Their combined efforts lead to the discovery of the Abbe sine condition.
Theoretically, the Abbe sine condition could greatly improve how well lenses could be made.
They collaborated and in 1886, produced a new type of glass that could fully use the Abbe sine condition.
At this point another coordinate transformation can be proposed ( " i " . " e " ., the Abbe sine condition ) relating the object plane wavenumber spectrum to the image plane wavenumber spectrum as
However, the complete theory of the Abbe sine condition shows that if a lens is corrected for coma and spherical aberration, as all good photographic objectives must be, the second principal plane becomes a portion of a sphere of radius centered about the focal point ".
By virtue of this, high magnification systems, which typically have small values of ? max ( by the Abbe sine condition ), can have more blur in the image, owing to the broader PSF . The size of the PSF is proportional to the magnification, so that the blur is no worse in a relative sense, but it is definitely worse in an absolute sense.
This is another way of writing the Abbe sine condition, which simply reflects Heisenberg's uncertainty principle for Fourier transform pairs, namely that as the spatial extent of any function is expanded ( by the magnification factor, " M " ), the spectral extent contracts by the same factor, " M ", so that the " space-bandwidth product " remains constant.
Bandwidth truncation causes a ( fictitious, mathematical, ideal ) point source in the object plane to be blurred ( or, spread out ) in the image plane, giving rise to the term, " point spread function . " Whenever bandwidth is expanded or contracted, image size is typically contracted or expanded accordingly, in such a way that the space-bandwidth product remains constant, by Heisenberg's principle ( Scott [ 1998 ] and Abbe sine condition ).
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