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Principles of Radiometry and Photometry 259

Thus, if the angle subtended by the source is moderate, we can substi-
tute into Eq. 12.7 and write
H N (12.8)

If the point X does not lie on the “axis” (the normal through the cen-
ter of the circular source), then the irradiance would be subject to the
same factors outlined in the discussion of the “cosine-fourth” rule in
Sec. 9.7. Thus, if the line from the point X to the center of the circle
makes an angle to the normal, the irradiance at X is given by
4
H H cos (12.9)
0
where H 0 is the irradiance along the normal given by Eq. 12.7 or 12.8
and H is the irradiance at X (measured in a plane parallel to the
source). (See the note in Example 12.1 regarding the inaccuracy of the
cosine-fourth rule when the angles and are large.)
It is apparent that Eqs. 12.8 and 12.9 may be used in combination
to calculate the irradiance produced by any conceivable source config-
uration, to whatever degree of accuracy that time (or patience) allows.

12.6 The Radiometry of Images

The conservation of radiance
When a source is imaged by an optical system, the image has a radiance,
and it may be treated as a secondary source of radiation. However, one
must always keep in mind that the radiance of an image differs from
the radiance of an ordinary source in that the radiance of an image
exists only within the solid angle subtended from the image by the
clear aperture (or exit pupil) of the optical system. Outside of this
angle, the radiance of the image is zero.
At first consideration the conservation of radiance (or brightness)
seems quite counterintuitive. Ordinarily, the solid angle of radiation
accepted by an optical system from a source is quite small, as is the
fraction of the total power which passes through the lens and forms
the image. It is difficult to accept that the image formed by this small
fraction of the source power will have the same radiance as does the
source. We can easily demonstrate this, using only the first-order
optics from Chap. 2.
Let us assume a small source of radiance N with an area A. The
source thus has an intensity of AN. The source is imaged by an optical
system with an area P which is located a distance S from the source.
2
The solid angle subtended by the lens from the source is thus P/S ,
and the power intercepted by the lens and formed into the image is
2
ANP/S .