At 04:33 04.12.02, John A. Lind wrote:
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).
OK, we agree about where the cos^3(theta) term come from. If we stick to
the simple lens approximation, how is theta defined and found? It's all
defined by focal lenght and the film format, image circle doesn't enter
into the equation at all. So for *this approximation* any lens of a given
focal length will show the same falloff, because the angles are always the
same.
For complex lenses I'm sure the lens designer can do all sorts of funny
things to circumvent cos^4 - but you argue that a larger image circle will
give less falloff. To me this also implies that a smaller image circle
would give more falloff. So I make up an example where I reduce the image
circle by vignetting, and I have at least managed to convince myself that
this doesn't do anything to the rate of falloff.
I don't want to insult you, but I think you confuse vignetting and falloff.
One would want to vignette as much of the light falling outside of the film
area as possible (and as early as possible), to not have it bounce around
inside the lens or camera. For practical reasons you can't create a lens
hood that goes to infinity, so the object causing vignetting will have to
be placed somewhere out of focus. This of course means that the shadowing
effect will be gradual, so you have the choice of vignetting inside the
film frame at full aperture, or an image circle that is always a bit larger
than the film frame (and as a side effect causes a loss in contrast). Then
the larger image circle helps avoid uneven illimination from vignetting,
but the falloff would still be the same in either case.
It would be interesting if someone could compare a couple of tests of
lenses from different manufacturers. I believe some tests measure falloff
at a few apertures, and lenses of similar focal lengths would show *about*
the same degree of falloff if I'm not mistaken. I can't be proven right,
but any large deviations can certainly prove me wrong.
Thomas Bryhn
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