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 ?

• As far as I know (I may be wrong) the Arago spot shows up only in the Fresnel regime. I'm going to guess that Fraunhofer diffraction from a circular stop does not exhibit a bright spot at the center. Alternatively, it may simply be the case that one is never truly in the Fraunhofer limit, and that a faint bright spot on a dark background is easier to observe than a small dimming against a bright background. Commented Feb 24, 2016 at 18:54
• You aren't missing anything. The apertures are boundary conditions of the wave equation and the functional that maps the boundary conditions to the solutions is NOT linear and additive. That's part of the reason why boundary value problems are seriously hard, much harder than initial value problems that can be solved with a simple Green's function. Commented Feb 24, 2016 at 20:35
• @garyp you are exactly right, and if the OP had bothered to look at the wikipedia page, he'd have seen that right off. Commented Feb 24, 2016 at 20:47
• Look up Babinet's principle; there are many different ways to prove it. It is covered in all of the standard optics texts. Commented Feb 24, 2016 at 21:20
• @The record, last night I gave an answer which I now think is wrong. I said that the rings of interference around a disk's shadow in the Arago spot are caused by the light source (a point source in the original experiment), as though, if you remove the disk, the outer rings would stay as part of an Airy disk. But I just watched a vid of the coin moving, and with it the outer rings. I now see that the area of diffracted light from the disk is greater than I thought, and that the outer rings are caused by interference between light-waves from the source and from the opposite edge of the disk. Commented May 5, 2016 at 16:28

According to wikipedia's article about the Poisson-Arago spot it is a phenomenon occurring in Fresnel diffraction, which is a different limit from Fraunhofer diffraction.

From the question:

In other words, when the Fraunhofer criteria are met, complementary obstacles produce complementary diffraction patterns.

If I believe the OP, and also this article seems to assume the Fraunhofer limit, the contradiction is resolved. However I don't completely understand which limits Babinet's principle applies in. Therefore I will only claim that this is a possible answer, which relies upon Babinet's principle only applying in the Fraunhofer limit.

We don't even need to consider the applicability of Babinet's principle though. I think the confusion might simply come from that the OP used an argument about the interference pattern in Fraunhofer diffraction and added the Arago spot into the mix, which does not actually appear there, i.e. the diffraction pattern in the Fraunhofer limit does not have a spot in the middle. So there is certainly no contradiction in this limit.

I think you are missing things in both points.

1) There is no reason why the sum of diffracted patterns would give a uniform intensity. Why? Because waves are made of amplitudes, but patterns are made of intensities, that is: squared amplitudes. Two waves that would interfere destructively, give non-zero intensities when taken individually. In fact, they give the same intensity pattern when taken individually, because the only way to interfere destructively is to be in phase opposition (same absolute value of amplitude, but opposite signs).

This observation is the gist of Babinet's principle in Fraunhofer conditions: in any direction other than that of the incident plane wave, the intensity is zero, so the complementary obstacles yield the same pattern in those directions. In the direction of the plane wave, the intensity is non-zero; when one obstacle is finite and the other is infinite (as is the case here), the finite one (circular stop) yields full intensity, and the infinite one (hole) yields zero intensity: the energy passing through the hole is negligible as compared to that passing around the stop (we are in Fraunhofer conditions, at infinity, a finite energy gives infinitesimally small intensity).

2) The Poisson-Arago spot is a specific phenomenon observed in Fresnel regime.

About the Fraunhofer regime: it is not only a limiting case. You can produce experimentally a Fraunhofer diffraction pattern. How? Use a convergent lens. The convergent lens conjugates infinity in its object space to the focal plane in the image space. In other words: put a lens after a diffraction obstacle, and you will observe the Fraunhofer diffraction pattern on a screen in its focal plane, where coordinate $x$ on the screen matches direction $θ$ according to $x=f'\,\tan θ$.

With the circular stop, you will observe a bright dot at the center. This is not the Poisson-Arago spot. This dot (of theoretically zero width, unlike the Poisson-Arago spot) is the conjugate of the $θ=0$ direction in the Fraunhofer pattern, and as I explained above in this direction the finite obstacle gives maximal intensity (of course experimentally the lens is finite, has aberrations, the incident wave doesn't have infinite width... so the intensity of the dot is not infinite).

You're mostly correct, about how you expect both patterns together to produce uniform intensity corresponding to a plane wave (this is described as Babinet's principle. What you're forgetting is that it's the sum of the amplitude, including a phase. A circular hole and the complementary shape will produce identical patterns but opposite phase, except for additional background plane wave.