Page 299 - Modern Optical Engineering The Design of Optical Systems

278 Chapter Twelve

brightness of the image and the solid angle subtended by the image.
Thus, for points near the lens, the illumination is given by

E TB (12.19)

which the reader will recognize as Eq. 12.8 rewritten in photometric
symbols with a transmission constant (T) added. B is the brightness
of source S (since the brightness of an image equals the brightness of the
object) and is the solid angle subtended by the image. (We have tac-
itly assumed to be small.) Now for a point at the lens, it is obvious
that the solid angle subtended by the image S′ is exactly equal to the
solid angle subtended by the source S from the lens. Since S′ is at
infinity, this angle will not change as we shift our reference point a
short distance along the axis away from the lens, and the illumination
will remain constant in this region. However, at a distance
D (lens diameter)/ , the source image will subtend the same angle as
the diameter of the lens, and for points more distant than D, the size
of the solid angle subtended by the source of illumination will be
limited by the lens diameter. This solid angle will obviously be equal
2
to (area of lens)/d and the illumination beyond distance D will fall off
with the square of the distance (d) to the lens. Thus, the equations gov-
erning the illumination produced by a searchlight are

lens diameter
D (12.20)

for d ! D: E TB (a constant) (12.21)
TB (lens area)
for d " D: E (12.22)
d 2

The general technique used here is applicable to almost any illumina-
tion problem, and we can restate it in general terms as follows:

To determine the illumination at a point, the size and position of the
source image, as seen from the point, are calculated. The pupils and win-
dows of the system (again, as seen from the point) are determined. Then
the illumination at the point is the product of the system transmission,
the source brightness and the solid angle subtended by that area of the
source which can be seen from the point through the pupils and windows
of the system, multiplied by the cosine of the angle of incidence.

Note that for points (which lie within the beam) beyond the critical
distance D, the searchlight acts as if it were a source of a diameter
equal to that of the searchlight lens and a brightness TB. As mentioned