Page 77 - Modern Optical Engineering The Design of Optical Systems

60 Chapter Four

Etendue
Note that etendue or throughput, used in radiometry and radiative
transfer considerations, is the pupil aperture area times the solid
angle field of view, or the object/image area times the solid angle of the
acceptance/imaging cone, and is thus related to the square of the opti-
cal invariant.

Example 4.1
We can apply the invariant to the calculation made in Example 3.1 by
assuming that only the axial ray has been traced. The axial ray slope at
the object was 0.0333 . . . and the corresponding computed slope at the
image was found to be 0.047555. . . . Since the object and image were
both in air of index 1.0, we can find the image height from Eq. 4.16,
h′ h′ nu 1.0 ( 0.0333 . . . )
m
h 20 n′u′ 1.0 ( 0.047555 . . . )
20 ( 0.0333)
h′
0.047555)
h′ 14.0187

This value agrees with the height found in Example 3.1 by tracing a
ray from the tip of the object to the tip of the image. The saving of
time by the elimination of the calculation of this extra ray indicates
the usefulness of the invariant.

Image height for object at infinity
Another useful expression is derived when we consider the case of a
lens with its object at infinity. At the first surface the invariant is

Inv y n (0) y nu y nu
p 1 p 1 p
since the “axial” ray from an infinitely distant object has a slope angle
u of zero. At the image plane y p is the image height h′, and y for the
“axial” ray is zero; thus
Inv h′n′u′ (0) n′u′ h′n′u′
p
Equating the two expressions for Inv, we get

h′n′u′ y nu (4.18)
1 p
ny
1
h′ u
p
n′u′