
126 OPTICS FOR ENGINEERS
The derivative is easy because the equation is quadratic in q, and the result is
q
opt
=−
2
n
2
−1
p
n +2
, (5.45)
leading to
x
(
x
1
)
=
x
3
1
s
(
0
)
8f
3
−np
2
n +2
+
n
n −1
. (5.46)
Now we can see in Equation 5.42 that lenses of different shapes will have different aber-
rations, and with the optimal q from Equation 5.45, we can use Equations 5.40 as the second
constraint on our designs.
The optimal lens shape given by Equation 5.45 does not depend on x
1
, so the best choice of
surface curvatures is independent of the aperture. However, the amount of aberration depends
strongly on the aperture. Writing Equation 5.46, in terms of the f -number,