\cr \alpha}_P= \bmatrix{a_{11}-D a_{12}&D'(-a_{11}+D a_{21})+a_{12}-Da_{22}\cr a_{21}&a_{22}-D'a_{21}}_{PQ}\bmatrix{y'\cr \alpha'}_Q . \eqno(22)$$ (22)

In Fig. 10 we had D ' = 2 cm and D = 3 cm.

We are now in a position to discuss the first two cardinal points (and planes), the first and second focal points (and planes).