Editing Arago spot (section)

== Experimental aspects ==
=== 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.

=== Finite source size and spatial coherence ===

The main reason why the Arago spot is hard to observe in circular shadows from conventional light sources is that such light sources are bad approximations of point sources. If the wave source has a finite size ''S'' then the Arago spot will have an extent that is given by ''Sb''/''g'', as if the circular object acted like a lens.<ref name="sommerfeld"/> At the same time the intensity of the Arago spot is reduced with respect to the intensity of the undisturbed wave front. Defining the relative intensity <math>I_\text{rel}</math>as the intensity divided by the intensity of the undisturbed wavefront, the relative intensity for an extended circular source of diameter w can be expressed exactly using the following equation:<ref name=":0">{{Cite journal|last1=Reisinger|first1=T|last2=Leufke|first2=P M|last3=Gleiter|first3=H| last4=Hahn|first4=H| date=2017-03-14|title=On the relative intensity of Poisson's spot|journal=New Journal of Physics|volume=19| issue=3| pages=033022| doi=10.1088/1367-2630/aa5e7f|bibcode=2017NJPh...19c3022R|issn=1367-2630|doi-access=free}}</ref>
<math display="block">I_\text{rel}(w) = J_0^2\left(\frac{w R \pi}{g \lambda}\right) + J_1^2\left(\frac{w R \pi}{g \lambda}\right)</math>
where <math>J_0</math>and <math>J_1</math>are the Bessel functions of the first kind. <math>R</math> is the radius of the disc casting the shadow, <math>\lambda</math> the wavelength and <math>g</math> the distance between source and disc. For large sources the following asymptotic approximation applies:<ref name=":0" />
<math display="block">I_\text{rel}(w) \approx \frac{2 g \lambda }{\pi^2 w R}</math>

=== Deviation from circularity ===

If the cross-section of the circular object deviates slightly from its circular shape (but it still has a sharp edge on a smaller scale) the shape of the point-source Arago spot changes. In particular, if the object has an ellipsoidal cross-section the Arago spot has the shape of an [[evolute]].<ref name="coulson1922"/> Note that this is only the case if the source is close to an ideal point source. From an extended source the Arago spot is only affected marginally, since one can interpret the Arago spot as a [[point-spread function]]. Therefore, the image of the extended source only becomes washed out due to the convolution with the point-spread function, but it does not decrease in overall intensity.

=== 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.