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[OM] cos^4 explanation

Subject: [OM] cos^4 explanation
From: William Sommerwerck <williams@xxxxxxxxxx>
Date: Sun, 30 Dec 2001 15:33:07 -0800
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.

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.)

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. *

As for compensation, shooting an evenly illuminated image-less surface should
work. However, its values should _divide_ those of the picture, not multiply
them. Otherwise you'd make the falloff even worse.

cos^4 falloff and vignetting are usually considered separate effects, the
latter caused by some physical obstruction.

* This has the effect of reducing the illumination drop that occurs at closer
focusing distances, especially when shooting closeups. The opposite effect
occurs in a telephoto lens because the exit pupil sits closer to the film plane
-- close-ups require _more_ exposure compensation.



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