Aha! Now I see why you think the way you do. But that doesn't seem like a proper model for how an extender works. I'm sorry I wasn't very clear. I think the following sounds much more reasonable:
Let's say "perfect" extender merely magnifies the central field given by the lens (may it be 1.4x or 2.0x), without otherwise modifying it. If we focus on resolution, then it should not change the resolution in terms of smallest resolved detail. That is, the point-spread function (PSF)
would not change in angular size; on the sensor, on the other hand, the PSF would be magnified by the given magnification factor (1.4 or 2.0), so in terms of resolution per pixel, the PSF would be larger by the same factor (and thus the image appear less sharp, if the PSF is resolved by the pixel density of the sensor, of course).
Now what happens if the extender is not perfect, but adds its own fuziness? You seem to imply that the magnification factor of the PSF will change, but that doesn't seem reasonable. More realistically, the PSF of the extender will be convolved
with the PSF of the lens; if the PSFs can be crudely approximated by Gaussians
, the lens PSF has the full width at half maximum (FWHM)
of A, and the extender a PSF FWHM B, then the combined PSF would have a FWHM C = sqrt(A^2 + B^2). (in reality the PSFs are far more complicated than simple Gaussians for non-diffraction limited optics, but this gives an approximate scaling behaviour)
Back to your example. Let's say the PSF of the 70-200/2.8L II (FWHM=2) is twice that of the 200/2L (FWHM=1), and that the extender MkII (FWHM=1) is twice that of MkIII (FWHM=0.5). Then the improvement from using the 2.0x MkIII instead of the MkII on the 200/2L would be
sqrt((2*1)^2+1^2)/sqrt((2*1)^2+0.5^2) = 1.0847, or an improvement of 8.5%.
For the 70-200/2.8L II, on the other hand, the improvement would be much smaller:
sqrt((2*2)^2+1^2)/sqrt((2*2)^2+0.5^2) = 1.0445, or an improvement of 4.5%.
Simply put, the greater the imperfections of the lens, the more they mask the small imperfections added by the extender
. That is why I think it's a better to use as good lens as possible when testing for any potential differences in resolution generated by the extenders.
I don't know how the PSFs of the extenders compare to the PSFs of the lenses, but I would expect their contribution to be relatively small. That is to say, there is not much room for improvement, as the "lens flaws" themselves dominate the detoriation of resolution per pixel with magnification.
Finally, there is an easy way to measure PSFs. Just point the lens to a point source, and the resulting image will be the PSF. Unless you have a diffraction-limited telescope of aperture > 5 cm, bright stars at night make excellent point sources. Just take care to expose for short enough time or use a tracker to compensate for the Earth's rotation, otherwise stars will be motion blurred. A properly dimmed laser-pointer from a distance could also work, I guess.