Page 322 - Modern Optical Engineering The Design of Optical Systems

Optical System Layout 301

a 4 scope of 10-in length; we have not, however, included the eye
relief in these equations. To resolve this situation numerically, we
would now assume some reasonable value for f o , then proceed to test
various values of f e , selecting the value of f e which yields the desired
value for the eye relief R. Since R is not a critical dimension, a graphic
solution (after a few values of f e have been tried), plotting R versus f e
would be quite adequate for our purpose. Repeating the process for
several additional values of f o would then indicate the range of solutions
available.
To arrive at a solution analytically, we would proceed as follows: a
principal ray, starting at the center of the objective lens with some
arbitrary slope angle would be ray-traced by thin-lens equations (4.1,
4.2, and 4.3), using the symbolic values for the spacings and lens powers
derived from the three equations immediately preceding. The symbolic
values for the powers and spacings involved would thus be:

f o (L f o f e)
First airspace f s f
o 1 o
(Mf f )
e o
1 (Mf f ) 2
e
o
Erector power
r
f Mf f (L f f )
r e o o e
Mf e (L f o f e)
Second airspace s f f
2 e e
(Mf f )
e o
1
Eyelens power
e
f
e
The expression for the final intercept length of this ray, l′ e y e /u′ e
is then equated to the eye relief R, and a solution for f e expressed in
terms of f o , M, L, and R is extracted. As can be imagined, the procedure
is lengthy and the probability of making an error in the derivation is
approximately unity for the first few attempts. Careful work and fre-
quent checking are not only advisable, they are mandatory. When the
smoke has cleared away, one finds that
2
2
M RL f o (M R L)
f e
2
M (R L) f o (M 1) 2
and that for any chosen value for f o (which is less than L and more
than zero), a set of powers and spacings can be obtained which will
satisfy our original conditions for power M, length L, and eye relief R.
We are now faced, regardless of whether we have arrived via num-
bers or symbols, with the problem of determining what is a suitable