Editing Arago spot (section)

=== Surface roughness of circular object ===

The Arago spot is very sensitive to small-scale deviations from the ideal circular cross-section. This means that a small amount of surface roughness of the circular object can completely cancel out the bright spot. This is shown in the following three diagrams which are simulations of the Arago spot from a 4&nbsp;mm diameter disc ({{nowrap|1=''g'' = ''b'' = 1&nbsp;m}}):

[[File:poissonspot simulation d4mm lateral cor10.jpg|200px]]
[[File:poissonspot simulation d4mm lateral cor50.jpg|200px]]
[[File:poissonspot simulation d4mm lateral cor100.jpg|200px]]

The simulation includes a regular sinusoidal corrugation of the circular shape of amplitude 10&nbsp;μm, 50&nbsp;μm and 100&nbsp;μm, respectively. Note, that the 100&nbsp;μm edge corrugation almost completely removes the central bright spot.

This effect can be best understood using the [[Fresnel zone|Fresnel zone concept]]. The field transmitted by a radial segment that stems from a point on the obstacle edge provides a contribution whose phase is tight to the position of the edge point relative to Fresnel zones. If the variance in the radius of the obstacle are much smaller than the width of Fresnel zone near the edge, the contributions form radial segments are approximately in phase and [[Interference (wave propagation)|interfere]] constructively. However, if random edge corrugation have amplitude comparable to or greater than the width of that adjacent Fresnel zone, the contributions from radial segments are no longer in phase and cancel each other reducing the Arago spot intensity.

The adjacent Fresnel zone is approximately given by:<ref name="reisinger2009"/>
<math display="block">\Delta r \approx \sqrt{r^2 + \lambda \frac{g b}{g + b}} - r.</math>

The edge corrugation should not be much more than 10% of this width to see a close to ideal Arago spot. In the above simulations with the 4&nbsp;mm diameter disc the adjacent Fresnel zone has a width of about 77&nbsp;μm.