Sharpness at the limits of the "Depth of Field" based on a diffraction model[NOTE: DOF = Depth of field; COF = Circle of confusion; MTF = Modulation transfer function]
The following table shows the calculated sharpness at the limits of depth of field for 25 and 35 micron circles of confusion. These calculations are based on diffraction theory, not simple geometric optics. The simple geometric optics approch would predict a constant sharpness at the DOF limits. On film this would be around 58 lp/mm for a 25 micron COF and about 45 lp/mm for a 35 micron COF, using the resolution criteria detailed below. The resolution values given in the table below are the result of multiplying the diffraction limited defocus MTF function by the MTF of a typical sharp slide film, thus they represent lp/mm on film. "Resolution" was taken to be the average of the two values obtained for on film MTF of 2% and 5%. Values for "in focus" resolution are also given. A perfect (diffraction limited) lens is assumed. This is reasonable for objects close to the center of the frame, but not for objects close to the edge where residual aberrations will be present.
Another way of putting this is that these
are the best sharpness values you could expect on a sharp slide film for
objects at infinity and HFD/2 when the lens is focused at the HFD (Hyperfocal
distance) and the objects are not close to the edges of the frame. See
also my Hyperfocal Distance page.
It's clear that there is an optimum aperture for sharpness at the DOF limits, which is a function of the COF from which the DOF limits are calculated. Smaller values of COF give sharper images at the DOF limits, but of course the calculated DOF will be smaller for smaller COF values, so this should be no surprise.
Sharpness at the limits of the DOF falls off at small apertures because of increasing diffraction effects (ignored by the simple geometric optics model). It falls off at large apertures due to increasing OPD (optical path difference) for the defocused image at the DOF limits - also ignored in the simple geometric optics model.
All this of course, is theoretical. Real lenses have residual aberrations and curved fields which will modify the theoretical results. In practice you might not see much difference between shots at f11, f16 and f22, though you might start to notice a drop in overall sharpness at f32.
Note that a COF of 35 microns corresponds to the DOF markings on many lenses. A COF value of 25 microns then corresponds to the DOF markings on these same lenses shifted by 1 stop (i.e. read the DOF limits for f11, but use these as the 25 micron COF based DOF limits for f16).
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