Page 301 - Modern Optical Engineering The Design of Optical Systems

280 Chapter Twelve

could achieve would be that of the diffuser, which would be considerably
less than that of the lamp. The function of the condenser is to image the
source in the pupil of the projection lens so that the lens aperture has the
same brightness as the source. When this is done, the screen is illumi-
nated according to Eq. 12.11, where the solid angle is that subtended by
the source image (in the projection lens) from the screen. It is apparent
that the maximum value for the screen illumination is limited by the size
of the projection lens aperture. Therefore, the maximum screen illumi-
nation is achieved when the image of the source completely fills the aper-
ture of the lens. This is required for all points within the field of view, and
the condenser diameter must be sufficiently large so that it does not
vignette, if maximum illumination at the edge of the picture is required.
In this regard, note that the ray from the corner of the film to the oppo-
site edge of the lens aperture is the most demanding. The cosine-fourth
rule will, of course, reduce the illumination at points off the axis.
From the above, one might conclude that with a condenser of suffi-
cient magnification, the image of a very small source could be magni-
fied enough to fill the pupil of the projection lens. The necessary
illuminating cone angle is determined by the film gate and its dis-
tance to the lens pupil (i.e., to the image of the source). In Chap. 2 we
found that the magnification was given by m h′/h u/u′. The Abbe
sine condition uses m sin u/sin u′ for systems of reasonable image
quality. Since u′ in this case is fixed by the film gate, it is apparent
that a large magnification will require a large value of u. The largest
value that u can have is 90° with a sine of one; this establishes the
limit on the magnification that can be attained. This limit can be
expressed as

P
! 1.0 (12.24)

nS
where P is the aperture of the projection lens, is the half-field angle
of projection, n is the index in which the source is immersed (usually
n 1.0 for air), and S is the size of the source. It is impossible for
Eq. 12.24 to exceed a value of 1.0; a value of 0.5 is typical of many sys-
tems. Note that a value of 0.5 corresponds to a working speed of f/1.0
and that a value of 1.0 would require a working speed of f/0.5. (Eq. 12.24
is analogous to Eq. 13.22 for detector systems.)
When the source is irregular in shape, as in “V” filament lamps for
example, the solid angle for Eq. 12.11 is determined just as one might
expect, by dividing the area of the actual image of the filament by the
square of the distance to the screen. Condenser design is discussed in
Sec. 17.4.