Editing Arago spot (section)

=== Intensity and size ===

[[File:poissonspot setup treisinger.jpg|400px|thumb|right|Arago spot experiment. A point source illuminates a circular object, casting a shadow on a screen. At the shadow's center a bright spot appears due to [[diffraction]], contradicting the prediction of [[geometric optics]].]]

For an ideal [[point source]], the intensity of the Arago spot equals that of the undisturbed [[wave front]]. Only the width of the Arago spot intensity peak depends on the distances between source, circular object and screen, as well as the source's wavelength and the diameter of the circular object. <!--The shape of the Poisson spot's intensity distribution will in fact be similar for experimental setups with the same Fresnel number.--> This means that one can compensate for a reduction in the source's [[wavelength]] by increasing the distance between the circular object and screen or reducing the circular object's diameter.

The lateral intensity distribution on the screen has in fact the shape of a squared [[Bessel function|zeroth Bessel function of the first kind]] when close to the [[optical axis]] and using a [[plane wave|plane wave source]] (point source at infinity):<ref name="harvey1984"/>
<math display="block">U(P_1, r) \propto J_0^2 \left(\frac{\pi r d}{\lambda b}\right)</math>
where
* ''r'' is the distance of the point ''P''<sub>1</sub> on the screen from the optical axis
* ''d'' is the diameter of circular object
* ''λ'' is the wavelength
* ''b'' is the distance between circular object and screen.

The following images show the radial intensity distribution of the simulated Arago spot images above:

[[File:poissonspot simulation d4mm lateral.jpg|200px]]
[[File:poissonspot simulation d2mm lateral.jpg|200px]]
[[File:poissonspot simulation d1mm lateral.jpg|200px]]

The red lines in these three graphs correspond to the simulated images above, and the green lines were computed by applying the corresponding parameters to the squared Bessel function given above.