Yes,
That lens design technique largely mitigates on of the cosine factors
but the others remain.
I lifted the following from a PNET post:
"The idea that the illumination falloff of the lenses is solely due to
the inverse square law is incorrect. The inverse square law is only one
of three reasons that the illumination from a normal (non-fisheye) lens
decreases off-axis. The first reason is the inverse square law, which
brings in two factors of the cosine of the angle of the ray. The second
reason, assuming the film is flat, is the angle the light makes to the
film spreads the light over a larger area -- this is the same effect
that causes the seasons. The slanted rays effect brings in another
factor of cosine. If you look at most lenses, as you tilt the lens away
from looking at it straight on, the aperture becomes elliptical. This
reduction in the area of the aperture brings in another factor of
cosine. The net result is that most lenses have an illumination falloff
going as the fourth power of cosine of the angle of the ray to the
optical axis, rather than the second power that using only the inverse
square law would predict.
Some lenses (e.g., Super-Angulon, Grandaon, Nikkor-SW, etc.) use an
optical trick to retain an almost circular appearance of the aperture
off-axis. This regains most of a factor of cosine, improving the
illumination to close to a cosine to the power of three behavior. A
clue that a lens falls into this category is large front and rear
elements with a narrow center.
If someone doesn't believe these geometric arguments, then they should
try comparing cosine to the third and fourth laws to the illumination
curves published by Schneider and Rodenstock. They will find that the
curves generally follow a third or fourth power of cosine law fairly
well. This doesn't include the curves for the lenses wide open, for
which mechanical vignetting makes the falloff worse.
Modern 65 mm lens intended for 4x5 use are of the type using the
tilting pupil trick and so should illumination behavior of about cosine
to the third. An example is the illumination curve for the 65 mm f5.6
Super-Angulon, available from Schneider's website. The corner of a 4x5
negative is 103/2 = 76.5 mm from the center, which is u/u_max = 89.8%
for u_max = 85.2 mm, as specified by the datasheet. For the lens at f22
and focused on infinity, the correct curve is the higher solid one,
which shows about 25% illumination at u/u_max = 89.8%. Theta to this
point is inverse tangent (76.5 / 65) = 49.6 degrees. The cosine of this
angle is 0.648. Applying only the inverse square law, the predicted
illumination is 0.42 or 1.25 stops, agreeing with the calculation above
from the path lengths. Using cosine to the third, the prediction is
0.27, which is very close to the value read from the graph. This is 1.9
stops.
A 65 mm lens designed for medium format would probably not use the
tilting pupil trick. If it could cover 4x5, the illumination would be
cosine to the fourth, which at the corner of a 4x5 film would a factor
of 0.18 of the central illumination, for a falloff of 2.5 stops. "
Perhaps it is better with a concrete example---there is a typo in it
however in one of the numbers.
Mike
Thank you for greatly extending and technifying my simple and
incomplete explanation Mike.
However, I always thought that the Slyusarev effect (apparently
increasing pupil size)
was there precisely to counter the fact that the aperture becomes
elliptical
when viewed off-axis, so that it all "evens out".
--
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