Page 42 - Modern Optical Engineering The Design of Optical Systems

Gaussian Optics: The Cardinal Points 25

arrow O′A′. A plane object normal to the axis is imaged as a plane, also
normal to the axis. See Sec. 4.5 for a tilted object.
A third ray could be constructed from O to the first nodal point; this
ray would appear to emerge from the second nodal point and would be
parallel to the entering ray. If the object and image are both in air, the
nodal points coincide with the principal points, and such a ray is
drawn from O to P 1 and from P 2 to O′, as indicated by the dashed line
in Fig. 2.3.
At this point in our discussion, it is necessary to adopt a convention
for the algebraic signs given to the various distances involved. The
following conventions are used by most workers in the field of optics.
There is nothing sacrosanct about these conventions, and many optical
workers adopt their own, but the use of some consistent sign conven-
tions is a practical necessity.
1. Heights above the optical axis are positive (e.g., OA and P 2 B).
Heights below the axis are negative (P 1 C and A′O′).
2. Distances measured to the left of a reference point are negative; to
the right, positive. Thus P 1 A is negative and P 2 A′ is positive.
3. The focal length of a converging lens is positive and the focal length
of a diverging lens is negative.

Image position
Figure 2.4 is identical to Fig. 2.3 except that the distances have been
given single letters; the heights of the object and image are labeled h
and h′, the focal lengths are f and f′, the object and image distances
(from the principal planes) are s and s′, and the distances from focal
point to object and image are x and x′, respectively. According to our
sign convention, h, f, f′, x′, and s′ are positive as shown, and x, s, and
h′ are negative. Note that the primed symbols refer to dimensions
associated with the image and the unprimed symbols to those associ-
ated with the object.

Figure 2.4 The ray sketch of Fig. 2.3 with the distances labeled as focal length or object
and image distances.