Page 202 - Modern Optical Engineering The Design of Optical Systems

Stops, Apertures, Pupils and Diffraction 185

Cosine-to-the-fourth
For off-axis image points, even when there is no vignetting, the illumina-
tion is usually lower than for the image point on the axis. Figure 9.9 is
a schematic drawing showing the relationship between exit pupil and
image plane for point A on axis and point H off axis. The illumination
at an image point is proportional to the solid angle which the exit pupil
subtends from the point.
The solid angle subtended by the pupil from point A is the area of
the exit pupil divided by the square of the distance OA. From point H,
the solid angle is the projected area of the pupil divided by the square
of the distance OH. Since OH is greater than OA by a factor equal to
1/cos , this increased distance reduces the illumination by a factor of
2
cos . The exit pupil is viewed obliquely from point H, and its projected
area is reduced by a factor which is approximately cos . (This is a fair
approximation if OH is large compared to the size of the pupil; for
high-speed lenses used at large obliquities, it may be subject to signifi-
cant errors. See Example A in Chap. 12 for an exact expression.)
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Thus the illumination at point H is reduced by a factor of cos . This
is, however, true for illumination on a plane normal to the line OH
(indicated by the dashed line in Fig. 9.9). We want the illumination in
the plane AH. An illumination of x lumens per square foot on the
dashed plane will be reduced on plane AH because the same number
of lumens is spread over a greater area in plane AH. The reduction
factor is cos , and combining all the factors we find that

4
Illumination at H cos (illumination at A) (9.5)
The importance of this effect on wide-angle lenses can be judged from
4
4
the fact that cos 30° 0.56, cos 45° 0.25, and cos 60° 0.06. It
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can be seen that the illumination on the film in a wide-angle camera
may fall off quite rapidly.

Figure 9.9 Relationship between
exit pupil and image points,
used to demonstrate that the
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illumination at H is cos times
that at A.