Assume two obstacles are complementary. For example, a circular aperture and a circular stop of same radii. My theoretical analysis leads to contradictory conclusions about such situations.
1) Assuming incident light is a plane wave consisting of N sources of Huygens wavelets, obstacle 1 keeps a subset of those wavelets (blocking the others) while the obstacle 2 keeps the complementary subset of wavelets. In other words, the SUM of the diffracted patterns is expected to be a uniform intensity on the screen (all the wavelets = same as what we have when there is NO obstacle). In other words, when the Fraunhofer criteria are met, complementary obstacles produce complementary diffraction patterns.
2) Experiment shows that a circular stop and a circular aperture indeed produce opposite fringes, but the circular stop's diffraction pattern has the Poisson-Arago spot in the middle, while the circular aperture doesn't have a corresponding dark spot in the middle. Therefore these patterns are NOT complementary.
What am I missing ? I somehow have the feeling that patterns are complementary "everywhere except the center" because there is something ill-defined about using Huygens wavelets to predict the center of the pattern... Anyone has input on this ?