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Optical System Considerations 57

Figure 4.3 A two-component system operating at finite conjugates.

when we are given the required system magnification, the positions of
the two components, and the object-to-image distance (neglecting the
spaces between the principal planes of the components). Thus, knowing
s, s′, d, and the magnification m, we wish to determine the powers (or
focal lengths) of the two components, which are given by

(ms md s′)
(4.9)
A
msd
(d ms s′)
(4.10)
B
ds′
In the second type of problem we are faced with the inverse case, in
that we know the component powers, the desired object-to-image dis-
tance, and the magnification; we must determine the locations for
the two components. Under these circumstances the mathematics
result in a quadratic relationship, and thus there may be two solu-
tions, one solution, or no solution (i.e., an imaginary solution). The
following quadratic equation (Eq. 4.11) in d (the spacing) is first
2
solved for d [using the standard equation x 5 s2 b 6 2b 2 4acd/2a
to solve 0 ax bx c].
2
2
2
(m 1) f A f B
0 d dT T (f f ) (4.11)
A B
m
Then s and s′ are easily determined from
(m 1) d T
s (4.12)
(m 1) md
A
s′ T s d (4.13)