At 23:33 12/30/01, William Sommerwerck wrote:
As I am persona non grata with this group, I have not posted in many a
moon. However, I can't let the discussion about cos^4 falloff pass without
mention.
Don't quite know why . . . but whatever it was is apparently water long
under the bridge . . . welcome back.
There are four cosine-related sources of falloff in a simple lens. (It's
been a few years since I read a Modern Photography article about this, so
the following might not be completely correct.) The first two are the
off-axis narrowing of the lens's entrance and exit pupils. The third is
the spreading of a point source as it strikes the film at an angle. The
fourth -- I think -- has something to do with the inverse-square law of
illumination. But I'm not sure. (Yes, that would make it cos^5. I told you
I didn't remember all of it.)
Note: this only applies to rectilinear lenses, *not* fisheyes!
You have the causes and almost had the math. The first is the aperture
(entrance pupil) appearing as an ellipse (approximately with polygonal
apertures) to off-axis light sources. Therefore less light is collected
off-axis than on-axis. This is cos(theta), where theta is the off-axis
angle. The next _three_ are light spreading from a solid angle in space to
an area on film (magnification) from a patch near the image edge as
compared to one in the middle. The light collected from the same solid
angle near a corner is spread out over more film than the light collected
from the same size solid angle near the middle. This is cos^3(theta). The
two of these must be multiplied together to get the complete falloff which
becomes cos^4(theta).
The cos^4 rule applies only to simple lenses. Complex lenses -- especially
wide-angle retrofocus designs -- can break this "rule." They do it principally
by enlarging the off-axis areas of the entrance and exit pupils. (You can see
this as you turn a wide-angle lens.) I believe there is another source of
improvement obtained by moving the exit pupil farther from the film plane than
it "should" be. *
Yes, exactly! This is why some lenses exhibit less "cos^4 falloff" than
others even though they have the same angle of acceptance, the same focal
length, and the same speed. The designs in some compensate for it [more]
compared to the designs in others.
Note that as a typical lens is gradually stopped down, the off-axis ray
paths are gradually eliminated, the most off-axis one being eliminate
first. This is why the effect diminishes if the lens is stopped down at or
near its narrowest aperture. Someone else mentioned photographing an
evenly illuminated "gray card" and using this as a "filter" in PhotoShop to
compensate for the falloff. This could be done, however you would have to
do this at each aperture setting for each lens.
-- John
< This message was delivered via the Olympus Mailing List >
< For questions, mailto:owner-olympus@xxxxxxxxxxxxxxx >
< Web Page: http://Zuiko.sls.bc.ca/swright/olympuslist.html >
|