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Douglas-Stinson

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I am attempting to calculate the Fresnel difraction pattern from different diameter circular apertures for specific source to aperture and aperture to sensor distances. I'm generally following the procedure given in Klaus D. Mielenz "Algorithms for Fresnel Diffraction at Rectabngual and Circular Apertures", Journal of the National Institute of Standards and Technology, 103,497 (1998). But the formulas in that paper seem identical to those in Born and Wolf. The solutions are given in terms of functions Lommel functions V(u,v) and U(u,v), where U is used where v>u (essentially in the shadow region) and V is used when v<u. The paper gives closed form solutions for v=0 (on axis) and v=u.

When I implemented these solutions in MathCAD, I get reasonable answers using the U functions and they match the closed form results for v=u.

However, when I calculate the results using the V functions, I get much larger numbers and they don't match the closed form solutions at v=0 or v=u.

Has anyone else attempted these calculations and can tell me were I've gone wrong?

Calculations in attached file.

When I implemented these solutions in MathCAD, I get reasonable answers using the U functions and they match the closed form results for v=u.

However, when I calculate the results using the V functions, I get much larger numbers and they don't match the closed form solutions at v=0 or v=u.

Has anyone else attempted these calculations and can tell me were I've gone wrong?

Calculations in attached file.