At 12:40 12/3/02, Thomas Bryhn wrote in part:
True, there are lens designs that have better falloff characteristics than
the simple lens approximation implies, but for any given lens design or
geometry the falloff can't be reduced by introducing mechanical vignetting
(= making the image circle smaller).
I may have not been clear enough with what I wrote. It was intended to
express just the opposite.
The cos^4(theta) falloff for a simple rectilinear lens, where "theta" is
the off-axis ray path angle, is caused by two effects which are multiplied
together to arrive at the combined effect. Cos(theta) falloff is
attributed to the effect you write of from the aperture becomimg more
football shaped the farther off-axis a ray path becomes. Some lens designs
mitigate this by making the apparent entrance pupil (aperture as viewed
from the front) enlarge/tip some for off-axis ray paths.
The rest of the effect, a cos^3(theta) falloff is the spreading of light
gathered from a solid angle to a flat piece of film. The light gathered by
the lens over a solid angle spreads out more (is magnified more) nearer the
edge and in the corners than in the center. The farther from the center,
the greater the spreading. It is how a rectilinear lens maps flat planes
in space to a flat film plane from a position that is effectively a point
in space, or at least very nearly a point. This is the effect that making
the image circle somewhat larger than the minimum required can
mitigate. Make it too large though, and the lens suffers from loss of
contrast and risk of flare from the extra light bouncing around inside the
light box (region between lens rear element and film plane).
Multiply the two effects together for the entire cos^4(theta) falloff. I'm
not a lens designer either, but this is what I recall from several decades
ago about practical lenses when studying basic optical physics as an
undergraduate.
-- John
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