Page 61 - Modern Optical Engineering The Design of Optical Systems

44 Chapter Three

l′ 3 199.6846 as the spacing between surfaces #3 and #4, Eq. 3.17 can
be used to calculate y 4 , which is h′.
Similarly, Eq. 3.17 can be used to determine the initial slope angle
u 1 by regarding the object plane as surface zero and rearranging the
equation to solve for u′ 0 u 1 as shown below:

n′ u′
0
0
y y t
1 0 0 n′
0
y y 0 h y 1
1
u′ u
0 1
t l
0 1
Note that all paraxial rays from a given object point will intersect the
paraxial focus image plane at exactly the same point.

3.4 Calculation of the Focal Points
and Principal Points

In general, the focal lengths of an optical system can easily be calculated
by tracing a ray parallel to the optical axis (i.e., with an initial slope
angle u equal to zero) completely through the optical system. Then the
effective focal length (ef l) is minus the ray height at the first surface
divided by the ray slope angle u′ k after the ray emerges from the last
surface. Similarly, the back focal length (bf l) is minus the ray height
at the last surface divided by u′ k . Using the customary convention that
the data of the last surface of the system are identified by the sub-
script k, we can write
y 1
ef l (3.19)
u′
k
y k
bf l (3.20)
u′
k
The cardinal points of a single lens element can be readily deter-
mined by use of the raytracing formulas given in the preceding section.
The focal point is the point where the rays from an infinitely distant
axial object cross the optical axis at a common focus. As indicated, this
point can be located by tracing a ray with an initial slope (u 1 ) of zero
through the lens and determining the axial intercept.
The reader may wish to test his or her understanding and skill at
raytracing by calculating the focal lengths of the doublet lens of Fig. 3.4.
The results should be: