Matrix
Techniques in Optics
Deducing
the Properties of an Optical System from its Matrix
Suppose we have multiplied the various matrices for an optical system and have come up with the following for the composite system:
and we have made the matrix unimodular, i.e. AD - BC = 1. Let us investigate the implications of the composite matrix having one of the elements equal to zero.
Let's start with that unimodular requirement - remember that if the individual matrices are unimodular, so will be the product. The matrices for the drift space and thin lens are unimodular. The other two have determinants equal to 1/n if we go from air to n. However, we would expect them to occur in the sequence air-to-n-to-air in which case the product will have unity determinant.
If the matrix is unimodular and A (or D) is zero, then B = 1/C. Similary if B (or C) is zero then A = 1/D.
A = 0
This means that :
or that all parallel rays coming into the system end up at the same vertical point. Furthermore, if the incident angle is zero then the rays are brought to focus at:
B = 0
This means that
or that all the rays leaving the point y1 (no matter what angle) meet at the same point y2. So we have object points focused on image points and the magnification is A.
C = 0
This means that
or that parallel rays incident at the incident angle will emerge parallel at the emergent angle.
D = 0
This means that
or that all rays leaving the
same point (independent of angle) will emerge parallel at angle
.