Page 63 - Modern Optical Engineering The Design of Optical Systems

46 Chapter Three

And the final slope is found by Eq. 2.31a:

n′ u′ n′ u′ y (n′ n ) c
2 2 1 1 2 2 2 2
(n 1)
(n 1) y c y 1 tc (1 n) c
1 1 1 n 1 2
(n 1)

(1.0)u′ u′ y (n 1) c c tc c
2 2 1 1 2 1 2 n
Thus the power (or reciprocal focal length) of the element is
expressed as
1 u′ (n 1)

2 (n 1) c c tc c (3.21)
1 2
1
2
f y n
1
or, if we substitute c 1/R,
1 1 1 t (n 1)

(n 1) (3.21a)
f R R R R n
1 2 1 2
The back focal length can be found by dividing y 2 by u′ 2 to get
ft (n 1)
y 2
bf l f (3.22)
u′ nR
2 1
The distance from the second surface to the second principal point is
just the difference between the back focal length and the effective focal
length (see Fig. 3.6); this is obviously the last term of Eq. 3.22.
The above procedure has located the second principal point and
second focal point of the lens. The “first” points are found simply by
substituting R 1 for R 2 and vice versa.
The focal points and principal points for several shapes of elements
are diagramed in Fig. 3.7. Notice that the principal points of an
equiconvex or equiconcave element are approximately evenly spaced
within the element. In the plano forms, one principal point is always
at the curved surface and the other is about one-third of the way into
the lens. In the meniscus forms, one of the principal points is completely
outside the lens; in extreme meniscus shapes, both the principal points
may lie outside the lens and their order may be reversed from that
shown. Note well that the focal points of the negative elements are in
reversed order compared to a positive element.
If the lens element is not immersed in air, we can derive a similar
expression for it. Assuming that the object medium has an index of n 1 ,
the lens index is n 2 , and the image medium has an index of n 3 , then the