# Arago Spot - Experimental Aspects - The Circular Object's Surface Roughness

The Circular Object's Surface Roughness

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 mm diameter disc (g = b = 1 m):

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

This effect can be best understood using the Fresnel zone concept. The circular object blocks a certain number of Fresnel zones. The Fresnel zone that begins with the edge of the circular object is the only one that contributes to the Arago spot. All the Fresnel zones that are further out destructively interfere with each other and thus cancel. Random edge corrugation whose amplitude is of the same order as the width of that adjacent Fresnel zone reduces the Arago spot intensity. Contributions from the parts of the edge whose radius has been increased by the corrugation to about the width of the adjacent Fresnel zone now destructively interfere with those contributions from the parts which have not been affected by the corrugation.

The adjacent Fresnel zone is approximately given by:

$Delta r approx sqrt{r^2 + lambda frac{g b}{g+b}} - r$

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 mm diameter disc the adjacent Fresnel zone has a width of about 77 µm.