I need to work out the shape of a diffraction pattern from 6 "small" circular apertures arranged in a hexagonal shape with side length a. I have not been given the radius of the circles. All I know is that one circular aperture gives a diffraction amplitude proportional to the first order Bessel function (but we haven't studied these, so I don't really know what that is) and inversely proportional to the radius in the Fraunhofer domain.

My approach this far has been to model the pattern as the convolution of six delta functions and the function for the single aperture. I am however not getting anything useful just some very messy integrals.

So is there anybody here who knows how to solve this problem? What will the pattern look like? Many thanks! :)


1 Answer 1


Consider the aperture the convolution of six point apertures with a circle. The six points give a pattern that is the sum of patterns due to each pair of apertures. Finally the diffraction pattern is the product of the diffraction of one circle and the superposition of the cosine functions you get from all the pairs of holes (with different spacing and directions... a little bit messy bit not too bad).

  • $\begingroup$ Thank you for replying! A follow up question: How comes we can treat the holes as 3 simple pairs? Will not all of them influence eachother? $\endgroup$
    – Jhonny
    Nov 23, 2017 at 16:02
  • $\begingroup$ They will - but superposition should still work $\endgroup$
    – Floris
    Nov 23, 2017 at 19:38

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