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In this article,, the 3D-coherence properties of light speckles are studied at different distances from the source. What I don't understand is the distinction drawn between what the authors call Airy speckles and gaussian speckles. In particular, it's not clear under which experimental conditions Airy or gaussian speckles are produced. How should I choose my setup so that I can know a priori that the speckles will be Airy or gaussian?

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Airy speckle or Airy pattern is a pattern difraction resulting of an uniform incident light wave illuminating a circular operture.

See this graph

Note that the central speckle (Airy disk) can be approximated, under some conditions, by a Gaussian speckle.

Apparently, a Gaussian beam with diameter of D focused through an aperture of diameter D will have a focal profile that is nearly Gaussian.

Note that a gaussian beam - without diffraction - gives a gaussian spot which size is increasing, while axial distance $z$ from the point $z=0$ of minimum transveral size (waist size) increases.

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