Page 79 - Modern Optical Engineering The Design of Optical Systems

62 Chapter Four

At a location in the system where we know the data of all three rays,
we can determine the scaling constants by a simultaneous solution of
Eqs. 4.20 and 4.21, and get

A ( y u u y )/(u y y u ) (4.22)
3 2 3 2 2 1 2 1
B ( u y y u )/(u y y u ) (4.23)
3 1 3 1 2 1 2 1
Typically, the axial and principal rays are chosen as rays 1 and 2. The
third ray is often defined in object space, although it may be defined any-
where in the system where the data of all three rays are known. The
scaling factors A and B are calculated from Eqs. 4.22 and 4.23. Then,
after putting these values for A and B into Eqs. 4.20 and 4.21, the data
of ray 3 in image space can be found by inserting the image space data
of rays 1 and 2 into the resulting equations. (Assuming that the data of
ray 3 in image space is what is desired.)

Focal length determination
For example, if we have the raytrace data for the axial ray #1 and
oblique ray #2 and define ray 3 as having u 0 and y 1 in object
3 3
space, Eqs. 4.20 and 4.21 can give us the final ray height and slope in
image space for ray 3. Then, with the primed data (y′ and u′) indicat-
ing the values in image space, we get:
efl 1/u ′ (y u u y )/(u u ′ u u ′) (4.24)
3 1 2 1 2 1 2 2 1
bfl y ′ /u ′ (u y ′ u y ′)/(u u ′ u u ′) (4.25)
3 3 2 1 1 2 1 2 1 1
If we reverse the whole process and set u ′ 0 and y ′ = 1, we can get
3 3
the (normally negative) front focal length
ffl y /u ( u ′y )/(u u ′ u u ′) (4.26)
3 3 2 1 1 2 2 1

Formula for two specific rays
The above formulas are perfectly general. If we select certain rays to
trace, a simplified expression can often be derived. For example, the
OSLO reference manual gives the following variation on this scheme:
If we start ray #1 at the foot of the object, and have it strike the first
surface at y 1 ≡ y 3 , and if we start ray #2 at height h on the object and
send it through the center of the first surface, then the ray coordinates
in object space (i.e., at the object) are:
for ray #1: y 1 0 and u 1 y 3 /s,
for ray #2: y 2 h and u 2 h/s