Page 78 - Modern Optical Engineering The Design of Optical Systems

Optical System Considerations 61

which is useful for systems where the object and image are not in air.
If both object and image are in air, we set n n′ 1.0, and recalling
that f y 1 /u′, we find

h′ u f (4.19)
p
tan U f (for nonparaxial rays)
P

Telescopic magnification
If we evaluate the invariant at the entrance and exit pupils of a system,
y p is (by definition) equal to zero, and the invariant becomes

Inv ynu y′n′u′
p p
where y is the pupil semidiameter, and u p is the angular half field of
view. For an afocal system we can equate the invariant at the entrance
and exit pupils and then solve for the afocal (or telescopic) angular
magnification to get
u′ yn
p
MP
u
p y′n′
which indicates that the telescopic magnification is equal to the ratio
of entrance pupil diameter to exit pupil diameter (assuming that n n′).
This is discussed further in Chap. 13.

Data of a third ray from two traced rays
As one might suspect from the preceding, an optical system is actually
completely defined by the paraxial raytrace data of any two unrelated
rays, i.e., rays with different axial intersections.
Paraxial raytrace data may be scaled. In other words the ray
heights and slopes can be multiplied or divided by a scaling constant;
the result is a new raytrace. The new ray will still intersect the axis
at the same point(s) as the old, but the ray heights and slopes will be
different.
If we treat raytrace data as just a set of equations, or equalities, it
is apparent that since one may add or subtract equalities and thereby
obtain another equality, one can add the scaled data (ray height or
slope) of two rays and get the data for a third ray. If A and B are scaling
constants, we can express the data of the third ray as the sum of the
scaled data of rays 1 and 2.
y Ay By (4.20)
3 1 2
u 3 Au Bu (4.21)
1 2