Page 281 - Modern Optical Engineering The Design of Optical Systems

Principles of Radiometry and Photometry 261

cylindrical lens systems can be handled by substituting the solid angle
2
for sin ′ (just as in Eq. 12.8); image points off the optical axis are
subject to the cosine-fourth law in addition to any losses due to
vignetting (Eq. 12.9 and Sec. 9.7).
The similarity between the equations for the irradiance produced by
a diffuse source and by an optical system makes it apparent that,
when it is viewed from the image point, the aperture of the optical
system takes on the radiance of the object it is imaging. This is an
extremely useful concept; for radiometric purposes, a complex optical
system can often be treated as if it consisted solely of a transmission
loss and an exit pupil with the same radiance as the object. Similarly,
when an optical system produces an image of a source, the image can
be treated as a new source of the same radiance (less transmission
losses). Of course, the direction that radiation is emitted from the
image is limited by the aperture of the system.
When an object is so small that its image is a diffraction pattern (Airy
disk), then the preceding techniques, which apply to extended sources,
cannot be used. Instead, the power intercepted by the optical system,
reduced by transmission losses, is spread into the diffraction pattern.
To determine the irradiance (or the radiance) of the image, we note
that 84 percent of the power intercepted and transmitted by the lens is
concentrated into the central bright spot (the Airy disk). A precise deter-
mination of irradiance requires that one integrate the relative irradiance-
times-area product over the central disk and equate this to 84 percent of
the image power. If P is the total power in the Airy pattern, H 0 the irra-
diance at the center of the pattern, and z the radius of the first dark ring,
a numerical integration of Eq. 9.14 over the central disk yields

0.84P 0.72H z 2
0
Rearranging and substituting the value of z given by Eq. 9.16, we get

P NA 2
H 1.17 P
0
z 2
where is the wavelength and NA is n′ sin U′, the numerical aperture.
The irradiance for points not at the center of the pattern is then found
by Eq. 9.14. Note that the preceding assumes a circular aperture; for
rectangular apertures, the process would be based on Eq. 9.12.

Example 12.1
In Fig. 12.4, A is a circular source with a radiance of 10 W ster 1 cm 2
radiating toward plane BC. The diameter of A subtends 60° from point B.
The distance AB is 100 cm and the distance BC is 100 cm. An optical
system at D forms an image of the region about point C at E. Plane BC