Why is a flower shaped diffraction occluder the best solution for controlling diffraction? According to this TED talk by Jeremy Kasdin, Nasa is planning to spend $1bn on a "Starshade" project, where a giant flower shaped metal eclipser 20 meters wide is placed 50k kilometers in front of a space telescope, to fit the telescope diameter.
The idea is to occlude a star and photograph it's exoplanets.
The above solution is surreal. Why can't they control the diffraction of the light around the occlusion circle with a refractive material, to direct it outwards?
I suggest that they can design a round black occluder with soft edges overlaid with a refractive material that deflects the light away from the centre, similar to a lense.
Why do the angles and shapes at the edge of the flower shaped occluder have to be very precise in order to control diffraction?
 A: Whatever the shape of the shield is, at the telescope you're going to see the Fourier transform of it. With a simple disk shaped shield you'll see ringing artefacts at the edges and these will cause the light from the star to spill round the shield potentially hiding the planets.
Generally speaking Gaussian profiles are good for this, because the Fourier transform of a Gaussian is just another Gaussian and there is no ringing. The petals at the edge of the disk are designed to cut the transmitted intensity in an approximately Gaussian curve. I don't have the kit to hand to calculate 2D Fourier transforms, but I can show you how this works in 1D. Suppose our shield is a simple disk, i.e. a top hat function in 1D, then the Fourier transform looks like this:

The blue line, $f(x)$, is the profile of the shield and the magenta line, $g(k)$, is the Fourier transform. Note how the ringing spreads the light well outside the shield. Now suppose use a Gaussian edge to the shield. If I make the half width of the Gaussian 0.05 (in the arbitrary coordinates I've used) then the transform changes to:

Note how the ringing is reduced, and if I increase the Gaussian width to 0.1 the first maximum is almost completely eliminated:

Now, be a little cautious about taking the above graphs too literally as a guide to the performance of the shield. These are 1D plots remember, and to calculate the performance of the shield you'd need to do a 2D Fourier transform. Nevertheless it does show the basic principle of how feathering the edges of the shield improves its performance.
You ask about the precise shape of the petals. To be honest I don't know to what extent the width of the petals matters. Their shape matters because it will control the profile at the edge of the shield, and the length determines the overall width of the feathered area. I don't think it makes a lot of difference if you use lots of narrow petals of fewer wider ones. I would guess fewer wider ones is technically easier given that you've got to deploy this thing automatically in space.
