Page 280 - Modern Optical Engineering The Design of Optical Systems

260 Chapter Twelve

The lens will form an image with a magnification M, and the area of
2
the image will thus be AM . The image distance will be MS, and the
2
solid angle subtended by the lens from the image will be P/M S ster.
2
2
Thus the power in the image (ANP/S ) is spread over the image area
2
2
(AM ) and exists only over the solid angle (P/M S ). The image radi-
2
ance is power per unit area per solid angle; combining the expressions
above, we get (neglecting any transmission losses)
Image radiance power/area solid angle
2
(ANP/S )

2
2
2
(AM ) (P/M S )
We can cancel A, P, S, and M, leaving us with
Image radiance N (the object radiance)
which is a statement of the conservation of radiance (or brightness).
The conservance of radiance (or brightness/luminance) states that
the radiance of an image formed by an optical system is equal to the
radiance of the object, mutiplied by the transmission of the system.
More precisely, the radiance divided by the square of the index is the
invariant quantity. Thus (with the object and image both in air) we
have
N′ tN (12.10a)

and more generally,
N′ tN (n′/n) 2 (12.10b)
where N and N′ are the radiance and n and n′ are the indices of object
and image space respectively, and t is the system transmission.
Another way of expressing Eq. 12.10a is: In air, the radiance of an image
cannot exceed the radiance of the object. Note that the index factor
(n′/n) can also be applied to Eq. 12.11 for the irradiance H.
2

The irradiance of an image
By the application of exactly the same integration technique used in
Sec. 12.5, it can be shown that the irradiance produced in the plane of
an image is given by

2
H T N sin ′ watt/cm 2 TN (for small angles) (12.11)
2
where T is the system transmission, N (W ster 1 cm ) is the object
radiance, and ′ is the half angle subtended by the exit pupil of the
optical system from the image. Small or noncircular exit pupils and