Any numerical procedure should reproduce the simple systems for which the answer is known analytically. Unfortunately, few diffraction problems have been solved in closed form. From our point of view, an analytic theory of a circular aperture containing at least one aberration would be acceptable.
A solution to an otherwise perfect circular aperture with defocusing aberration appears in Chapter 8 of Born and Wolf. It involves what are called the Lommel functions to perform a version of the integral in Eq. B.4
ASYMM output
25 APERTURE output
ASYMM output
25 APERTURE output
(Born and Wolf 1980, pp. 438-439). Two solutions are each written as an infinite series having its own region of applicability, one inside geometric shadow and one outside of it. These series were programmed into a MathCadâ„¢ document and each was graphed in the region where it was supposed to work (not shown). Results were indistinguishable from the results of APERTURE. The tests went up to 12 wavelengths defocusing aber ration and were limited to Bessel functions up to order 80.
Again this verification is reassuring, but incomplete. It tests the defocusing component of the aberration function (which appears in a separate term of both algorithms) but leaves the other aberrations alone. However, the central loops of ASYMM and APERTURE do not distinguish between defocusing and other aberrations. If a mistake were being made, it would have to be in the preparatory statements of both programs. Furthermore, any supposed error would have to leave the defocusing aberration untarnished in both procedures.
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