Page 278 - Modern Optical Engineering The Design of Optical Systems

258 Chapter Twelve

where N is the radiance of the source. Since the distance from dA to X
is S/cos , and the radiation arrives at an angle , the incremental irra-
diance at X produced by dA is
3
N dA cos
4
cos

dH J cos (12.5)
0 S 2 S 2
The same irradiance is produced by each incremental area making up
a ring of radius r and a width dr, so that we can substitute the area of
the ring, 2 r dr, for dA in Eq. 12.5 to get the incremental irradiance
from the ring.
4
2 r dr N cos
dH (12.6)
S 2
To simplify the integration, we substitute
r S tan
dr S sec d
2
into Eq. 12.6 to get
2
4
2 S tan S sec d N cos
dH
S 2
2 N tan cos d 2 N sin cos d
2
Integrating to determine the irradiance from the entire source, we get
H
m 2 N sin cos d 2 N

2
sin
0 2 0
2
H N sin watt/cm 2 (12.7)
m
where H is the irradiance produced at a point by a circular source of
radiance N W ster 1 cm 2 which subtends an angle of 2 m from the
point (when the point is on the “axis” of the source). Note well that m
is the angle defined by the source diameter.
Unfortunately noncircular sources do not readily yield to analysis.
However, small noncircular sources may be approximated with a fair
degree of accuracy by noting that the solid angle subtended by the
source from X is
2
sin
2 (1 cos ) 2
(1 cos )
and for small values of , cos approaches unity and

sin
2